A heuristic wave equation parameterizing BEC dark matter halos with a quantum core and an isothermal atmosphere

Abstract

The Gross–Pitaevskii–Poisson equations that govern the evolution of self-gravitating Bose–Einstein condensates, possibly representing dark matter halos, experience a process of gravitational cooling and violent relaxation. We propose a heuristic parametrization of this complicated process in the spirit of Lynden-Bell’s theory of violent relaxation for collisionless stellar systems. We derive a generalized wave equation that was introduced phenomenologically in Chavanis (Eur Phys J Plus 132:248, 2017) involving a logarithmic nonlinearity associated with an effective temperature $T_\mathrm{eff}$ and a damping term associated with a friction $\xi$. These terms can be obtained from a maximum entropy production principle and are linked by a form of Einstein relation expressing the fluctuation-dissipation theorem. The wave equation satisfies an H-theorem for the Lynden-Bell entropy and relaxes towards a stable equilibrium state which is a maximum of entropy at fixed mass and energy. This equilibrium state represents the most probable state of a Bose–Einstein condensate dark matter halo. It generically has a core-halo structure. The quantum core prevents gravitational collapse and may solve the core-cusp problem. The isothermal halo leads to flat rotation curves in agreement with the observations. These results are consistent with the phenomenology of dark matter halos. Furthermore, as shown in a previous paper (Chavanis in Phys Rev D 100:123506, 2019), the maximization of entropy with respect to the core mass at fixed total mass and total energy determines a core mass–halo mass relation which agrees with the relation obtained in direct numerical simulations. We stress the importance of using a microcanonical description instead of a canonical one. We also explain how our formalism can be applied to the case of fermionic dark matter halos.

Graphic abstract

Appendices

Appendix A: Properties of the Wigner distribution

In this Appendix, we check that the density $\rho$, velocity $\mathbf{u}$ and pressure tensor $P_{ij}$ obtained from the Wigner DF (43) coincide with the expressions obtained from the Schrödinger equation (10).

To that purpose, let us first establish useful identities. Using Eqs. (15) and (16), we get

$$\begin{aligned} \rho =|\psi |^2=\psi \psi ^* \end{aligned}$$

(A1)

and

$$\begin{aligned} \mathbf{u}=\frac{\nabla S}{m}=\frac{\hbar }{2im\rho }(\psi ^*\nabla \psi -\psi \nabla \psi ^*). \end{aligned}$$

(A2)

We also note that

$$\begin{aligned} \int \mathrm{d}{} \mathbf{v}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } =(2\pi )^3 \delta \left( \frac{m \mathbf{y}}{\hbar }\right) =\frac{(2\pi \hbar )^3}{m^3}\delta (\mathbf{y}). \end{aligned}$$

(A3)

We are now ready to compute the first moments of the Wigner DF

$$\begin{aligned} f(\mathbf{r},\mathbf{v},t)= & {} \frac{m^3}{(2\pi \hbar )^3} \int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) . \end{aligned}$$

(A4)

Integrating Eq. (A4) over $\mathbf{v}$ and using Eq. (A3), we obtain

$$\begin{aligned} \rho \equiv \int f\, \mathrm{d}{} \mathbf{v}=|\psi |^2, \end{aligned}$$

(A5)

which coincide with Eq. (A1).

Multiplying Eq. (A4) by $\mathbf{v}$ and integrating over $\mathbf{v}$, we get

$$\begin{aligned}&\int f \mathbf{v}\, \mathrm{d}{} \mathbf{v}=\frac{m^3}{(2\pi \hbar )^3} \int \mathrm{d}{} \mathbf{v}d\mathbf{y}\, \mathbf{v} \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\qquad \times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nonumber \\&\quad =\frac{m^3}{(2\pi \hbar )^3} \int \mathrm{d}{} \mathbf{v}\mathrm{d}{} \mathbf{y}\, \frac{\partial }{\partial \mathbf{y}}\left( \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar }\right) \frac{\hbar }{im} \nonumber \\&\qquad \times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nonumber \\&\quad =-\frac{m^3}{(2\pi \hbar )^3} \int \mathrm{d}{} \mathbf{v}\mathrm{d}{} \mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar }\frac{\hbar }{im} \nonumber \\&\qquad \times \frac{\partial }{\partial \mathbf{y}}\left[\psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \right], \end{aligned}$$

(A6)

where we have used an integration by parts to obtain the last equality. Calculating the derivative of the term in brackets, and using Eq. (A3), we obtain

$$\begin{aligned} \rho \mathbf{u}\equiv \int f \mathbf{v}\, \mathrm{d}{} \mathbf{v}=\frac{\hbar }{2im}\left( \psi ^*\nabla \psi -\psi \nabla \psi ^*\right) , \end{aligned}$$

(A7)

which coincide with Eq. (A2).

Multiplying Eq. (A4) by $v_iv_j$, integrating over $\mathbf{v}$, and proceeding as in Eq. (A6), we obtain

$$\begin{aligned}&\int f v_iv_j\, \mathrm{d}{} \mathbf{v}=\frac{m^3}{(2\pi \hbar )^3} \int \mathrm{d}{} \mathbf{v}d\mathbf{y}\, v_iv_j \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\qquad \times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nonumber \\&\quad =\frac{m^3}{(2\pi \hbar )^3} \int \mathrm{d}{} \mathbf{v}\mathrm{d}{} \mathbf{y}\, \frac{\partial ^2}{\partial y_i\partial y_j}\left( \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar }\right) \left( \frac{\hbar }{im}\right) ^2 \nonumber \\&\qquad \times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nonumber \\&\quad =\frac{m^3}{(2\pi \hbar )^3} \int \mathrm{d}{} \mathbf{v}\mathrm{d}{} \mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar }\left( \frac{\hbar }{im}\right) ^2 \nonumber \\&\qquad \times \frac{\partial ^2}{\partial y_i\partial y_j}\left[\psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \right]. \end{aligned}$$

(A8)

Calculating the second derivative of the term in brackets, and using Eq. (A3), we get

$$\begin{aligned} \int f v_iv_j\, \mathrm{d}{} \mathbf{v}= & {} -\frac{\hbar ^2}{4m^2}\biggl ( \psi ^*\frac{\partial ^2\psi }{\partial x_i\partial x_j}-\frac{\partial \psi ^*}{\partial x_j}\frac{\partial \psi }{\partial x_i}\nonumber \\&-\frac{\partial \psi }{\partial x_j}\frac{\partial \psi ^*}{\partial x_i}+\psi \frac{\partial ^2\psi ^*}{\partial x_i\partial x_j}\biggr ). \end{aligned}$$

(A9)

The pressure tensor can be written as

$$\begin{aligned} P_{ij}\equiv \int f (\mathbf{v}-\mathbf{u})_i (\mathbf{v}-\mathbf{u})_j\, \mathrm{d}{} \mathbf{v}=\int f v_iv_j\, \mathrm{d}{} \mathbf{v}-\rho u_iu_j. \end{aligned}$$

(A10)

Using Eqs. (A7) and (A9), we obtain after simplification

$$\begin{aligned} P_{ij}= & {} \frac{\hbar ^2}{4m^2}\biggl (\frac{\psi ^*}{\psi }\frac{\partial \psi }{\partial x_i}\frac{\partial \psi }{\partial x_j} +\frac{\psi }{\psi ^*}\frac{\partial \psi ^*}{\partial x_i}\frac{\partial \psi ^*}{\partial x_j}\nonumber \\&-\psi ^*\frac{\partial ^2\psi }{\partial x_i\partial x_j}-\psi \frac{\partial ^2\psi ^*}{\partial x_i\partial x_j}\biggr ). \end{aligned}$$

(A11)

The quantum pressure tensor obtained from the Schrödinger equation (10) can be written as [see Eq. (23)]

$$\begin{aligned} P_{ij}^{Q}=\frac{\hbar ^2}{4m^2}\left( \frac{1}{\rho } \partial _i\rho \partial _j\rho -\partial _i\partial _j\rho \right) . \end{aligned}$$

(A12)

Using Eq. (A1), we get after simplification

$$\begin{aligned} P_{ij}^Q= & {} \frac{\hbar ^2}{4m^2}\biggl (\frac{\psi ^*}{\psi }\frac{\partial \psi }{\partial x_i}\frac{\partial \psi }{\partial x_j} +\frac{\psi }{\psi ^*}\frac{\partial \psi ^*}{\partial x_i}\frac{\partial \psi ^*}{\partial x_j}\nonumber \\&-\psi ^*\frac{\partial ^2\psi }{\partial x_i\partial x_j}-\psi \frac{\partial ^2\psi ^*}{\partial x_i\partial x_j}\biggr ). \end{aligned}$$

(A13)

Comparing Eqs. (A11) and (A13) we see that

$$\begin{aligned} P_{ij}=P_{ij}^{Q}. \end{aligned}$$

(A14)

Let us now check that the expression of the energy in the Wigner representation is the same as in the Schrödinger representation. In terms of the Wigner DF, the total energy is given by

$$\begin{aligned} E=\int f \frac{v^2}{2}\, \mathrm{d}{} \mathbf{r}d\mathbf{v}+\frac{1}{2}\int \rho \Phi \, \mathrm{d}{} \mathbf{r}. \end{aligned}$$

(A15)

According to Eqs. (A10) and (A14), it can be rewritten as

$$\begin{aligned} E=\int \rho \frac{\mathbf{u}^2}{2}\, d\mathbf{r}+\frac{1}{2}\int P_{ii}^{Q}\, \mathrm{d}{} \mathbf{r}+\frac{1}{2}\int \rho \Phi \, \mathrm{d}{} \mathbf{r}. \end{aligned}$$

(A16)

Using Eq. (G.22) of [93] establishing that $\int P_{ii}^Q\, \mathrm{d}{} \mathbf{r}=2\Theta _Q$, we find that

$$\begin{aligned} E=\int \rho \frac{\mathbf{u}^2}{2}\, \mathrm{d}{} \mathbf{r}+\int \rho \frac{Q}{m}\, \mathrm{d}{} \mathbf{r}+\frac{1}{2}\int \rho \Phi \, \mathrm{d}{} \mathbf{r}, \end{aligned}$$

(A17)

which is in agreement with Eq. (28) obtained from the Schrödinger equation. From this equivalence, we can directly conclude that the Wigner–Poisson equations conserve the energy.

Finally, we derive the Wigner equation (44). Taking the time derivative of the Wigner DF (A4), we get

$$\begin{aligned}&\frac{\partial f}{\partial t}=\frac{m^3}{(2\pi \hbar )^3} \int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\quad \times \Biggl [\psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \frac{\partial \psi }{\partial t}\left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nonumber \\&\quad +\frac{\partial \psi ^*}{\partial t}\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \Biggr ]. \end{aligned}$$

(A18)

The next step is to substitute the Schrödinger equation (10) into the term in brackets of Eq. (A18). The contribution of the potential term in the Schrödinger equation is

$$\begin{aligned}&\left( \frac{\partial f}{\partial t}\right) _\mathrm{pot}=\frac{i m^4}{(2\pi )^3\hbar ^4}\int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\qquad \times \left[\Phi \left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) - \Phi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \right]\nonumber \\&\qquad \times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nonumber \\&\quad =\frac{im^4}{(2\pi )^3\hbar ^4}\int d\mathbf{y}\mathrm{d}{} \mathbf{v}'\, \mathrm{e}^{im(\mathbf{v}-\mathbf{v}')\cdot \mathbf{y}/\hbar } \nonumber \\&\qquad \times \left[\Phi \left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) - \Phi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \right]f(\mathbf{r},\mathbf{v}',t).\nonumber \\ \end{aligned}$$

(A19)

The second equality can be checked directly by substituting Eq. (A4), integrating over $\mathbf{v}'$, and using Eq. (A3). The contribution of the kinetic term in the Schrödinger equation is

$$\begin{aligned}&\left( \frac{\partial f}{\partial t}\right) _\mathrm{kin}^{(1)}=\frac{i m^2}{2(2\pi )^3\hbar ^2}\int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\qquad \times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \Delta _\mathbf{r}\psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nonumber \\&\quad =-\frac{i m^2}{(2\pi )^3\hbar ^2}\int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\qquad \times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \nabla _\mathbf{y}\cdot \nabla _\mathbf{r}\psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) . \end{aligned}$$

(A20)

Integrating by parts, we get

$$\begin{aligned}&\left( \frac{\partial f}{\partial t}\right) _\mathrm{kin}^{(1)}= \frac{i m^2}{2(2\pi )^3\hbar ^2}\int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\quad \times \nabla \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \cdot \nabla \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nonumber \\&\quad -\frac{ m^3}{(2\pi )^3\hbar ^3}{} \mathbf{v} \cdot \int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\quad \times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \nabla \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) . \end{aligned}$$

(A21)

Making the same operations with the kinetic term of the complex conjugate Schrödinger equation and adding the two results, we obtain

$$\begin{aligned}&\left( \frac{\partial f}{\partial t}\right) _\mathrm{kin}= -\frac{ m^3}{(2\pi \hbar )^3}{} \mathbf{v}\cdot \int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\quad \times \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \nabla \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nonumber \\&\quad -\frac{ m^3}{(2\pi \hbar )^3}{} \mathbf{v} \cdot \int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\quad \times \psi \left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) \nabla \psi ^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) =-\mathbf{v}\cdot \frac{\partial f}{\partial \mathbf{r}}. \end{aligned}$$

(A22)

The second equality can be checked directly by substituting Eq. (A4). Summing Eqs. (A19) and (A22), we obtain the Wigner equation (44).

Remark

When $\hbar \rightarrow 0$, we can make the approximation $\Phi (\mathbf{r}\pm \mathbf{y}/2,t)\simeq \Phi (\mathbf{r},t)\pm \nabla \Phi (\mathbf{r},t)\cdot \mathbf{y}/2$ in the integral of the Wigner equation (44) and we obtain

$$\begin{aligned}&\frac{\partial f}{\partial t}+\mathbf{v}\cdot \frac{\partial f}{\partial \mathbf{r}}\nonumber \\&\quad =\frac{im^4}{(2\pi \hbar )^3\hbar }\nabla \Phi \cdot \int \mathrm{e}^{im(\mathbf{v}-\mathbf{v}')\cdot \mathbf{y}/\hbar } \mathbf{y} f(\mathbf{r},\mathbf{v}',t)\, d\mathbf{y}\mathrm{d}{} \mathbf{v}'\nonumber \\&\quad =\frac{m^3}{(2\pi \hbar )^3}\nabla \Phi \cdot \frac{\partial }{\partial \mathbf{v}}\int \mathrm{e}^{im(\mathbf{v}-\mathbf{v}')\cdot \mathbf{y}/\hbar } f(\mathbf{r},\mathbf{v}',t)\, d\mathbf{y}\mathrm{d}{} \mathbf{v}'\nonumber \\&\quad =\nabla \Phi \cdot \frac{\partial }{\partial \mathbf{v}}\int \delta (\mathbf{v}-\mathbf{v}') f(\mathbf{r},\mathbf{v}',t)\, \mathrm{d}{} \mathbf{v}'\nonumber \\&\quad =\nabla \Phi \cdot \frac{\partial f}{\partial \mathbf{v}}. \end{aligned}$$

(A23)

This returns the Vlasov equation (C1). The quantum correction to the Vlasov equation (i.e. the difference between the Vlasov and the Wigner equation) is of order $\hbar ^2$. Similarly, let us assume that orbit crossing has not yet occurred and let us take the limit $\hbar \rightarrow 0$ in the Wigner DF (43). Substituting Eq. (14) into Eq. (43), making the approximations $\rho (\mathbf{r}\pm \mathbf{y}/2,t)\simeq \rho (\mathbf{r},t)$ and $S(\mathbf{r}\pm \mathbf{y}/2,t)\simeq S(\mathbf{r},t)\pm \nabla S(\mathbf{r},t)\cdot \mathbf{y}/2$, and using Eq. (16), we get

$$\begin{aligned} f(\mathbf{r},\mathbf{v},t)= & {} \frac{m^3}{(2\pi \hbar )^3} \rho \int d\mathbf{y}\, \mathrm{e}^{im(\mathbf{v}-\mathbf{u})\cdot \mathbf{y}/\hbar }\nonumber \\= & {} \rho (\mathbf{r},t) \delta ( \mathbf{v}-\mathbf{u}(\mathbf{r},t)). \end{aligned}$$

(A24)

This returns the single-speed solution from Eq. (C13). For finite $\hbar /m$, the DF f is proportional to a Gaussian in $\mathbf{v}$ of thickness $\hbar /m$.

Appendix B: Generalized GPP equations

In this Appendix, we recall the basic properties of the generalized GPP equations introduced in [93]. We also discuss the difference between the canonical and microcanonical formulation.

1.1 1. Basic properties

Let us consider the generalized GPP equations

$$\begin{aligned}&i\hbar \frac{\partial \psi }{\partial t}=-\frac{\hbar ^2}{2m}\Delta \psi + m\left[\Phi +\frac{d V}{d|\psi |^2}\right]\psi \nonumber \\&\quad +\,mT_\mathrm{eff}\ln (|\psi |^2)\psi -i\frac{\hbar }{2}\xi \left[\ln \left( \frac{\psi }{\psi ^*}\right) -\left\langle \ln \left( \frac{\psi }{\psi ^*}\right) \right\rangle \right]\psi , \nonumber \\ \end{aligned}$$

(B1)

$$\begin{aligned}&\Delta \Phi =4\pi G |\psi |^2, \end{aligned}$$

(B2)

with a potential $V(|\psi |^2)$. Introducing the enthalpy (see Appendix B5)

$$\begin{aligned} h(\rho )=V'(\rho ), \end{aligned}$$

(B3)

we can rewrite the generalized GP equation (B1) as

$$\begin{aligned}&i\hbar \frac{\partial \psi }{\partial t}=-\frac{\hbar ^2}{2m}\Delta \psi + m\left[\Phi +h(|\psi |^2)\right]\psi \nonumber \\&\quad +mT_\mathrm{eff}\ln (|\psi |^2)\psi -i\frac{\hbar }{2}\xi \left[\ln \left( \frac{\psi }{\psi ^*}\right) -\left\langle \ln \left( \frac{\psi }{\psi ^*}\right) \right\rangle \right]\psi .\nonumber \\ \end{aligned}$$

(B4)

The stationary solutions of the generalized GPP equations (B1) and (B2) are of the form

$$\begin{aligned} \psi (\mathbf{r},t)=\phi (\mathbf{r})\mathrm{e}^{-iEt/\hbar }, \end{aligned}$$

(B5)

where $\phi (\mathbf{r})=\sqrt{\rho (\mathbf{r})}$ and E (Eigenenergy) are real. These quantities are determined by the Eigenvalue problem

$$\begin{aligned}&-\frac{\hbar ^2}{2m}\Delta \phi +m\left[\Phi +h(\rho )\right]\phi +mT_\mathrm{eff}\ln (\rho )\phi =E\phi , \end{aligned}$$

(B6)

$$\begin{aligned}&\Delta \Phi =4\pi G \phi ^2. \end{aligned}$$

(B7)

Dividing Eq. (B6) by $\phi$, we get

$$\begin{aligned} Q+m\Phi +mh(\rho )+mT_\mathrm{eff}\ln \rho =E, \end{aligned}$$

(B8)

where Q is the quantum potential defined by Eq. (21).

Using the Madelung [50] transformation (see Sect. 2.2), the hydrodynamic equations corresponding to the generalized GPP equations (B1) and (B2) are

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{u})=0,\nonumber \\ \end{aligned}$$

(B9)

$$\begin{aligned} \frac{\partial S}{\partial t}&+\frac{1}{2m}(\nabla S)^2+m[\Phi +T_\mathrm{eff}\ln \rho +h(\rho )]\nonumber \\&+\,Q+\xi (S-\langle S\rangle )=0,\end{aligned}$$

(B10)

$$\begin{aligned}&\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{m}\nabla Q-\frac{1}{\rho }\nabla P-\nabla \Phi -\frac{1}{\rho }\nabla P_\mathrm{th}-\xi \mathbf{u}, \nonumber \\ \end{aligned}$$

(B11)

$$\begin{aligned}&\Delta \Phi =4\pi G \rho ,\nonumber \\ \end{aligned}$$

(B12)

where the barotropic equation of state $P(\rho )$ is determined by the relation (see Appendix B5)

$$\begin{aligned} h'(\rho )=\frac{P'(\rho )}{\rho }. \end{aligned}$$

(B13)

For a general polytropic equation of state of the form

$$\begin{aligned} P_\mathrm{poly}=K\rho ^{\gamma }, \end{aligned}$$

(B14)

the enthalpy and the potential are given by

$$\begin{aligned} h_\mathrm{poly}(\rho )=\frac{K\gamma }{\gamma -1}\rho ^{\gamma -1},\quad V_\mathrm{poly}(\rho )=\frac{K}{\gamma -1}\rho ^{\gamma }. \end{aligned}$$

(B15)

In particular, for the standard self-interacting BEC described by the polytropic equation of state

$$\begin{aligned} P_\mathrm{int}(\rho )=\frac{2\pi a_s\hbar ^2}{m^3}\rho ^2, \end{aligned}$$

(B16)

we have

$$\begin{aligned} h_\mathrm{int}(\rho )=\frac{4\pi a_s\hbar ^2}{m^3}\rho ,\quad V_\mathrm{int}(\rho )=\frac{2\pi a_s\hbar ^2}{m^3}\rho ^2. \end{aligned}$$

(B17)

For the Lynden-Bell pressure of zero-point energy

$$\begin{aligned} P_\mathrm{LB}^{(0)}=\frac{1}{5}\left( \frac{3}{4\pi \eta _0}\right) ^{2/3}\rho ^{5/3}, \end{aligned}$$

(B18)

we have

$$\begin{aligned}&h_\mathrm{LB}^{(0)}(\rho )=\frac{1}{2}\left( \frac{3}{4\pi \eta _0}\right) ^{2/3}\rho ^{2/3}, \end{aligned}$$

(B19)

$$\begin{aligned}&V_\mathrm{LB}^{(0)}(\rho )=\frac{3}{10}\left( \frac{3}{4\pi \eta _0}\right) ^{2/3}\rho ^{5/3}. \end{aligned}$$

(B20)

For the Lynden-Bell isothermal equation of state

$$\begin{aligned} P_\mathrm{th}=\rho T_\mathrm{eff}, \end{aligned}$$

(B21)

the enthalpy and the potential are given by

$$\begin{aligned} h_\mathrm{th}(\rho )=T_\mathrm{eff}\ln \rho ,\quad V_\mathrm{th}(\rho )=T_\mathrm{eff}\rho (\ln \rho -1). \end{aligned}$$

(B22)

An equilibrium state of the quantum damped Euler equations (B9)–(B12) satisfies the condition of quantum hydrostatic equilibrium

$$\begin{aligned} \frac{\rho }{m}\nabla Q+\nabla P+\rho \nabla \Phi +\nabla P_\mathrm{th}=\mathbf{0}. \end{aligned}$$

(B23)

This equation is equivalent to Eq. (B8). Indeed, dividing Eq. (B23) by $\rho$, using Eq. (B13) and integrating, we obtain Eq. (B8), where E appears as a constant of integration.

1.2 2. Microcanonical model

We introduce the mass

$$\begin{aligned} M=\int \rho \, \mathrm{d}{} \mathbf{r} \end{aligned}$$

(B24)

and the energy

$$\begin{aligned} E_\mathrm{tot}=\Theta _c+\Theta _Q+U+W+\Theta _\mathrm{th}, \end{aligned}$$

(B25)

which includes the classical kinetic energy

$$\begin{aligned} \Theta _c=\int \rho \frac{\mathbf{u}^2}{2}\, \mathrm{d}{} \mathbf{r}, \end{aligned}$$

(B26)

the quantum kinetic energy

$$\begin{aligned} \Theta _Q=\frac{1}{m}\int \rho Q\, \mathrm{d}{} \mathbf{r}, \end{aligned}$$

(B27)

the internal energy

$$\begin{aligned} U=\int V(\rho )\, \mathrm{d}{} \mathbf{r}, \end{aligned}$$

(B28)

the gravitational energy

$$\begin{aligned} W=\frac{1}{2}\int \rho \Phi \, \mathrm{d}{} \mathbf{r}, \end{aligned}$$

(B29)

and the thermal energy

$$\begin{aligned} \Theta _\mathrm{th}=\frac{3}{2}M T_\mathrm{eff}. \end{aligned}$$

(B30)

We also introduce the entropy (up to an additive constant)

$$\begin{aligned} S=-\int \rho (\ln \rho -1)\, \mathrm{d}{} \mathbf{r}+\frac{3}{2}M\ln T_\mathrm{eff}. \end{aligned}$$

(B31)

The functionals (B26)–(B29) are justified in [93] and the functionals (B30) and (B31) are justified in Appendix D3.

In the microcanonical situation, the total energy $E_\mathrm{tot}$ is fixed and the temperature $T_\mathrm{eff}(t)$ evolves in time so as to satisfy the constraint from Eq. (B25) yielding

$$\begin{aligned} \frac{3}{2}M T_\mathrm{eff}(t)=E_\mathrm{tot}-\Theta _c(t)-\Theta _Q(t)-U(t)-W(t). \end{aligned}$$

(B32)

The generalized GPP equations (B1) and (B2) with the time-dependent temperature $T_\mathrm{eff}(t)$ given by Eq. (B32)³⁷ conserve the energy (by construction) and satisfy an H-theorem for the entropy (B31). This can be proven as follows. From Eq. (B31), we have

$$\begin{aligned} \dot{S}=-\int \ln \rho \frac{\partial \rho }{\partial t}\, \mathrm{d}{} \mathbf{r}+\frac{3}{2}M\frac{\dot{T}_\mathrm{eff}}{T_\mathrm{eff}}. \end{aligned}$$

(B34)

Using the equation of continuity (B9) and integrating by parts, we get

$$\begin{aligned} \dot{S}=-\int \mathbf{u}\cdot \nabla \rho \, d\mathbf{r}+\frac{3}{2}M\frac{\dot{T}_\mathrm{eff}}{T_\mathrm{eff}}. \end{aligned}$$

(B35)

On the other hand, taking the time derivative of Eq. (B32), and using Eqs. (B26)–(B29), we obtain

$$\begin{aligned} \frac{3}{2}M \dot{T}_\mathrm{eff}=-\int \left( \frac{\mathbf{u}^2}{2}+\frac{Q}{m}+h+\Phi \right) \frac{\partial \rho }{\partial t}\, \mathrm{d}{} \mathbf{r}-\int \rho \mathbf{u}\cdot \frac{\partial \mathbf{u}}{\partial t}\, \mathrm{d}{} \mathbf{r}, \end{aligned}$$

(B36)

where we have used the identities of Appendix C of [93]. Using the equation of continuity (B9), the damped quantum Euler equation (B11), and recalling the identity of vector analysis $(\mathbf{u}\cdot \nabla )\mathbf{u}=\nabla (\mathbf{u}^2/2)-\mathbf{u}\times (\nabla \times \mathbf{u})$ and the fact that the flow is irrotational ($\nabla \times \mathbf{u}=\mathbf{0}$) since $\mathbf{u}=\nabla S/m$, we get after simplification (using straightforward integrations by parts)

$$\begin{aligned} \frac{3}{2}M \dot{T}_\mathrm{eff}=\int \mathbf{u}\cdot \nabla P_\mathrm{th}\, \mathrm{d}{} \mathbf{r}+\xi \int \rho \mathbf{u}^2\, \mathrm{d}{} \mathbf{r}. \end{aligned}$$

(B37)

Combining Eqs. (B35) and (B37), and using Eq. (B21), we finally obtain

$$\begin{aligned} \dot{S}=\frac{\xi }{T_\mathrm{eff}}\int \rho \mathbf{u}^2\, \mathrm{d}{} \mathbf{r}\ge 0. \end{aligned}$$

(B38)

Therefore, the entropy increases monotonically. At equilibrium (where $\dot{S}=0$), we get $\mathbf{u}=\mathbf{0}$ leading to the quantum equation of hydrostatic equilibrium (B23).

We can easily show that the generalized GPP equations in the microcanonical ensemble relax towards an equilibrium state that maximizes the entropy at fixed mass and energy. First of all, an extremum of entropy at fixed mass and energy is determined by the variational principle

$$\begin{aligned} \delta S+\alpha \delta M=0, \end{aligned}$$

(B39)

where $\alpha =\mu /mT_\mathrm{eff}$ is a Lagrange multiplier taking into account the conservation of mass [the conservation of energy is taken into account in Eq. (B32)]. Using

$$\begin{aligned} \delta S=-\int \ln \rho \, \delta \rho \, d\mathbf{r}+\frac{3M}{2T_\mathrm{eff}}\delta T_\mathrm{eff} \end{aligned}$$

(B40)

and

$$\begin{aligned} \frac{3}{2}M \delta T_\mathrm{eff}=-\int \left( \frac{Q}{m}+h+\Phi \right) \delta \rho \, \mathrm{d}{} \mathbf{r}, \end{aligned}$$

(B41)

we obtain

$$\begin{aligned} mT_\mathrm{eff}\ln \rho +Q+mh+m\Phi =\mu , \end{aligned}$$

(B42)

which is equivalent to Eq. (B8) with $\mu =E$. This shows that the Eigenenergy E coincides with the chemical potential $\mu$. Taking the gradient of Eq. (B42) and using Eq. (B13), we recover the equation of quantum hydrostatic equilibrium (B23). Therefore, an equilibrium state of the generalized GPP equations is an extremum of entropy at fixed mass and energy. On the other hand, by proceeding as explained at the end of Appendix D2, we can show that only entropy maxima (not minima or saddle points) at fixed mass and energy are dynamically stable [127]. From these results, we conclude that the generalized GPP equations with a time-dependent effective temperature relax towards an equilibrium state which maximizes the entropy at fixed mass and energy:

$$\begin{aligned} \max \ \lbrace {S}\, |\, M, E_\mathrm{tot} \,\, \mathrm{fixed} \rbrace . \end{aligned}$$

(B43)

Proceeding as in Appendix G of [93], we obtain the damped quantum virial theorem

$$\begin{aligned} \frac{1}{2}\ddot{I}+\frac{1}{2}\xi \dot{I}=2(\Theta _c+\Theta _\mathrm{th}+\Theta _Q)+3\int P\, \mathrm{d}{} \mathbf{r}+W, \end{aligned}$$

(B44)

where $I=\int \rho r^2\, \mathrm{d}{} \mathbf{r}$ is the moment of inertia. According to Eq. (B25), we have $\Theta _c+\Theta _\mathrm{th}+\Theta _Q=E_\mathrm{tot}-U-W$ so we can rewrite Eq. (B44) as³⁸

$$\begin{aligned} \frac{1}{2}\ddot{I}+\frac{1}{2}\xi \dot{I}=2E_\mathrm{tot}-2U-W+3\int P\, \mathrm{d}{} \mathbf{r}. \end{aligned}$$

(B45)

For a polytropic equation of state, the internal energy satisfies the identity [see Eqs. (B14) and (B15)]

$$\begin{aligned} U_\mathrm{poly}=\frac{1}{\gamma -1}\int P_\mathrm{poly}\, \mathrm{d}{} \mathbf{r}, \end{aligned}$$

(B46)

and the damped quantum virial theorem becomes

$$\begin{aligned} \frac{1}{2}\ddot{I}+\frac{1}{2}\xi \dot{I}=2E_\mathrm{tot}+(3\gamma -5)U_\mathrm{poly}-W. \end{aligned}$$

(B47)

We note that the term involving the internal energy vanishes for the particular index $\gamma =5/3$ which corresponds to the Lynden-Bell pressure of zero-point energy (B18).

Remark

In the strong friction limit $\xi \rightarrow +\infty$, we can neglect the inertial term (l.h.s.) in the damped quantum Euler equation (B11) and get

$$\begin{aligned} \mathbf{u}=-\frac{1}{\xi \rho } \left( T_\mathrm{eff}\nabla \rho +\nabla P+\frac{\rho }{m}\nabla Q+\rho \nabla \Phi \right) . \end{aligned}$$

(B48)

Substituting this relation into the continuity equation (B9) we obtain the quantum Smoluchowski–Poisson equations

$$\begin{aligned}&\xi \frac{\partial \rho }{\partial t}=\nabla \cdot \left( T_\mathrm{eff}\nabla \rho +\nabla P+\frac{\rho }{m}\nabla Q+\rho \nabla \Phi \right) , \end{aligned}$$

(B49)

$$\begin{aligned}&\Delta \Phi =4\pi G\rho . \end{aligned}$$

(B50)

Since $|\mathbf{u}|=O(1/\xi )$ in the strong friction limit, we can neglect the classical kinetic energy $\Theta _c$ in Eq. (B32). Therefore, the evolution of the effective temperature is given by

$$\begin{aligned} \frac{3}{2}M T_\mathrm{eff}(t)=E_\mathrm{tot}-\Theta _Q(t)-U(t)-W(t). \end{aligned}$$

(B51)

The H-theorem takes the form

$$\begin{aligned} \dot{S}=\frac{1}{\xi T_\mathrm{eff}}\int \frac{1}{\rho }\left( T_\mathrm{eff}\nabla \rho +\nabla P+\frac{\rho }{m}\nabla Q+\rho \nabla \Phi \right) ^2\, \mathrm{d}{} \mathbf{r}\ge 0, \end{aligned}$$

(B52)

and we have the same general properties as those discussed after Eq. (B38). On the other hand, the overdamped quantum virial theorem is given by

$$\begin{aligned} \frac{1}{2}\xi \dot{I}=2(\Theta _\mathrm{th}+\Theta _Q)+3\int P\, \mathrm{d}{} \mathbf{r}+W. \end{aligned}$$

(B53)

Using $\Theta _\mathrm{th}+\Theta _Q=E_\mathrm{tot}-U-W$, we can rewrite Eq. (B53) as

$$\begin{aligned} \frac{1}{2}\xi \dot{I}=2E_\mathrm{tot}-2U-W+3\int P\, d\mathbf{r}. \end{aligned}$$

(B54)

For a polytropic equation of state, using Eq. (B46), the overdamped quantum virial theorem becomes

$$\begin{aligned} \frac{1}{2}\xi \dot{I}=2E_\mathrm{tot}+(3\gamma -5)U_\mathrm{poly}-W. \end{aligned}$$

(B55)

1.3 3. Canonical model

In the canonical ensemble, the temperature $T_\mathrm{eff}$ is fixed and the appropriate thermodynamic potential is the free energy

$$\begin{aligned} F=E_\mathrm{tot}-T_\mathrm{eff}S. \end{aligned}$$

(B56)

Using Eqs. (B25) and (B31), we obtain (up to an additive constant)

$$\begin{aligned} F=\Theta _c+\Theta _Q+U+W+U_\mathrm{th}, \end{aligned}$$

(B57)

where

$$\begin{aligned} U_\mathrm{th}=T_\mathrm{eff}\int \rho (\ln \rho -1)\, \mathrm{d}{} \mathbf{r} \end{aligned}$$

(B58)

is the internal energy associated with the thermal pressure. This returns the free energy introduced in [93].

The generalized GPP equations (B1) and (B2) with a fixed temperature $T_\mathrm{eff}$ satisfy an H-theorem for the free energy (B57). Indeed, using calculations similar to the previous ones (see also Appendix D of [93]) we obtain

$$\begin{aligned} \dot{F}=-\xi \int \rho \mathbf{u}^2\, \mathrm{d}{} \mathbf{r}\le 0. \end{aligned}$$

(B59)

Therefore, the free energy decreases monotonically. At equilibrium (where $\dot{F}=0$), we get $\mathbf{u}=\mathbf{0}$ leading to the equation of quantum hydrostatic equilibrium (B23).

We can easily show that the generalized GPP equations in the canonical ensemble relax towards an equilibrium state that minimizes the free energy at fixed mass. First of all, an extremum of free energy at fixed mass is determined by the variational principle

$$\begin{aligned} \delta F-\frac{\mu }{m}\delta M=0, \end{aligned}$$

(B60)

where $\mu /m$ is a Lagrange multiplier taking into account the conservation of mass. Using calculations similar to the previous ones (see also section 3.4 of [93]) we obtain Eq. (B42) which is equivalent to Eq. (B8) with $\mu =E$ or to the quantum equation of hydrostatic equilibrium (B23). Therefore, an equilibrium state of the generalized GPP equations is an extremum of free energy at fixed mass. On the other hand, by proceeding as explained at the end of Appendix D2 we can show that only minima (not maxima or saddle points) of free energy at fixed mass are dynamically stable [127]. From these results, we conclude that the generalized GPP equations with a constant effective temperature relax towards an equilibrium state which minimizes the free energy at fixed mass:

$$\begin{aligned} \min \ \lbrace {F}\, |\, M \,\, \mathrm{fixed} \rbrace . \end{aligned}$$

(B61)

The damped quantum virial theorem is given by Eq. (B44) or, equivalently, by

$$\begin{aligned} \frac{1}{2}\ddot{I}+\frac{1}{2}\xi \dot{I}=2(\Theta _c+\Theta _Q)+3M T_\mathrm{eff}+3\int P\, \mathrm{d}{} \mathbf{r}+W, \end{aligned}$$

(B62)

where $T_\mathrm{eff}$ is constant.

Remark

In the strong friction limit $\xi \rightarrow +\infty$, we obtain the quantum Smoluchowski–Poisson equations (B49) and (B50) where $T_\mathrm{eff}$ is constant. The H-theorem now reads

$$\begin{aligned} \dot{F}=-\frac{1}{\xi }\int \frac{1}{\rho }\left( T_\mathrm{eff}\nabla \rho +\nabla P+\frac{\rho }{m}\nabla Q+\rho \nabla \Phi \right) ^2\, \mathrm{d}{} \mathbf{r}\le 0. \end{aligned}$$

(B63)

On the other hand, the overdamped quantum virial theorem is given by Eq. (B53) or, equivalently, by

$$\begin{aligned} \frac{1}{2}\xi \dot{I}=2\Theta _Q+3M T_\mathrm{eff}+3\int P\, \mathrm{d}{} \mathbf{r}+W, \end{aligned}$$

(B64)

where $T_\mathrm{eff}$ is constant.

1.4 4. Numerical algorithm

A stationary solution of the ordinary GPP equations

$$\begin{aligned}&i\hbar \frac{\partial \psi }{\partial t}=-\frac{\hbar ^2}{2m}\Delta \psi + m\left[\Phi +\frac{d V}{d|\psi |^2}\right]\psi , \end{aligned}$$

(B65)

$$\begin{aligned}&\Delta \Phi =4\pi G |\psi |^2, \end{aligned}$$

(B66)

is of the form $\psi (\mathbf{r},t)=\phi (\mathbf{r})\mathrm{e}^{-iEt/\hbar }$, where $\phi (\mathbf{r})=\sqrt{\rho (\mathbf{r})}$ and E (Eigenenergy) are real. These quantities are determined by the Eigenvalue problem

$$\begin{aligned}&-\frac{\hbar ^2}{2m}\Delta \phi +m\left[\Phi +h(\rho )\right]\phi =E\phi , \end{aligned}$$

(B67)

$$\begin{aligned}&\Delta \Phi =4\pi G \phi ^2. \end{aligned}$$

(B68)

Dividing Eq. (B67) by $\phi$, we get

$$\begin{aligned} Q+m\Phi +mh(\rho )=E, \end{aligned}$$

(B69)

where Q is the quantum potential defined by Eq. (21).

Using the Madelung transformation, the ordinary GPP equations (B65) and (B66) are equivalent to the hydrodynamic equations

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{u})=0, \end{aligned}$$

(B70)

$$\begin{aligned}&\frac{\partial S}{\partial t}+\frac{1}{2m}(\nabla S)^2+m\Phi +mh(\rho )+Q=0, \end{aligned}$$

(B71)

$$\begin{aligned}&\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{m}\nabla Q-\frac{1}{\rho }\nabla P-\nabla \Phi , \end{aligned}$$

(B72)

$$\begin{aligned}&\Delta \Phi =4\pi G \rho . \end{aligned}$$

(B73)

A stationary solution of the quantum Euler–Poisson equations (B70)–(B73) satisfies the condition of quantum hydrostatic equilibrium

$$\begin{aligned} \frac{\rho }{m}\nabla Q+\nabla P+\rho \nabla \Phi =\mathbf{0}. \end{aligned}$$

(B74)

This equation is equivalent to Eq. (B69) as can be seen by taking the gradient of Eq. (B69) and using Eq. (B13).

On the other hand, a stationary solution of the ordinary GPP equations (B65) and (B66) is an extremum of energy

$$\begin{aligned} E_\mathrm{tot}=\Theta _c+\Theta _Q+U+W \end{aligned}$$

(B75)

at fixed mass M. Indeed, writing the variational principle as

$$\begin{aligned} \delta E_\mathrm{tot}-\frac{\mu }{m}\delta M=0, \end{aligned}$$

(B76)

where $\mu /m$ is a Lagrange multiplier (global chemical potential) taking into account the mass constraint, we get

$$\begin{aligned} Q+m\Phi +mh(\rho )=\mu , \end{aligned}$$

(B77)

which is equivalent to Eq. (B69) with $E=\mu$, and to Eq. (B74). Furthermore, it can be shown that an equilibrium state is dynamically stable if, and only if, it is a minimum of energy at fixed mass [52].³⁹ Therefore, a stable stationary solution of the GPP equations is determined by the minimization problem

$$\begin{aligned} \min \ \lbrace {E_\mathrm{tot}}\, |\, M \,\, \mathrm{fixed} \rbrace . \end{aligned}$$

(B78)

We stress that the ordinary GPP equations do not relax towards the stationary state that minimizes the energy at fixed mass (ground state) since these equations are reversible and the energy is conserved. They rather experience a process of gravitational cooling and violent relaxation (on a coarse-grained scale) towards a quasistationary state with a core-halo structure (see Sect. 2.3). The characterization of this core-halo state was the topic of this paper.

Independently from the physical problem treated in this paper, it is an interesting mathematical problem in itself to be able to construct stationary solutions of the ordinary GPP equations (B65) and (B66). However, it is difficult in practice to numerically solve the nonlinear Eigenvalue problem defined by Eqs. (B67) and (B68) and make sure that the solution is dynamically stable. Interestingly, the generalized GPP equations (B1) and (B2) with $T_\mathrm{eff}=0$ provide a useful numerical algorithm to reach that goal. Indeed, we have shown in Appendix B3 that these equations satisfy an H-theorem and that they relax towards a stationary state that minimizes the free energy F defined by Eq. (B57) at fixed mass M. Since F with $U_\mathrm{th}=0$ coincides with $E_\mathrm{tot}$, this equilibrium state solves the minimization problem (B78). Therefore, it is a dynamically stable steady state of the ordinary GPP equations (B65) and (B66). Consequently, the generalized GPP equations (B1) and (B2) with $T_\mathrm{eff}=0$ provide a useful numerical algorithm to construct stable stationary solutions of the ordinary GPP equations (B65) and (B66). By construction, the equilibrium solution reached by the generalized GPP equations (B1) and (B2) with $T_\mathrm{eff}=0$, which are true relaxation equations, is guaranteed to be a stable stationary solution of the ordinary GPP equations (B65) and (B66).

Remark

Instead of solving the generalized GPP equations (B1) and (B2), we may equivalently solve the quantum damped Euler equations (B9)–(B12) or the simpler (diffusive) quantum Smoluchowski–Poisson equations (B49) and (B50) with $T_\mathrm{eff}=0$ which also relax towards a stationary solution that satisfies the minimization problem (B78).

1.5 5. General identities for a cold gas

For a cold ($T=0$) gas, the first principle of thermodynamics

$$\begin{aligned} d\left( \frac{u}{\rho }\right) =-P d\left( \frac{1}{\rho }\right) +T d\left( \frac{s}{\rho }\right) , \end{aligned}$$

(B79)

where u is the density of internal energy, s the density of entropy, $\rho$ the mass density, and P the pressure reduces to

$$\begin{aligned} d\left( \frac{u}{\rho }\right) =-P d\left( \frac{1}{\rho }\right) =\frac{P}{\rho ^2}d\rho . \end{aligned}$$

(B80)

Introducing the enthalpy per particle

$$\begin{aligned} h=\frac{P+u}{\rho }, \end{aligned}$$

(B81)

we get

$$\begin{aligned} du=h d\rho \quad \mathrm{and}\quad dh=\frac{dP}{\rho }. \end{aligned}$$

(B82)

Comparing Eq. (B81) with the Gibbs–Duhem relation at $T=0$:

$$\begin{aligned} u=-P+Ts+\frac{\mu }{m} \rho \quad \Rightarrow \quad \frac{\mu }{m}=\frac{P+u}{\rho }, \end{aligned}$$

(B83)

we see that the enthalpy $h(\mathbf{r})$ is equal to the local chemical potential $\mu (\mathbf{r})$ by unit of mass: $h(\mathbf{r})=\mu (\mathbf{r})/m$. On the other hand, for a barotropic gas for which $P=P(\rho )$, the foregoing equations can be written as

$$\begin{aligned}&P(\rho )=-\frac{d(u/\rho )}{d(1/\rho )}=\rho ^2\left[\frac{u(\rho )}{\rho }\right]'=\rho u'(\rho )-u(\rho ),\nonumber \\ \end{aligned}$$

(B84)

$$\begin{aligned}&P'(\rho )=\rho u''(\rho ),\quad h(\rho )=\frac{P(\rho )+u(\rho )}{\rho }, \end{aligned}$$

(B85)

$$\begin{aligned}&h(\rho )=u'(\rho ),\quad h'(\rho )=\frac{P'(\rho )}{\rho }, \end{aligned}$$

(B86)

$$\begin{aligned}&u(\rho )=\rho \int ^{\rho } \frac{P(\rho ')}{{\rho '}^2}\, d\rho '. \end{aligned}$$

(B87)

Comparing Eq. (B86) with Eqs. (B3) and (B13), we see that the potential $V(\rho )$ represents the density of internal energy

$$\begin{aligned} u(\rho )=V(\rho ). \end{aligned}$$

(B88)

We then have

$$\begin{aligned} P(\rho )= & {} \rho ^2\left[\frac{V(\rho )}{\rho }\right]'=\rho V'(\rho )-V(\rho ), \end{aligned}$$

(B89)

$$\begin{aligned} P'(\rho )= & {} \rho V''(\rho ),\quad h(\rho )=\frac{P(\rho )+V(\rho )}{\rho }, \end{aligned}$$

(B90)

$$\begin{aligned} h(\rho )= & {} V'(\rho ),\quad h'(\rho )=\frac{P'(\rho )}{\rho }, \end{aligned}$$

(B91)

$$\begin{aligned} V(\rho )= & {} \rho \int ^{\rho } \frac{P(\rho ')}{{\rho '}^2}\, d\rho '. \end{aligned}$$

(B92)

The squared speed of sound is

$$\begin{aligned} c_s^2=P'(\rho )=\rho V''(\rho ). \end{aligned}$$

(B93)

Appendix C: Classical collisionless self-gravitating systems

In this Appendix and in the following one, we discuss certain aspects of the dynamical evolution of classical collisionless self-gravitating systems to facilitate the comparison with the dynamical evolution of quantum self-gravitating systems treated in the main text.

1.1 1. Vlasov equation

A classical collisionless self-gravitating system (such as a stellar system or such as CDM) is governed by the Vlasov–Poisson equations

$$\begin{aligned}&\frac{\partial f}{\partial t}+\mathbf{v}\cdot \frac{\partial f}{\partial \mathbf{r}}-\nabla \Phi \cdot \frac{\partial f}{\partial \mathbf{v}}=0, \end{aligned}$$

(C1)

$$\begin{aligned}&\Delta \Phi =4\pi G\int f\, \mathrm{d}{} \mathbf{v}, \end{aligned}$$

(C2)

where $f=f(\mathbf{r},\mathbf{v},t)$ is the six-dimensional DF. The Vlasov equation is also known as the collisionless Boltzmann equation [7]. It states that, in the absence of encounters (“collisions”) between the particles, the density (DF) of a “fluid” particle is conserved when we follow its motion in phase space, i.e., $Df/Dt=0$ where $D/Dt=\partial /\partial t+\mathbf{v}\cdot \partial /\partial \mathbf{r}-\nabla \Phi \cdot \partial /\partial \mathbf{v}$ is the material derivative (Stokes operator). As a result, the Vlasov–Poisson equations conserve the energy $E=(1/2)\int f v^2\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}+(1/2)\int \rho \Phi \, \mathrm{d}{} \mathbf{r}$, the mass $M=\int f \, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}$ and an infinite class of Casimir integrals of the form $\int h(f)\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}$ where h(f) is an arbitrary function of f [91, 94].

Remark

The Vlasov equation can be viewed as the expression of the Liouville theorem in the individual phase space. Under the circumstance in which stellar encounters can be ignored, each star can be idealized as an independent conservative system described by the Hamiltonian $H=\mathbf{v}^2/2+\Phi (\mathbf{r},t)$ yielding the equations of motion $\mathrm{d}{} \mathbf{r}/dt=\mathbf{v}$, $d\mathbf{v}/dt=-\nabla \Phi$. The equation of continuity $\partial f/\partial t+\nabla _6\cdot (f \mathbf{U}_6)=0$, where $\nabla _6=(\partial _\mathbf{r},\partial _\mathbf{v})$ is a generalized nabla operator and $\mathbf{U}_6=(\mathbf{v},-\nabla \Phi )$ a generalized velocity field, and the fact that the flow in phase space in incompressible, $\nabla _6\cdot \mathbf{U}_6=0$, leads to the Liouville equation $\partial f/\partial t+\mathbf{U}_6\cdot \nabla _6 f=0$, which is the Vlasov equation.

1.2 2. Hydrodynamics of the Vlasov equation: Jeans equations

From the Vlasov equation, we can derive a system of hydrodynamic equations called the Jeans equations.⁴⁰ By integrating the Vlasov equation (C1) over the velocity, we get the continuity equation (expressing the local mass conservation)

$$\begin{aligned} \frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{u})=0, \end{aligned}$$

(C3)

where we have introduced the local density

$$\begin{aligned} \rho =\int f\, \mathrm{d}{} \mathbf{v} \end{aligned}$$

(C4)

and the local velocity

$$\begin{aligned} \mathbf{u}=\langle \mathbf{v}\rangle =\frac{1}{\rho }\int f \mathbf{v}\, \mathrm{d}{} \mathbf{v}. \end{aligned}$$

(C5)

Then, multiplying the Vlasov equation (C1) by $\mathbf{v}$ and integrating over the velocity, we obtain the momentum equation

$$\begin{aligned} \frac{\partial }{\partial t}(\rho u_i)+\frac{\partial }{\partial x_j}\int f v_iv_j\, d\mathbf{v}=-\rho \frac{\partial \Phi }{\partial x_i}. \end{aligned}$$

(C6)

Introducing the difference $\mathbf{w}=\mathbf{v}-\mathbf{u}$ between the velocity $\mathbf{v}$ of a particle and the mean local velocity $\mathbf{u}(\mathbf{r},t)$, and using the fact that $\langle \mathbf{w}\rangle =\mathbf{0}$, we get

$$\begin{aligned} \int f v_iv_j\, \mathrm{d}{} \mathbf{v}=\rho u_i u_j+\int f w_i w_j\, \mathrm{d}{} \mathbf{v}. \end{aligned}$$

(C7)

Therefore, the momentum equation (C6) takes the form

$$\begin{aligned} \frac{\partial }{\partial t}(\rho u_i)+\frac{\partial }{\partial x_j}(\rho u_i u_j)=-\partial _j P_{ij}-\rho \frac{\partial \Phi }{\partial x_i}, \end{aligned}$$

(C8)

where we have introduced the pressure tensor ($-P_{ij}$ is the stress tensor)

$$\begin{aligned} P_{ij}=\rho \langle w_iw_j\rangle =\int f (\mathbf{v}-\mathbf{u})_i (\mathbf{v}-\mathbf{u})_j\, \mathrm{d}{} \mathbf{v}. \end{aligned}$$

(C9)

It can be written as

$$\begin{aligned} P_{ij}=\rho (\langle v_i v_j\rangle - \langle v_i \rangle \langle v_j\rangle ). \end{aligned}$$

(C10)

Using the continuity equation (C3), we obtain the identity

$$\begin{aligned} \frac{\partial }{\partial t}(\rho \mathbf{u})+\nabla (\rho \mathbf{u}\otimes \mathbf{u})=\rho \left[\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}\right]. \end{aligned}$$

(C11)

As a result, the momentum equation (C8) can be rewritten as

$$\begin{aligned} \frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\partial _jP_{ij}-\nabla \Phi . \end{aligned}$$

(C12)

These equations are essentially those for a compressible fluid which is supported by pressure in the form of a velocity dispersion. These equations are not closed because the pressure tensor $P_{ij}$ depends on the DF which is not explicitly known in general. Actually, we can build up an infinite hierarchy of equations by introducing higher and higher moments of the velocity. In general, there is no simple way to close this hierarchy of Jeans equations except in the single speed case (see Appendix C3). In more general cases, some approximations must be introduced.

1.3 3. Single-speed solution: pressureless Euler equations

The Vlasov–Poisson equations admit a particular solution of the form

$$\begin{aligned} f(\mathbf{r},\mathbf{v},t)=\rho (\mathbf{r},t)\delta (\mathbf{v}-\mathbf{u}(\mathbf{r},t)). \end{aligned}$$

(C13)

This is called the single-speed solution because there is a single velocity attached to any given point $\mathbf{r}$ at time t. It corresponds to the “dust model” where the pressure is zero because there is no thermal motion (at a given location all the particles have the same velocity). The density $\rho (\mathbf{r},t)$ and the velocity $\mathbf{u}(\mathbf{r},t)$ satisfy the pressureless Euler equations

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{u})=0, \end{aligned}$$

(C14)

$$\begin{aligned}&\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\nabla \Phi , \end{aligned}$$

(C15)

$$\begin{aligned}&\Delta \Phi =4\pi G\rho . \end{aligned}$$

(C16)

These equations are exact. They can be deduced from the Jeans equations (C3) and (C12) by closing the hierarchy with the condition $P_{ij}=0$ obtained by substituting Eq. (C13) into Eq. (C9). They correspond to very particular initial conditions where the particles at a given location all have the same velocity. However, there is a well-known difficulty with the solution (C13). Even if one starts with a DF of the form of Eq. (C13) then, after a finite time, the solution of the Vlasov–Poisson equations becomes multi-stream because of particle crossing. This leads to the formation of caustics (singularities) in the density field at shell-crossing. Therefore, Eq. (C13) ceases to be valid. This phenomenon renders the pressureless hydrodynamical description (C14)–(C16) useless beyond the first time of crossing when the fast particles cross the slow ones. Therefore, after shell-crossing, the pressureless Euler equations are not defined anymore and we must come back to the original Vlasov–Poisson equations, or to the Jeans equations, because we need to account for a velocity dispersion. Indeed, the velocity field becomes multi-valued even if, initially, it is single-valued. The pressureless Euler equations are only valid until shell crossing and they fail as soon as orbit crossing (multistreaming) occurs.

Different attempts to cure this problem have been proposed in order to continue using a hydrodynamical model.

(i) A first heuristic possibility to avoid multi-streaming is to introduce a viscosity term $\nu \Delta \mathbf{u}$ in the momentum equation (C15) yielding the Navier–Stokes equation

$$\begin{aligned} \frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\nabla \Phi +\nu \Delta \mathbf{u}. \end{aligned}$$

(C17)

In order for the diffusion term to have a smoothing effect only in the regions where particle-crossing is about to occur, the viscosity $\nu$ should be small. More precisely, the limit $\nu \rightarrow 0$ should be taken, which is different from setting $\nu =0$.⁴¹

(ii) A second heuristic possibility is to introduce a pressure term $(-1/\rho )\nabla P$ in the momentum equation (C15) yielding

$$\begin{aligned} \frac{\partial \mathbf {u}}{\partial t} + (\mathbf {u} \cdot \nabla ) \mathbf {u} = -\nabla \Phi - \frac{1}{\rho } \nabla P, \end{aligned}$$

(C18)

where P is the fluid pressure, a local quantity given by a specified equation of state which takes into account velocity dispersion. This amounts to closing the hierarchy of Jeans equations with the isotropy ansatz $P_{ij}=P(\rho ) \delta _{ij}$. In this manner, there is no shell-crossing singularities. The velocity dispersion giving rise to the pressure can be a consequence of the multi-streaming or it can be already present in the initial condition. We need $P\rightarrow 0$ at large scales to recover the CDM model and $P\ne 0$ at small scales to avoid singularities.⁴²

(iii) A third heuristic possibility, proposed by Widrow and Kaiser [144], is to replace the Vlasov equation by a wave equation having the form of a Schrödinger equation with an effective Planck constant $\hbar _\mathrm{eff}$ controlling the spatial resolution.⁴³ Using the Madelung transformation, this prescription leads, instead of Eq. (C15), to a momentum equation of the form

$$\begin{aligned} \frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\nabla \Phi -\frac{1}{\rho }\partial _jP_{ij}^{Q}, \end{aligned}$$

(C19)

with an effective “quantum” pressure tensor

$$\begin{aligned} P_{ij}^{Q}=-\frac{\hbar _\mathrm{eff}^2}{4m^2}\rho \, \partial _i\partial _j\ln \rho =\frac{\hbar _\mathrm{eff}^2}{4m^2}\left( \frac{1}{\rho } \partial _i\rho \partial _j\rho -\partial _i\partial _j\rho \right) . \end{aligned}$$

(C20)

Interestingly, the effective quantum pressure tensor $P_{ij}^Q$ bares some resemblance with the Jeans pressure tensor $P_{ij}$ of Eq. (C10) with the velocity average operators replaced by density gradients (a nonlocal quantity). This amounts to closing the hierarchy of Jeans equations with the condition $P_{ij}=P_{ij}^Q$.⁴⁴ Therefore, although the system is classical, the procedure of Widrow and Kaiser [144] amounts to closing the Jeans equations by introducing an effective quantum potential of order $O(\hbar _\mathrm{eff}^2/m^2)$. When $\hbar _\mathrm{eff}/m\rightarrow 0$ the effective quantum pressure $P_{ij}^Q$ tends to zero at large scales while $P_{ij}^Q\ne 0$ at small scales (in line with the fact that quantum mechanics is negligible at large scales and important at small scales in BECDM), thereby preventing singularities. This is exactly what is required. The effective quantum pressure tensor acts as a regularizer of caustics and singularities in classical solutions. The quantum pressure also replaces the role of the viscosity in the adhesion model (see footnote 41) [145]. Therefore, the Schrödinger method can handle multiple streams in phase space.

(iv) A fourth possibility is to use the heuristic kinetic theory of violent relaxation developed by Chavanis et al. [94] (see also the Appendix of [84]). In that case, the pressureless Euler equation (C15) is replaced by the damped Euler equation

$$\begin{aligned} \frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\nabla \Phi -\frac{1}{\rho }\nabla P_\mathrm{LB}-\xi \mathbf{u}, \end{aligned}$$

(C21)

where $P_\mathrm{LB}$ is the Lynden-Bell pressure (or a generalized pressure) and $\xi$ is a friction coefficient accounting for nonlinear Landau damping.

In these different hydrodynamic models, we can go beyond the first time of crossing. Therefore, these hydrodynamic models are defined for all times. The solution of these hydrodynamic equations is expected to remain close to the solution of the Vlasov–Poisson equations for all times provided that $\nu \rightarrow 0$, $P\rightarrow 0$, and $\hbar _\mathrm{eff}/m\rightarrow 0$ in these respective models. We note that the limits $\nu ,P,\hbar _\mathrm{eff}/m\rightarrow 0$ are crucially different from taking $P=\nu =\hbar _\mathrm{eff}/m=0$.

Remark

The Vlasov–Poisson equations (C1) and (C2) are valid for all times. Similarly, the Jeans equations (C3)–(C12), which are equivalent to the Vlasov equation, are valid for all times but they are not closed (we have to consider the whole hierarchy of equations). By contrast, the pressureless Euler–Poisson equations (C14)–(C16) are valid only until the first time of crossing. Before that time they coincide with the Vlasov–Poisson equations and after that time they break down. The modified Euler–Poisson equations (C17)–(C21) are valid for all times. If $\nu \rightarrow 0$, $P\rightarrow 0$ and $\hbar _\mathrm{eff}\rightarrow 0$, they are expected to be close to the Vlasov–Poisson equations. In particular, Eqs. (C19)–(C20) are equivalent to the Schrödinger and Wigner equations, which are themselves equivalent to the Vlasov equation when $\hbar _\mathrm{eff}\rightarrow 0$.

Appendix D: Violent relaxation of classical collisionless self-gravitating systems

The Vlasov–Poisson equations (C1) and (C2) describing classical collisionless stellar systems are known to experience a process of violent relaxation [91] caused by the rapid fluctuations of the strongly varying gravitational potential at the early stage of galaxy formation. While the fine-grained DF ${f}(\mathbf{r},\mathbf{v},t)$ always evolves in time, forming intermingled filaments in phase space at smaller and smaller scales, the coarse-grained DF $\overline{f}(\mathbf{r},\mathbf{v},t)$, which smooths out this intricate filamentation, rapidly relaxes towards a quasistationary state $\overline{f}_\mathrm{QSS}(\mathbf{r},\mathbf{v})$. This process takes place on a very short timescale of the order of the dynamical time $t_D$. For $t>t_D$, the evolution of ${f}(\mathbf{r},\mathbf{v},t)$ occurs on scales smaller than the coarse-graining mesh. This complicated dynamics is associated with phase mixing, violent relaxation and nonlinear Landau damping. As a result, the coarse-grained DF $\overline{f}(\mathbf{r},\mathbf{v},t)$ does not satisfy the Vlasov equation. Phase-space correlations introduce an effective “collision” term $\mathcal{C}[\overline{f}]$ on the right hand side of the coarse-grained Vlasov equation. This collision term drives the relaxation of the coarse-grained DF towards the quasistationary state. The determination of the quasistationary state $\overline{f}_\mathrm{QSS}(\mathbf{r},\mathbf{v})$ and of the collision term $\mathcal{C}[\overline{f}]$ is a problem of fundamental interest but also of great difficulty because of the very nonlinear nature of the process. Here, we tackle this problem through a thermodynamical approach. We use a MEP to determine the quasistationary state $\overline{f}_\mathrm{QSS}(\mathbf{r},\mathbf{v})$ and we use a MEPP to determine the collision term $\mathcal{C}(\overline{f})$. At a general level, the MEP determines the most probable equilibrium state of the system and the MEPP determines the most probable evolution of the system [146].

1.1 1. Maximum entropy principle

In a seminal paper, Lynden-Bell [91] argued that the coarse-grained DF $\overline{f}(\mathbf{r},\mathbf{v},t)$ violently relaxes towards a quasistationary state $\overline{f}_\mathrm{QSS}(\mathbf{r},\mathbf{v})$ which maximizes a suitable mixing entropy while taking into account all the constraints of the Vlasov equation.⁴⁵ In the two-level approximation of the theory, where the fine-grained DF ${f}(\mathbf{r},\mathbf{v},t)$ takes only two values $f=\eta _0$ and $f=0$,⁴⁶ the constraints reduce to the conservation of mass and energy and the Lynden-Bell entropy reads

$$\begin{aligned} S=-\eta _0\int \biggl \lbrace {\overline{f}\over \eta _{0}}\ln {\overline{f}\over \eta _{0}}+\biggl (1-{\overline{f}\over \eta _{0}}\biggr )\ln \biggl (1-{\overline{f}\over \eta _{0}}\biggr ) \biggr \rbrace \, \mathrm{d}{} \mathbf{r} \mathrm{d}{} \mathbf{v}. \end{aligned}$$

(D1)

It can be obtained from a combinatorial analysis taking into account the specificities of the Vlasov equation. In particular, the coarse-grained DF must satisfy the inequality $\overline{f}(\mathbf{r},\mathbf{v},t)\le \eta _{0}$ arising from the incompressibility of the flow in phase space and the conservation of the DF on the fine-grained scale.⁴⁷ This constraint is similar to the Pauli exclusion principle in quantum mechanics (with another interpretation) and this is why the Lynden-Bell entropy (D1) resembles the Fermi–Dirac entropy of quantum mechanics. In this sense, the process of violent relaxation is similar in some respects to the collisional relaxation of self-gravitating fermionic particles. The Lynden-Bell DF corresponds to a fourth type of statistics corresponding to distinguishable particles experiencing an exclusion principle [91].

According to the MEP, the DF of the quasistationary state $\overline{f}_\mathrm{QSS}$ maximizes the Lynden-Bell entropy (D1) at fixed mass

$$\begin{aligned} M=\int \rho \, \mathrm{d}{} \mathbf{r} \end{aligned}$$

(D2)

and energy

$$\begin{aligned} E=\int {1\over 2}f v^{2} \, d\mathbf{r} d\mathbf{v}+{1\over 2}\int \rho \Phi \, d\mathbf{r}=K+W, \end{aligned}$$

(D3)

where K is the kinetic energy and W the potential (gravitational) energy. Introducing Lagrange multipliers and writing the variational principle under the form

$$\begin{aligned} \delta S-\beta \delta E+\alpha \delta M=0, \end{aligned}$$

(D4)

we find that the extrema of entropy at fixed mass and energy correspond to the Lynden-Bell DF

$$\begin{aligned} \overline{f}_\mathrm{LB}(\mathbf{r},\mathbf{v})=\frac{\eta _0}{1+\mathrm{e}^{\beta (v^2/2+\Phi (\mathbf{r}))-\alpha }}, \end{aligned}$$

(D5)

where $\epsilon ={v^{2}/2}+\Phi (\mathbf{r})$ is the energy of a particle by unit of mass. The Lagrange multipliers $\beta =1/T_\mathrm{eff}$ and $\alpha =\mu _\mathrm{eff}/T_\mathrm{eff}$ are the effective inverse temperature and the effective chemical potential (divided by the effective temperature). The Lynden-Bell DF (D5) which maximizes the Lynden-Bell entropy at fixed mass and energy is the most probable, or most mixed, state taking into account all the constraints of the Vlasov equation. The Lynden-Bell DF is similar to the Fermi–Dirac DF in quantum mechanics provided that we make the correspondence

$$\begin{aligned} \eta _0=\frac{gm^4}{h^3}, \end{aligned}$$

(D6)

where $g=2s+1$ is the multiplicity of the quantum states. In particular, the Lynden-Bell DF (D5) satisfies the constraint $\overline{f}_\mathrm{LB}(\mathbf{r},\mathbf{v})\le \eta _{0}$ which is similar to the Pauli exclusion principle in quantum mechanics. We note that the effective temperature $T_\mathrm{eff}$ has not the dimension of a temperature. It should rather be interpreted as a velocity dispersion. However, we shall use this notation which is more transparent. We also note that the mass m of the particles does not appear in the Lynden-Bell theory since it is based on the Vlasov equation for collisionless systems which is independent of the mass of the particles. In this sense, we can say that the temperature in Lynden-Bell’s theory is proportional to the mass of the particles [91].

From the Lynden-Bell DF (D5), one can determine the density and the pressure through the equations

$$\begin{aligned}&\rho =\int \overline{f}\, \mathrm{d}{} \mathbf{v}=\int _0^{+\infty } \frac{\eta _0}{1+\mathrm{e}^{\beta (v^2/2+\Phi (\mathbf{r}))-\alpha }} 4\pi v^2\, dv, \end{aligned}$$

(D7)

$$\begin{aligned}&P=\frac{1}{3}\int \overline{f} v^2\, d\mathbf{v}=\frac{1}{3}\int _0^{+\infty } \frac{\eta _0}{1+\mathrm{e}^{\beta (v^2/2+\Phi (\mathbf{r}))-\alpha }} \, 4\pi v^4\, dv.\nonumber \\ \end{aligned}$$

(D8)

They can be rewritten as

$$\begin{aligned}&\rho (\mathbf{r})=\frac{4\pi \eta _0\sqrt{2}}{\beta ^{3/2}}I_{1/2}\left[\mathrm{e}^{\beta \Phi (\mathbf{r})-\alpha }\right], \end{aligned}$$

(D9)

$$\begin{aligned}&P(\mathbf{r})=\frac{8\pi \eta _0\sqrt{2}}{3\beta ^{5/2}}I_{3/2}\left[\mathrm{e}^{\beta \Phi (\mathbf{r})-\alpha }\right], \end{aligned}$$

(D10)

where

$$\begin{aligned} I_n(t)=\int _0^{+\infty }\frac{x^n}{1+t\mathrm{e}^x}\, dx \end{aligned}$$

(D11)

denote the Fermi–Dirac integrals. Eliminating formally the gravitational potential $\Phi (\mathbf{r})$ between Eqs. (D9) and (D10), we see that the equation of state is barotropic: $P(\mathbf{r})=P[\rho (\mathbf{r})]$. Equations (D9) and (D10) determine the Lynden-Bell equation of state $P_\mathrm{LB}(\rho )$ in parametric form. The Lynden-Bell equation of state is formally similar to the Fermi–Dirac equation of state. Using Eq. (D8), the kinetic energy can be written as

$$\begin{aligned} K=\frac{3}{2}\int P_\mathrm{LB}(\rho )\, \mathrm{d}{} \mathbf{r}. \end{aligned}$$

(D12)

On the other hand, the Lynden-Bell DF (D5), and more generally any DF of the form $f=f(\epsilon )$, implies the condition of hydrostatic equilibrium (see, e.g., [42] and the Remark of Appendix F)

$$\begin{aligned} \nabla P+\rho \nabla \Phi =\mathbf{0}. \end{aligned}$$

(D13)

In the completely degenerate limit $\overline{f}_\mathrm{LB}\sim \eta _0$, the Lynden-Bell DF reduces to a step function

$$\begin{aligned} f_\mathrm{LB}(\mathbf{r},\mathbf{v})=\eta _0 H(\epsilon -\epsilon _\mathrm{LB}), \end{aligned}$$

(D14)

where H is the Heaviside function ($H(x)=1$ if $x<1$ and $H(x)=0$ if $x>1$) and $\epsilon _\mathrm{LB}$ is the Lynden-Bell energy, which is the counterpart of the Fermi energy. In that case, Eqs. (D7) and (D8) reduce to

$$\begin{aligned}&\rho =\int _0^{v_\mathrm{LB}} \eta _0 4\pi v^2\, dv=4\pi \eta _0 \frac{v_\mathrm{LB}^3}{3}, \end{aligned}$$

(D15)

$$\begin{aligned}&P=\frac{1}{3}\int _0^{v_\mathrm{LB}} \eta _0 v^2 4\pi v^2\, dv=\frac{4\pi \eta _0}{3} \frac{v_\mathrm{LB}^5}{5}, \end{aligned}$$

(D16)

where $v_\mathrm{LB}(\mathbf{r})=\sqrt{2(\epsilon _\mathrm{LB}-\Phi (\mathbf{r}))}$ is the Lynden-Bell velocity. Equations (D15) and (D16) lead to the equation of state

$$\begin{aligned} P=\frac{1}{5}\left( \frac{3}{4\pi \eta _0}\right) ^{2/3}\rho ^{5/3}. \end{aligned}$$

(D17)

This is a polytropic equation of state of index $n=3/2$ like the one arising in the theory of nonrelativistic white dwarf stars at $T=0$ corresponding to the ground state of the self-gravitating Fermi gas [120].

In the nondegenerate, or dilute, limit $\overline{f}_\mathrm{LB}\ll \eta _{0}$, the Lynden-Bell entropy becomes similar to the Boltzmann entropy

$$\begin{aligned} S\simeq -\int \overline{f} \left[\ln \left( \frac{\overline{f}}{\eta _{0}}\right) -1\right]\, \mathrm{d}{} \mathbf{r} \mathrm{d}{} \mathbf{v}, \end{aligned}$$

(D18)

the Lynden-Bell DF becomes similar to the Maxwell-Boltzmann DF

$$\begin{aligned} \overline{f}_\mathrm{LB}(\mathbf{r},\mathbf{v})\simeq \eta _0 \mathrm{e}^{\alpha -\beta [v^2/2+\Phi (\mathbf{r})]}, \end{aligned}$$

(D19)

and the Lynden-Bell equation of state becomes similar to the equation of state of an isothermal gas

$$\begin{aligned} P=\rho T_\mathrm{eff}. \end{aligned}$$

(D20)

This equation of state has been studied in relation to isothermal stars [120] and to the statistical mechanics of “collisional” self-gravitating systems relaxing under the effect of gravitational encounters [128]. Remarkably, the Lynden-Bell theory of violent relaxation explains how a collisionless self-gravitating system can “thermalize” on a very short timescale, much shorter that the collisional relaxation time $t_\mathrm{relax} \sim (N/\ln N)t_D$, without the need of “collisions” [91].

So far, we have assumed that the system is isolated so that it conserves the mass and the energy. This corresponds to the microcanonical ensemble. We now consider the canonical ensemble in which the temperature $T_\mathrm{eff}=1/\beta$ is fixed instead of the energy. In that case, the equilibrium state is obtained by minimizing the free energy $F=E-T_\mathrm{eff}S$ at fixed mass M or, equivalently, by maximizing the Massieu function $J=S-\beta E$ at fixed mass M. The variational principle determining the extrema of free energy at fixed mass reads

$$\begin{aligned} \delta J+\alpha \delta M=0. \end{aligned}$$

(D21)

Since $\beta$ is fixed, this variational principle for the first variations of the thermodynamical potential is equivalent to Eq. (D4) and it returns the Lynden-Bell DF (D5). Therefore, the equilibrium states are the same in the microcanonical and canonical ensembles (the extrema of entropy at fixed mass and energy coincide with the extrema of free energy at fixed mass). However, their thermodynamical stability (related to the sign of the second variations of the thermodynamical potential) may be different in the microcanonical and canonical ensembles. This is the notion of ensembles inequivalence for systems with long-range interactions [128,129,130,131]. An equilibrium state is thermodynamically stable in the microcanonical ensemble if it is a maximum of entropy at fixed mass and energy. This corresponds to

$$\begin{aligned}&\delta ^2 J=\delta ^2 S-\beta \delta ^2 E =-\int \frac{1}{\overline{f}(1-\overline{f}/\eta _0)}\frac{(\delta f)^2}{2}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}\nonumber \\&\quad -{1\over 2}\beta \int \delta \rho \delta \Phi \, d\mathbf{r}\le 0\qquad \end{aligned}$$

(D22)

for all perturbations $\delta f$ that conserve mass and energy at first order. An equilibrium state is thermodynamically stable in the canonical ensemble if it is a minimum of free energy at fixed mass. This corresponds to the inequality of Eq. (D22) for all perturbations $\delta f$ that conserve mass. We note that canonical stability implies microcanonical stability but the converse is wrong [132]. For example, it is shown in [102] that the core-halo solution with a negative specific heat is stable in the microcanonical ensemble while it is unstable in the canonical ensemble.

1.2 2. Maximum entropy production principle

We now consider the dynamical evolution of the coarse-grained DF $\overline{f}(\mathbf{r},\mathbf{v},t)$. Writing $f=\overline{f}+\delta f$ and $\Phi =\overline{\Phi }+\delta \Phi$, where $\delta f$ and $\delta \Phi$ denote fluctuations about the coarse-grained fields, and taking the local average of the Vlasov equation (C1), we get

$$\begin{aligned} \frac{\partial \overline{f}}{\partial t}+\mathbf{v}\cdot \frac{\partial \overline{f}}{\partial \mathbf{r}}-\nabla {\Phi }\cdot \frac{\partial \overline{f}}{\partial \mathbf{v}}=\frac{\partial }{\partial \mathbf{v}}\cdot \overline{\delta f\nabla \delta \Phi }. \end{aligned}$$

(D23)

This equation shows that the correlations of the fluctuations of the gravitational potential and DF create an effective “collision” term $\mathcal{C}[\overline{f}]$ in the r.h.s. of Eq. (D23). In Ref. [94] we obtained an explicit expression of this collision term by using heuristic arguments based on a MEPP.⁴⁸ We considered the general case where the fine-grained DF may take an arbitrary number of values. Below, we detail this procedure in the simpler case where the fined-grained DF takes only two values $f=0$ and $f=\eta _0$ as in the preceding section.

To apply the MEPP, we first write the relaxation equation for the coarse-grained DF under the form

$$\begin{aligned} \frac{\partial \overline{f}}{\partial t}+\mathbf{v}\cdot \frac{\partial \overline{f}}{\partial \mathbf{r}}-\nabla {\Phi }\cdot \frac{\partial \overline{f}}{\partial \mathbf{v}}=-\frac{\partial }{\partial \mathbf{v}}\cdot \mathbf{J}, \end{aligned}$$

(D24)

where $\mathbf{J}$ is the current to be determined. The form of Eq. (D24) ensures the conservation of mass provided that $\mathbf{J}$ decreases sufficiently rapidly for large $|\mathbf{v}|$. From Eqs. (D1), (D3) and (D24), we get

$$\begin{aligned}&\dot{S}=-\int \frac{1}{\overline{f}(1-\overline{f}/\eta _0)} \mathbf{J}\cdot \frac{\partial \overline{f}}{\partial \mathbf{v}}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}, \end{aligned}$$

(D25)

$$\begin{aligned}&\dot{E}=\int \mathbf{J}\cdot \mathbf{v}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}, \end{aligned}$$

(D26)

where we have used straightforward integrations by parts. Following the MEPP, we shall determine the optimal current $\mathbf{J}$ which maximizes the rate of entropy production (D25) while satisfying the conservation of energy $\dot{E}=0$. For this problem to have a solution, we shall also impose a limitation on the current $|\mathbf{J}|$ characterized by a bound $C(\mathbf{r},\mathbf{v},t)$ which exists but is not explicitly known, so that

$$\begin{aligned} {J^{2}\over 2f}\le C(\mathbf{r},\mathbf{v},t). \end{aligned}$$

(D27)

It can be shown by a convexity argument that reaching the bound (D27) is always favorable for increasing $\dot{S}$, so this constraint can be replaced by an equality. The variational problem can then be solved by introducing at each time t Lagrange multipliers $\beta (t)$ and $1/{\tilde{D}}(\mathbf{r},\mathbf{v},t)$ for the two constraints. The condition

$$\begin{aligned} \delta \dot{S}-\beta (t)\delta \dot{E}-\int {1\over {\tilde{D}}}\delta \biggl ({J^{2}\over 2f}\biggr )\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}=0 \end{aligned}$$

(D28)

yields an optimal current of the form

$$\begin{aligned} \mathbf{J}=-D\biggl [{\partial \overline{f}\over \partial \mathbf{v}}+\beta (t)\overline{f}(1-\overline{f}/\eta _0)\mathbf{v}\biggr ], \end{aligned}$$

(D29)

where we have set $\tilde{D}=D(1-\overline{f}/\eta _0)$ to avoid divergences when $\overline{f}\rightarrow \eta _0$.⁴⁹ The time evolution of the Lagrange multiplier $\beta (t)$ is determined by the conservation of energy $\dot{E}=0$, introducing Eq. (D29) into the constraint (D26). This yields

$$\begin{aligned} \beta (t)=-{\int D {\partial \overline{f}\over \partial \mathbf{v}}\cdot \mathbf{v}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}\over \int D \overline{f}(1-\overline{f}/\eta _0) v^{2}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}}. \end{aligned}$$

(D30)

Note that the optimal current (D29) can be written as

$$\begin{aligned} \mathbf{J}=-{\tilde{D}} \frac{\partial \alpha }{\partial \mathbf{v}}, \end{aligned}$$

(D31)

where

$$\begin{aligned} \alpha (\mathbf{r},\mathbf{v},t)\equiv \ln \left( \frac{\overline{f}/\eta _0}{1-\overline{f}/\eta _0}\right) +\beta (t)\left[\frac{v^2}{2}+\Phi (\mathbf{r},t)\right]\end{aligned}$$

(D32)

is a time-dependent chemical potential which is uniform at equilibrium according to Eq. (D5). The relation from Eq. (D31) then corresponds to the linear thermodynamics of Onsager [147, 148] where the currents are proportional to the gradients of the thermodynamic potentials that are uniform at statistical equilibrium. Therefore, the MEPP can be viewed as a variational formulation of Onsager’s linear thermodynamics [149].⁵⁰

Introducing the optimal current (D29) into Eq. (D24), we obtain the relaxation equation

$$\begin{aligned}&\frac{\partial \overline{f}}{\partial t}+\mathbf{v}\cdot \frac{\partial \overline{f}}{\partial \mathbf{r}}-\nabla {\Phi }\cdot \frac{\partial \overline{f}}{\partial \mathbf{v}}\nonumber \\&\quad =\frac{\partial }{\partial \mathbf{v}}\left[D\left( \frac{\partial \overline{f}}{\partial \mathbf{v}}+\beta (t) \overline{f}(1-\overline{f}/\eta _0)\mathbf{v}\right) \right]. \end{aligned}$$

(D33)

Morphologically, this relaxation equation has the form of a nonlinear Fokker–Planck equation or, more precisely, the form of a fermionic Kramers equation [149, 151]. The first term is a diffusion term and the second term is a friction term. The fermionic factor $\overline{f}(1-\overline{f}/\eta _0)$ takes into account the Lynden-Bell exclusion principle. The function $\beta (t)$ can be considered as a time dependent inverse temperature evolving with time so as to conserve the energy (microcanonical formulation).⁵¹ The friction coefficient $\xi$ satisfies a generalized Einstein relation: $\xi =D\beta$. Note that D is not determined by the MEPP since it is related to the unknown bound $C(\mathbf{r},\mathbf{v},t)$ in Eq. (D27).

It is straightforward to check that Eq. (D33) with the constraint (D30) satisfies an H-theorem for the Lynden-Bell entropy (D1). From Eq. (D25), we can write

$$\begin{aligned} \dot{S}= & {} -\int {1\over \overline{f}(1-\overline{f}/\eta _0)}{} \mathbf{J}\cdot \biggl [{\partial \overline{f}\over \partial \mathbf{v}}+\beta (t)\overline{f}(1-\overline{f}/\eta _0)\mathbf{v}\biggr ]\, d\mathbf{r}\mathrm{d}{} \mathbf{v}\nonumber \\&+\beta (t) \int \mathbf{J}\cdot \mathbf{v} \, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}.\qquad \end{aligned}$$

(D34)

The last integral vanishes due to the conservation of the energy (see Eq. (D26) with $\dot{E}=0$). Using Eq. (D29), we obtain

$$\begin{aligned} \dot{S}=\int {\mathbf{J}^{2}\over D\overline{f}(1-\overline{f}/\eta _0)}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}, \end{aligned}$$

(D35)

which is positive ($\dot{S}\ge 0$) provided that $D>0$. This proves the H-theorem. At equilibrium, we have $\dot{S}=0$ which implies $\mathbf{J}=\mathbf{0}$. Then, according to Eq. (D29), we obtain

$$\begin{aligned} {\partial \overline{f}\over \partial \mathbf{v}}+\beta \overline{f}(1-\overline{f}/\eta _0)\mathbf{v}=\mathbf{0}. \end{aligned}$$

(D36)

Integrating this equation with respect to $\mathbf{v}$, we get

$$\begin{aligned} \ln \left( \frac{\overline{f}/\eta _0}{1-\overline{f}/\eta _0}\right) +\beta {v^{2}\over 2}=A(\mathbf{r}), \end{aligned}$$

(D37)

where $A(\mathbf{r})$ is a constant of integration that may depend on $\mathbf{r}$. Taking the gradient of the foregoing equation with respect to $\mathbf{r}$, we find that

$$\begin{aligned} \frac{1}{\overline{f}(1-\overline{f}/\eta _0)}{\partial \overline{f}\over \partial \mathbf{r}}=\nabla A(\mathbf{r}). \end{aligned}$$

(D38)

On the other hand, since $\partial _t\overline{f}=0$ and $\mathbf{J}=\mathbf{0}$, the advection term in Eq. (D24) cancels out:

$$\begin{aligned} \mathbf{v}\cdot \frac{\partial \overline{f}}{\partial \mathbf{r}}-\nabla {\Phi }\cdot \frac{\partial \overline{f}}{\partial \mathbf{v}}=0. \end{aligned}$$

(D39)

Together with Eqs. (D36) and (D38), this implies the relation

$$\begin{aligned} \mathbf{v}\cdot (\nabla A+\beta \nabla \Phi )=0. \end{aligned}$$

(D40)

This relation must be true for all $\mathbf{v}$ so that

$$\begin{aligned} \nabla A+\beta \nabla \Phi =\mathbf{0}, \end{aligned}$$

(D41)

which can be integrated into

$$\begin{aligned} A(\mathbf{r})=-\beta \Phi (\mathbf{r})+\alpha , \end{aligned}$$

(D42)

where $\alpha$ is a constant. Substituting Eq. (D42) into Eq. (D37), we finally recover the Lynden-Bell DF (D5) with $\beta =\lim _{t\rightarrow +\infty }\beta (t)$. Therefore, the stationary solutions of the generalized Fokker–Planck equation (D33) are the Lynden-Bell DFs which extremize the Lynden-Bell entropy at fixed energy and mass. In addition, one can show that an equilibrium state is linearly dynamically stable if, and only if, it is a (local) maximum of S at fixed M.⁵² Indeed, considering the linear dynamical stability of a stationary solution of Eqs. (D30) and (D33), we can derive the general relation [151]

$$\begin{aligned} 2\lambda \delta ^{2}{J}=\delta ^{2}\dot{S}\ge 0, \end{aligned}$$

(D43)

connecting the growth rate $\lambda$ of the perturbation $\delta f\sim \mathrm{e}^{\lambda t}$ to the second order variations of the free energy (Massieu function) $J=S-\beta E$ and the second order variations of the rate of entropy production $\delta ^{2}\dot{S}$. Since the product $\lambda \delta ^{2}{J}$ is positive because $\delta ^{2}\dot{S}\ge 0$ according to Eq. (D35), we conclude that a stationary solution of Eqs. (D30) and (D33) is linearly dynamically stable ($\lambda <0$) if, and only if, it is an entropy maximum at fixed mass and energy ($\delta ^2 J<0$). This aesthetic formula shows the equivalence between dynamical and thermodynamical stability for the generalized Fokker–Planck equation (D33) with the constraint from Eq. (D30). Therefore, this equation can only relax towards maxima of S, not towards minima or saddle points.

A relaxation equation of the form of Eq. (D24) appropriate to the canonical ensemble can be obtained by maximizing the rate of free energy dissipation $\dot{J}=\dot{S}-\beta \dot{E}$ with the constraint (D27). The corresponding variational principle

$$\begin{aligned} \delta \dot{J}-\int {1\over {\tilde{D}}}\delta \biggl ({J^{2}\over 2f}\biggr )\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}=0 \end{aligned}$$

(D44)

again yields an optimal current of the form of Eq. (D29) now involving a constant inverse temperature $\beta$. This equation satisfies an H-theorem for the Lynden-Bell free energy. From Eqs. (D25) and (D26) we have

$$\begin{aligned} \dot{J}= & {} \dot{S}-\beta \dot{E}\nonumber \\= & {} -\int {1\over \overline{f}(1-\overline{f}/\eta _0)}{} \mathbf{J}\cdot {\partial \overline{f}\over \partial \mathbf{v}} \, \mathrm{d}{} \mathbf{r}d\mathbf{v}-\beta \int \mathbf{J}\cdot \mathbf{v} \, d\mathbf{r}\mathrm{d}{} \mathbf{v}.\nonumber \\ \end{aligned}$$

(D45)

Using Eq. (D29), we obtain

$$\begin{aligned} \dot{J}=\int {\mathbf{J}^{2}\over D\overline{f}(1-\overline{f}/\eta _0)}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}, \end{aligned}$$

(D46)

which is positive ($\dot{J}\ge 0$) provided that $D>0$. Therefore, the free energy F decreases monotonically until an equilibrium state of the form (D5) is reached (H-theorem). In the canonical ensemble, we can show that $2\lambda \delta ^{2}{J}=\delta ^{2}\dot{J}\ge 0$ [149]. Therefore, a stationary solution of the generalized Fokker–Planck equation (D33) with constant temperature is linearly stable if, and only if, it is a (local) minimum of free energy F at fixed mass. Therefore, this equation can only relax towards minima of F, not towards maxima or saddle points.

Remark

By using the MEPP (or the linear thermodynamics of Onsager), we have constructed a kinetic equation for the coarse-grained DF, Eq. (D33), which relaxes towards the Lynden-Bell DF (D5). This kinetic equation has the form of a fermionic Kramers equation. We stress that this thermodynamical approach is phenomenological in nature. It is also possible to derive a kinetic equation for the coarse-grained DF through a systematic procedure by developing a quasilinear theory of the Vlasov–Poisson equations [96,97,98,99,100,101]. This kinetic equation has the form of a fermionic Landau equation. Although it is satisfying to derive this equation from a systematic procedure, its domain of validity remains unclear (it may only describe a “gentle” relaxation). However, the fermionic Kramers equation can be obtained from the fermionic Landau equation by making a thermal bath approximation [96]. In that case, the diffusion coefficient (which is not determined by the MEPP) can be explicitly calculated. The fermionic Kramers equation with a time-dependent temperature and the fermionic Landau equation both satisfy an H-theorem for the Lynden-Bell entropy. We note, however, that the fermionic Landau equation conserves the energy locally while the fermionic Kramers equation with a time-dependent temperature conserves the energy globally. This difference affects the hydrodynamic equations derived from these kinetic equations: they involve either a viscosity (Landau) or a friction (Kramers) [127].⁵³ We finally note that it is possible to generalize the results of this Appendix to an arbitrary entropy of the form $S=-\int C(\overline{f})\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}$, where C is a convex function [100, 151]. This generalization may allow us to solve heuristically the problem of incomplete relaxation mentioned in Sect. 3.3 by considering entropic functionals different from the Lynden-Bell entropy that yield DFs with a finite mass. It can also allow us to deal with more complex situations than those considered here. In any case, the MEPP provides a numerical algorithm in the form of a generalized Fokker–Planck equation that can be used to construct dynamically stable stationary solutions of the Vlasov–Poisson equations, a nontrivial problem in itself [151]. Finally, to be complete, we note that a different kinetic approach of collisionless relaxation has been developed in [152] where the coarse-grained DF is defined as the convolution of the fine-grained DF by a Gaussian window.

1.3 3. Hydrodynamic equations

We now take the hydrodynamic moments of the coarse-grained Vlasov equation (D33). To make the results fully explicit, we consider the nondegenerate limit of the theory (the general case is treated in [94]). In that case, the coarse-grained Vlasov equation takes the form

$$\begin{aligned} \frac{\partial \overline{f}}{\partial t}+\mathbf{v}\cdot \frac{\partial \overline{f}}{\partial \mathbf{r}}-\nabla {\Phi }\cdot \frac{\partial \overline{f}}{\partial \mathbf{v}}=\frac{\partial }{\partial \mathbf{v}}\cdot \left[D\left( \frac{\partial \overline{f}}{\partial \mathbf{v}}+\beta (t) \overline{f}{} \mathbf{v}\right) \right], \end{aligned}$$

(D47)

which is similar to the classical Kramers equation. The evolution of the inverse effective temperature $\beta (t)=1/T_\mathrm{eff}(t)$ is given by

$$\begin{aligned} \beta (t)=-{\int D {\partial \overline{f}\over \partial \mathbf{v}}\cdot \mathbf{v}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}\over \int D \overline{f} v^{2}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}}. \end{aligned}$$

(D48)

Assuming that D is constant, and making an integration by parts, we get

$$\begin{aligned} \beta (t)=\frac{3M}{2K(t)}, \end{aligned}$$

(D49)

where

$$\begin{aligned} K(t)=\int f\frac{v^2}{2}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v} \end{aligned}$$

(D50)

is the total kinetic energy. Eq. (D49) can be rewritten as $K(t)=\frac{3}{2}MT_\mathrm{eff}(t)$. This relation shows that the effective temperature $T_\mathrm{eff}(t)=1/\beta (t)$ can be interpreted as the average kinetic temperature of the system, i.e., $T_\mathrm{eff}(t)=\langle T_\mathrm{kin}(\mathbf{r},t)\rangle$ where $\frac{3}{2}\rho T_\mathrm{kin}(\mathbf{r},t)=\int f \frac{v^2}{2}\, d\mathbf{v}$ is the local kinetic energy. One has $E=\frac{3}{2}MT_\mathrm{eff}+W$.

Taking the hydrodynamic moments of the coarse-grained Vlasov equation (D47), we obtain a system of equations similar to the Jeans equations (see Appendix C2) but including dissipative effects:

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{u})=0, \end{aligned}$$

(D51)

$$\begin{aligned}&\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\partial _jP_{ij}-\nabla \Phi -\xi \mathbf{u}, \end{aligned}$$

(D52)

where $\xi =D\beta$. They are called the damped Jeans equations. On the other hand, the effective collision term in Eq. (D47) provides a source of relaxation which allows us to close the hierarchy of moment equations. Indeed, we can compute the pressure tensor in Eq. (D52) by making a LTE approximation

$$\begin{aligned} \overline{f}_\mathrm{LTE}(\mathbf{r},\mathbf{v},t)=\frac{\rho (\mathbf{r},t)}{[2\pi T_\mathrm{eff}(t)]^{3/2}}\mathrm{e}^{-\frac{[\mathbf{v}-\mathbf{u}(\mathbf{r},t)]^2}{2T_\mathrm{eff}(t)}}. \end{aligned}$$

(D53)

In that case, we get

$$\begin{aligned} P_{ij}=P_\mathrm{th}\delta _{ij}\quad \mathrm{with}\quad P_\mathrm{th}=\rho T_\mathrm{eff}(t). \end{aligned}$$

(D54)

This leads to a system of hydrodynamic equations of the form

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{u})=0, \end{aligned}$$

(D55)

$$\begin{aligned}&\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\nabla P_\mathrm{th}-\nabla \Phi -\xi \mathbf{u}, \end{aligned}$$

(D56)

$$\begin{aligned}&\Delta \Phi =4\pi G\rho , \end{aligned}$$

(D57)

called the damped Euler equations. Using the LTE approximation (D53), the total energy (D3) can be written as

$$\begin{aligned} E=\int \rho \frac{\mathbf{u}^2}{2}\, d\mathbf{r}+\frac{3}{2}MT_\mathrm{eff}+W. \end{aligned}$$

(D58)

In the microcanonical ensemble where the energy E is fixed, this equation determines the evolution of the effective temperature $T_\mathrm{eff}(t)$. On the other hand, the Boltzmann-like entropy (D18) can be written (up to an additive constant) as

$$\begin{aligned} S=-\int \rho (\ln \rho -1)\, \mathrm{d}{} \mathbf{r}+\frac{3}{2}M\ln T_\mathrm{eff}. \end{aligned}$$

(D59)

This justifies the expressions of E and S used in Appendix B2. One can then show (like in Appendix B2) that the hydrodynamic equations (D55)–(D57) with a time-dependent effective temperature $T_\mathrm{eff}(t)$ determined by Eq. (D58) satisfy an H-theorem for the entropy S given by Eq. (D59). Analogously, in the canonical ensemble where the effective temperature $T_\mathrm{eff}$ is fixed, one can show (like in Appendix B3) that the hydrodynamic equations (D55)–(D57) with a constant effective temperature $T_\mathrm{eff}$ satisfy an H-theorem for the free energy $F=E-T_\mathrm{eff}S$. One can also derive explicit expressions of the virial theorem like in Appendices B2 and B3 (one simply has to take $P=Q=0$ in the equations of these Appendices).

Remark

We note that, in Eq. (D58), the effective temperature $T_\mathrm{eff}$ represents the thermal kinetic energy $\frac{3}{2}M T_\mathrm{eff}(t)=\int f \frac{w^2}{2}\, \mathrm{d}{} \mathbf{r}d\mathbf{v}$ where $\mathbf{w}=\mathbf{v}-\mathbf{u}(\mathbf{r},t)$ is the fluctuating velocity while the exact relation (D49) indicates that it should represent the total kinetic energy $\frac{3}{2}M T_\mathrm{eff}(t)=\int f \frac{v^2}{2}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}$ including the mean kinetic energy. This is an artefact of the LTE approximation (D53) which involves a uniform temperature $T_\mathrm{eff}(t)$ instead of a space-dependent temperature $T(\mathbf{r},t)$. We can solve this problem by replacing $T_\mathrm{eff}(t)$ by $T(\mathbf{r},t)$ in Eq. (D53) and introducing a hydrodynamic equation for the local temperature $T(\mathbf{r},t)$ (second moment) as in Ref. [94] (see Appendix D4). However, the LTE approximation (D53) becomes exact close to equilibrium or in the strong friction limit $\xi \rightarrow +\infty$ where $|\mathbf{u}|=O(1/\xi )$. Indeed, in the strong friction limit, we have

$$\begin{aligned} \mathbf{u}=-\frac{1}{\xi \rho } \left( T_\mathrm{eff}\nabla \rho +\rho \nabla \Phi \right) . \end{aligned}$$

(D60)

Substituting this relation into the continuity equation (D55) we obtain the Smoluchowski–Poisson equations

$$\begin{aligned} \xi \frac{\partial \rho }{\partial t}= & {} \nabla \cdot \left( T_\mathrm{eff}\nabla \rho +\rho \nabla \Phi \right) , \end{aligned}$$

(D61)

$$\begin{aligned} \Delta \Phi= & {} 4\pi G\rho . \end{aligned}$$

(D62)

Since $|\mathbf{u}|=O(1/\xi )$, we can neglect the kinetic energy $\Theta _c$ in Eq. (D58). Therefore, the evolution of the effective temperature $T_\mathrm{eff}(t)$ is given by the energy constraint

$$\begin{aligned} E=\frac{3}{2}M T_\mathrm{eff}+\frac{1}{2}\int \rho \Phi \, \mathrm{d}{} \mathbf{r}. \end{aligned}$$

(D63)

In the canonical ensemble, Eqs. (D61) and (D62) are valid with fixed $T_\mathrm{eff}$. One can then derive the H-theorems and the virial theorems like in the Remarks at the end of Appendices B2 and B3 (one simply has to take $P=Q=0$ in the equations of these Appendices). We refer to the series of papers initiated in [123] for a detailed study of these equations.

1.4 4. Inhomogeneous temperature

In the previous section, we have assumed for simplicity that the temperature (velocity dispersion) is uniform. In a more elaborate model (see Ref. [94]), we can take into account an inhomogeneous temperature by considering the second moment of the coarse-grained Vlasov equation and closing the hierarchy of hydrodynamic equations with the LTE approximation (D53) with $T_\mathrm{eff}(t)$ replaced by $T(\mathbf{r},t)$. This yields (see Refs. [94, 153] for details and generalizations)

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{u})=0, \end{aligned}$$

(D65)

$$\begin{aligned}&\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\nabla (\rho T)-\nabla \Phi -\xi \mathbf{u}, \end{aligned}$$

(D66)

$$\begin{aligned}&\frac{3}{2}\left( \frac{\partial T}{\partial t}+\mathbf{u}\cdot \nabla T\right) +T\nabla \cdot \mathbf{u}=-3\xi (T-T_\mathrm{eff}(t)),\nonumber \\ \end{aligned}$$

(D67)

where $T_\mathrm{eff}(t)=1/\beta (t)$ evolves according to Eq. (D49) so as to conserve the total energy

$$\begin{aligned} E=\int \rho \frac{\mathbf{u}^2}{2}\, d\mathbf{r}+\frac{3}{2}\int \rho T\, \mathrm{d}{} \mathbf{r}+W. \end{aligned}$$

(D68)

The pressure is $P=\rho T$ and the thermal energy is $\Theta _\mathrm{th}=\int f \frac{w^2}{2}\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}=\frac{3}{2}\int \rho T\, \mathrm{d}{} \mathbf{r}$. These equations conserve the mass and the energy and satisfy an H-theorem for the Boltzmann-like entropy

$$\begin{aligned} S=-\int \rho (\ln \rho -1)\, \mathrm{d}{} \mathbf{r}+\frac{3}{2}\int \rho \ln T\, \mathrm{d}{} \mathbf{r}. \end{aligned}$$

(D69)

As a result, they relax towards the equilibrium Boltzmann-like distribution with a uniform temperature $T=T_\mathrm{eff}$. The equation for the entropy density $\sigma =-\ln (\rho T^{-3/2})$ is

$$\begin{aligned} \frac{\partial \sigma }{\partial t}+\mathbf{u}\cdot \nabla \sigma =3\xi \frac{T_\mathrm{eff}(t)-T}{T}. \end{aligned}$$

(D70)

In the absence of dissipation ($\xi =0$) we recover the usual Euler equations [154] which conserve the entropy. In that case, $\rho T^{-3/2}$ is constant along the fluid (Lagrangian) trajectories, corresponding to the adiabatic law $P\rho ^{-5/3}=\mathrm{cst}$. The damped virial theorem is given by (see, e.g., Appendix G of [93])

$$\begin{aligned} \frac{1}{2}\ddot{I}+\frac{1}{2}\xi \dot{I}=2\Theta _c+3\int \rho T\, \mathrm{d}{} \mathbf{r}+W. \end{aligned}$$

(D71)

Using the energy conservation equation (D68), it can be rewritten as

$$\begin{aligned} \frac{1}{2}\ddot{I}+\frac{1}{2}\xi \dot{I}=2E-W. \end{aligned}$$

(D72)

In the canonical ensemble, $T_\mathrm{eff}$ is constant and the hydrodynamic equations satisfy an H-theorem for the free energy $F=E-T_\mathrm{eff}S$.

Appendix E: The classical limit $\hbar \rightarrow 0$

In this Appendix, we discuss how the quantum equations studied in the present paper pass to the limit $\hbar \rightarrow 0$.⁵⁴ Our discussion is essentially heuristic. A mathematically rigorous treatment of the classical limit $\hbar \rightarrow 0$ is difficult but would certainly be very valuable.

1.1 1. Classical versus quantum descriptions

In the collisionless regime, classical self-gravitating systems are described by the Vlasov–Poisson equations (C1) and (C2). The Vlasov equation is equivalent to the infinite hierarchy of Jeans equations (see Appendix C2). This is true for all times and for arbitrary initial conditions. Let us now assume that we start from a single-speed initial condition. Then, as long as the DF remains single-speed, the DF is given by Eq. (C13), the pressure tensor vanishes ($P_{ij}=0$), and the Jeans equations reduce to the pressureless Euler equations (C14) and (C15). This is true until shell crossing, after which the pressureless Euler equations develop singularities and the DF becomes multi-streamed. Before shell crossing the Vlasov equation is equivalent to the pressureless Euler equations but after shell crossing the pressureless Euler equations are not valid anymore. In that case $P_{ij}\ne 0$. Unfortunately, it is not possible to calculate the pressure tensor exactly so the Jeans equations are not closed. One has to come back to the Vlasov–Poisson equations. Some heuristic procedures to compute $P_{ij}$ approximately to continue using a hydrodynamical approach are described in Appendix C3.

In the collisionless regime, quantum self-gravitating systems made of condensed bosons are described by the Schrödinger–Poisson equations (10) and (11). The Schrödinger equation is equivalent to the quantum Euler equations (17)–(19). These equations are also equivalent to the Wigner equation (44), whose exact solution is given by Eq. (43), and to the quantum Jeans equations (see Sect. 3.4.1). For condensed bosons, $P_{ij}=P_{ij}^Q$, where $P_{ij}^Q$ is given by Eq. (23), and the quantum Jeans equations are closed (they reduce to the quantum Euler equations (17)–(19)). This is true for all times and for arbitrary initial conditions.

1.2 2. Comparison between the Wigner equation and the Vlasov equation

When $\hbar \rightarrow 0$, the Wigner equation (44) reduces to the Vlasov equation (C1) which corresponds to $\hbar =0$ (see Appendix A). On the other hand, the Wigner equation is equivalent to the Schrödinger equation (10) and to the quantum Euler equations (17)–(19). Therefore, the solution of the Vlasov equation ($\hbar =0$) is expected to be well-approximated by the solution of the Wigner, Schrödinger and quantum Euler equations with $\hbar \rightarrow 0$. In particular, $f(\mathbf{r},\mathbf{v},t)\simeq f_W(\mathbf{r},\mathbf{v},t)$ where $f_W$ is given by Eq. (43). This equivalence is valid for all times. Therefore, for what concerns the Wigner equation, the limit $\hbar \rightarrow 0$ is equivalent to $\hbar =0$.

1.3 3. Comparison between the quantum Euler equation and the pressureless Euler equations

When $\hbar \rightarrow 0$, the quantum Euler equations (17)–(19) seem to reduce to the pressureless Euler equations (C14) and (C15) which correspond to $P=\hbar =0$. However, this equivalence is only valid before shell crossing ($t<t_*$). After shell crossing ($t>t_*$), the quantum Euler equations are still valid while the pressureless Euler equations are not valid anymore (see Appendix C3). Therefore, for what concerns the quantum Euler equations (17)–(19), the limit $\hbar \rightarrow 0$ is different from $\hbar =0$. A small but finite value of $\hbar$ allows us to extend the solutions of the hydrodynamic equations past $t_*$ for all times.

1.4 4. Conclusion

The Vlasov equation is valid for all times. By contrast, the pressureless Euler equations are valid only for a single speed solution until the first time of crossing. Before that time, they are equivalent to the Vlasov equation but after that time they break down. The Schrödinger equation, the quantum Euler equations, and the Wigner equation are equivalent and they are valid for all times. When $\hbar \rightarrow 0$ their solutions are expected to be close to the solution of the Vlasov equation.

In conclusion, the Vlasov equation and the Schrödinger, quantum Euler, and Wigner equations with $\hbar \rightarrow 0$ are superior to the classical pressureless Euler equations. They take into account velocity dispersion whereas the classical pressureless fluid description does not. They can be used to describe multistreaming and caustics in the nonlinear regime, whereas the classical pressureless fluid equations break down in that regime. Indeed, the pressureless Euler equations develop shocks so they are not well-defined after the first shock. The velocity dispersion, or the quantum pressure, allow to regularize the dynamics and solve the problems of the classical pressureless hydrodynamic description.

Appendix F: The Vlasov–Bohm equation

Instead of using the rather complicated Wigner equation (44), one could consider the simpler Vlasov–Bohm equation

$$\begin{aligned} \frac{\partial f}{\partial t}+\mathbf{v}\cdot \frac{\partial f}{\partial \mathbf{r}}-\nabla \Phi \cdot \frac{\partial f}{\partial \mathbf{v}}-\frac{1}{m}\nabla Q\cdot \frac{\partial f}{\partial \mathbf{v}}=0. \end{aligned}$$

(F1)

This equation is similar to the classical Vlasov equation (C1) except that it includes the quantum potential Q from Eq. (21). We stress, however, that this equation is heuristic and that it is not expected to give an exact description of the dynamics. The Vlasov–Bohm–Poisson equations conserve the energy $E=(1/2)\int f v^2\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}+(1/2)\int \rho \Phi \, \mathrm{d}{} \mathbf{r}+(1/m)\int \rho Q\, \mathrm{d}{} \mathbf{r}$ and an infinite class of Casimir integrals of the form $\int h(f)\, \mathrm{d}{} \mathbf{r}\mathrm{d}{} \mathbf{v}$ where h(f) is an arbitrary function of f. Coarse-graining the Vlasov–Bohm equation like in Sect. 3.2, we obtain the fermionic Vlasov–Bohm-Kramers equation

$$\begin{aligned}&\frac{\partial \overline{f}}{\partial t}+\mathbf{v}\cdot \frac{\partial \overline{f}}{\partial \mathbf{r}}-\nabla \Phi \cdot \frac{\partial \overline{f}}{\partial \mathbf{v}}-\frac{1}{m}\nabla Q\cdot \frac{\partial \overline{f}}{\partial \mathbf{v}}\nonumber \\&\quad =\frac{\partial }{\partial \mathbf{v}}\cdot \left[D\left( \frac{\partial \overline{f}}{\partial \mathbf{v}}+\beta \overline{f}(1-\overline{f}/\eta _0)\mathbf{v}\right) \right]\end{aligned}$$

(F2)

instead of Eq. (46) [one can also obtain a fermionic Vlasov-Bohm–Landau equation instead of Eq. (49)]. This equation relaxes towards an equilibrium DF of the form

$$\begin{aligned} \overline{f}_\mathrm{LB}(\mathbf{r},\mathbf{v})=\frac{\eta _0}{1+\mathrm{e}^{\beta (v^2/2+\Phi (\mathbf{r})+Q/m)-\alpha }}, \end{aligned}$$

(F3)

which can be viewed as a Lynden-Bell DF incorporating the quantum potential. Since Q depends on the density itself this equation is a complicated integral equation. It implies the condition of quantum hydrostatic equilibrium from Eq. (103) (see the Remark below).

We can then take the hydrodynamic moments of the coarse-grained Vlasov–Bohm–Kramers equation (F2) as in Sect. 3.4. Instead of Eq. (57), we get

$$\begin{aligned}&\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\partial _jP_{ij}-\nabla \Phi \nonumber \\&\qquad -\frac{1}{m}\nabla Q-\frac{1}{\rho } D\beta \int \overline{f}(1-\overline{f}/\eta _0)\mathbf{v}\, \mathrm{d}{} \mathbf{v}, \end{aligned}$$

(F4)

in which the quantum potential appears explicity. Closing the hierarchy of quantum Jeans equations by making the LTE approximation from Eq. (64) to compute the pressure tensor, we obtain

$$\begin{aligned} \frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\nabla P_\mathrm{LB}-\frac{1}{m}\nabla Q-\nabla \Phi -\xi \mathbf{u}, \end{aligned}$$

(F5)

which coincides with Eq. (73). Therefore, at the level of the hydrodynamic equations, and within the assumptions made in our approach, the Vlasov–Bohm equation yields the same results as the Wigner equation. The reason is that our approach neglects collective effects (encapsulated in the dielectric function) where differences between the Wigner equation and the Vlasov–Bohm equation occur [127]. This approximation may be justified during the very nonlinear process of violent relaxation.

Remark

For spherically symmetric systems, the stationary solutions of the Vlasov–Bohm equation (F1) are of the form $f=f(\epsilon _Q)$ with $\epsilon _Q=v^2/2+\Phi +Q/m$. We can easily show that this relation implies the condition of quantum hydrostatic equilibrium from Eq. (103). Indeed, defining the density and the pressure by $\rho =\int f\, \mathrm{d}{} \mathbf{v}$ and $P=\frac{1}{3}\int f v^2\, \mathrm{d}{} \mathbf{v}$, we get

$$\begin{aligned} \nabla P= & {} \frac{1}{3}\int v^2\frac{\partial f}{\partial \mathbf{r}}\, d\mathbf{v}\nonumber \\= & {} \frac{1}{3}\left( \nabla \Phi +\frac{1}{m}\nabla Q\right) \int v^2 f'(\epsilon _Q)\, \mathrm{d}{} \mathbf{v}\nonumber \\= & {} \frac{1}{3}\left( \nabla \Phi +\frac{1}{m}\nabla Q\right) \int \frac{\partial f}{\partial \mathbf{v}}\cdot \mathbf{v}\, \mathrm{d}{} \mathbf{v}\nonumber \\= & {} -\left( \nabla \Phi +\frac{1}{m}\nabla Q\right) \int f\, \mathrm{d}{} \mathbf{v} \end{aligned}$$

(F6)

yielding

$$\begin{aligned} \nabla P+\rho \nabla \Phi +\frac{\rho }{m}\nabla Q=\mathbf{0}. \end{aligned}$$

(F7)

The effective collision term on the right hand side of Eq. (F2) selects the form of the equilibrium DF $f=f(\epsilon _Q)$ among all the possible stationary solutions of the Vlasov–Bohm equation. In the present case, the fermionic Kramers operator selects the Lynden-Bell DF from Eq. (F3).

Appendix G: Multistate systems

1.1 1. Hartree equations

We consider a system of N quantum particles in gravitational interaction. These particles may be fermions or bosons. Fundamentally, they are described by an N-body wave function $\Psi (\mathbf{r}_1,...,\mathbf{r}_N,t)$ satisfying the exact N-body Schrödinger equation [155]

$$\begin{aligned} i\hbar \frac{\partial \Psi }{\partial t}=-\frac{\hbar ^2}{2m}\Delta \Psi -\sum _{\alpha <\beta }\frac{Gm^2}{|\mathbf{r}_\alpha -\mathbf{r}_\beta |}\Psi . \end{aligned}$$

(G1)

When N is large, this equation is completely untractable. The aim of statistical mechanics and kinetic theory is to obtain simpler models described by one-body equations. Following Hartree [156, 157] we shall make a mean field approximation and ignore correlations among the particles. We thus assume that the N-body wave function $\Psi (\mathbf{r}_1,...,\mathbf{r}_N,t)$ can be written as a product of N one-body wave functions $\psi _{\alpha }(\mathbf{r},t)$ so that

$$\begin{aligned} \Psi (\mathbf{r}_1,...,\mathbf{r}_N,t)=\prod _{\alpha =1}^{N} \psi _\alpha (\mathbf{r}_\alpha ,t). \end{aligned}$$

(G2)

The mean field approximation is known to become exact for systems with long-range interactions (like self-gravitating systems) when $N\rightarrow +\infty$ in a proper thermodynamic limit [131]. In that case, the evolution of the quantum system is described by N coupled mean field Schrödinger equations of the form

$$\begin{aligned} i\hbar \frac{\partial \psi _{\alpha }}{\partial t}= & {} -\frac{\hbar ^2}{2m}\Delta \psi _{\alpha }+m\Phi \psi _{\alpha }, \end{aligned}$$

(G3)

$$\begin{aligned} \Delta \Phi= & {} 4\pi G\sum _{\alpha =1}^N p_{\alpha } |\psi _{\alpha }|^2, \end{aligned}$$

(G4)

where $p_{\alpha }$ is the occupation probability of state $\alpha$ such that the normalization condition $\sum _{\alpha }p_{\alpha }=1$ is fulfilled. We shall assume that $p_{\alpha }$ is constant or that it varies slowly with $\mathbf{r}$ and t. The total density is $\rho =\sum _{\alpha } p_{\alpha } |\psi _{\alpha }|^2$.

In the mean field approximation, each particle evolves in a self-consistent gravitational potential $\Phi$ that is produced by the particles themselves through the Poisson equation (G4). Equations (G3) and (G4) are called the Hartree (mean field) equations.

From the wavefunctions $\psi _{\alpha }(\mathbf{r},t)$ of the multistate system, we can define the Wigner DF by

$$\begin{aligned}&f(\mathbf{r},\mathbf{v},t)=\sum _{\alpha =1}^{N} \frac{m^3}{(2\pi \hbar )^3} p_{\alpha } \int d\mathbf{y}\, \mathrm{e}^{im\mathbf{v}\cdot \mathbf{y}/\hbar } \nonumber \\&\quad \times \psi _{\alpha }^*\left( \mathbf{r}+\frac{\mathbf{y}}{2},t\right) \psi _{\alpha }\left( \mathbf{r}-\frac{\mathbf{y}}{2},t\right) . \end{aligned}$$

(G5)

It satisfies the Wigner equation (44). The Wigner DF (G5) involves the density matrix $\rho (\mathbf{r},\mathbf{r}',t)=\sum _{\alpha =1}^N p_{\alpha }\psi _{\alpha }(\mathbf{r},t)\psi _{\alpha }^*(\mathbf{r},t)$ which appears in the von Neumann equation. The Hartree, Wigner and von Neumann equations are equivalent.

In the case of fermions, the wave function $\Psi$ is antisymmetric with respect to the exchange of any two variables $\mathbf{r}_\alpha$ and $\mathbf{r}_\beta$. The antisymmetry condition stems from the Pauli exclusion principle which establishes that two fermions cannot share the same position so that the probability density $|\Psi |^2$ vanishes as $\mathbf{r}_\alpha =\mathbf{r}_\beta$ (for a rigorous treatment, the spin of the fermions must be considered). A system of fermions is necessarily in a mixed quantum state to respect the Pauli exclusion principle. In the Hartree–Fock [158, 159] theory, the N-body wave function is expressed as

$$\begin{aligned} \Psi (\mathbf{r}_1,...,\mathbf{r}_N,t)=\frac{1}{\sqrt{N!}}\, \mathrm{det}[\psi _{\alpha }(\mathbf{r}_{\beta },t)]. \end{aligned}$$

(G6)

In other words, the wave function for many fermions is taken as a Slater determinant which is zero for two particles at the same position. This expression guarantees that the system obeys the Pauli exclusion principle. This leads to the Hartree–Fock equations corresponding to the Hartree Eqs. (G3) and (G4) with an additional exchange energy term introduced by Fock [158]. This exchange energy term can be simplified by using the Slater [160] approximation (see also Dirac [161] and Kohn–Sham [162]) which amounts to introducing a term $C_S \rho ^{1/3}\psi _{\alpha }$ in the right hand side of Eq. (G3), where $C_S$ is a positive constant in the gravitational case (it is negative in the electrostatic case). The Hartree–Fock equations neglect correlations. More general equations taking into account correlations (assuming that the fermions interact by microscopic forces in addition to the gravitational force) can be obtained through the density functional theory [155] initiated by Kohn and Sham [162].

A gas of bosons at $T=0$ forms a BEC in which all the bosons are in the same quantum state described by a single wave function $\psi (\mathbf{r},t)$. In the mean field approximation, the N-body wave function is expressed as

$$\begin{aligned} \Psi (\mathbf{r}_1,...,\mathbf{r}_N,t)=\prod _{\alpha =1}^{N} \psi (\mathbf{r}_\alpha ,t). \end{aligned}$$

(G7)

In that case, the evolution of the self-gravitating BEC is described by the Schrödinger–Poisson equations (10) and (11). This amounts to taking $\alpha =1$ and $p_{\alpha }=1$ in Eqs. (G3) and (G4). A system of bosons at $T=0$ is in a pure quantum state.

Remark

Actually, even in the case of BECs, we can have a quantum superposition of modes. Indeed, the wave function $\psi$ can be decomposed into Eigenfunctions of the Schrödinger–Poisson equations involving the fundamental state ($n=0$) and the excited states ($n>0$) as in Eq. (41). As a result, the multistate equations (G3) and (G4) are also relevant for bosons with this interpretation (in that case, $p_\alpha$ is the probability of mode $\alpha$ and N is the number of modes). The resulting equations can be called the Hartree (mean field) equations for bosons. In the case of fermions, the Hartree equations (G3) and (G4) are basically valid with $p_{\alpha }=1/N$ if $\alpha =1,...,N$ and $p_{\alpha }=0$ otherwise. However, as explained above, they can also be viewed as an expansion over the modes $\alpha$ of the system where $p_{\alpha }$ represents the probability of mode $\alpha$ and N represents the number of modes.⁵⁵ Below, we generalize the hydrodynamic representation of the SP equations to the case of a multistate system and show how a pressure term arises in the quantum Euler equations. This pressure term is similar to the one obtained from the approach of Sect. 3.

1.2 2. Madelung transformation

We can use the Madelung transformation to rewrite the Hartree–Fock equations (G3) and (G4) under the form of hydrodynamic equations for each state $\alpha$ (we shall use the Slater approximation $C_S \rho ^{1/3}\psi _{\alpha }$ to evaluate the exchange term for fermions). Let us write the wave function as

$$\begin{aligned} \psi _{\alpha }(\mathbf{r},t)=\sqrt{{\rho _{\alpha }(\mathbf{r},t)}} \mathrm{e}^{iS_{\alpha }(\mathbf{r},t)/\hbar }, \end{aligned}$$

(G8)

where $\rho _{\alpha }(\mathbf{r},t)$ is the density and $S_{\alpha }(\mathbf{r},t)$ is the action given by

$$\begin{aligned} \rho _{\alpha }=|\psi _{\alpha }|^2\quad \mathrm{and}\quad S_{\alpha }=-i\frac{\hbar }{2}\ln \left( \frac{\psi _{\alpha }}{\psi _{\alpha }^*}\right) . \end{aligned}$$

(G9)

Following Madelung, we introduce the velocity field

$$\begin{aligned} \mathbf{u}_{\alpha }=\frac{\nabla S_{\alpha }}{m}. \end{aligned}$$

(G10)

Substituting Eq. (G8) into Eqs. (G3) and (G4) and separating the real and the imaginary parts, we find that the Hartree–Fock equations are equivalent to hydrodynamic equations of the form

$$\begin{aligned}&\frac{\partial \rho _{\alpha }}{\partial t}+\nabla \cdot (\rho _{\alpha } \mathbf{u}_{\alpha })=0, \end{aligned}$$

(G11)

$$\begin{aligned}&\frac{\partial S_{\alpha }}{\partial t}+\frac{1}{2m}(\nabla S_{\alpha })^2+m\Phi +C_S\rho ^{1/3}+Q_{\alpha }=0, \end{aligned}$$

(G12)

$$\begin{aligned}&\frac{\partial \mathbf{u}_{\alpha }}{\partial t}+(\mathbf{u}_{\alpha }\cdot \nabla )\mathbf{u}_{\alpha }=-\frac{1}{m}\nabla Q_{\alpha }-\nabla \Phi -\frac{1}{\rho }\nabla P_\mathrm{Slater},\nonumber \\ \end{aligned}$$

(G13)

$$\begin{aligned}&\Delta \Phi =4\pi G\sum _{\alpha =1}^N p_{\alpha }\rho _{\alpha }, \end{aligned}$$

(G14)

where

$$\begin{aligned} Q_{\alpha }=-\frac{\hbar ^2}{2m}\frac{\Delta \sqrt{\rho _{\alpha }}}{\sqrt{\rho _{ \alpha }}}=-\frac{\hbar ^2}{4m}\left[\frac{\Delta \rho _{\alpha }}{\rho _{\alpha }}-\frac{1}{2}\frac{(\nabla \rho _{\alpha } )^2}{ \rho _{\alpha }^2} \right]\end{aligned}$$

(G15)

is the quantum potential and

$$\begin{aligned} P_\mathrm{Slater}=\frac{C_S}{4m}\rho ^{4/3} \end{aligned}$$

(G16)

is the Slater pressure. It corresponds to a polytrope of index $n=3$ with a polytropic constant $K_S=C_S/4m$. The Slater pressure is positive ($P_\mathrm{Slater}>0$) meaning that the exchange interaction is effectively repulsive in the gravitational case (it is attractive in the electrostatic case). Using the continuity equation (G11), we obtain the identity

$$\begin{aligned} \rho _{\alpha }\left[\frac{\partial \mathbf{u}_{\alpha }}{\partial t}+(\mathbf{u}_{\alpha }\cdot \nabla )\mathbf{u}_{\alpha }\right]=\frac{\partial }{\partial t}(\rho _{\alpha } \mathbf{u}_{\alpha })+\nabla (\rho _{\alpha } \mathbf{u}_{\alpha }\otimes \mathbf{u}_{\alpha }). \end{aligned}$$

(G17)

The quantum Euler equation (G13) can then be rewritten as

$$\begin{aligned}&\frac{\partial }{\partial t}(\rho _{\alpha } \mathbf{u}_{\alpha })+\nabla (\rho _{\alpha } \mathbf{u}_{\alpha }\otimes \mathbf{u}_{\alpha }) =-\frac{\rho _{\alpha }}{\rho }\nabla P_\mathrm{Slater}\nonumber \\&\quad -\rho _{\alpha }\nabla \Phi -\frac{\rho _{\alpha }}{m}\nabla Q_{\alpha }. \end{aligned}$$

(G18)

The foregoing equations are valid for each state $\alpha$. We now introduce the total density

$$\begin{aligned} \rho =\sum _{\alpha =1}^N p_{\alpha }\rho _{\alpha } \end{aligned}$$

(G19)

and the total velocity

$$\begin{aligned} \mathbf{u}=\frac{1}{\rho }\sum _{\alpha =1}^N p_{\alpha }\rho _{\alpha }{} \mathbf{u}_{\alpha }. \end{aligned}$$

(G20)

From Eqs. (G11), (G19) and (G20), we obtain the continuity equation

$$\begin{aligned} \frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{u})=0. \end{aligned}$$

(G21)

From Eqs. (G18), (G19) and (G20), we get

$$\begin{aligned}&\frac{\partial }{\partial t}(\rho \mathbf{u})+\sum _{\alpha =1}^N p_{\alpha }\nabla \left[\rho _{\alpha } (\mathbf{u}+\mathbf{w}_{\alpha })\otimes (\mathbf{u}+\mathbf{w}_{\alpha })\right]=\nonumber \\&\quad -\nabla P_\mathrm{Slater}-\rho \nabla \Phi -\sum _{\alpha =1}^N p_{\alpha }\frac{\rho _{\alpha }}{m}\nabla Q_{\alpha }, \end{aligned}$$

(G22)

where $\mathbf{w}_{\alpha }=\mathbf{u}_{\alpha }-\mathbf{u}$. Expanding the advection term in Eq. (G22) and using the identity

$$\begin{aligned} \sum _{\alpha =1}^Np_{\alpha }\rho _{\alpha }\mathbf{w}_{\alpha }=\sum _{\alpha =1}^N p_{\alpha }\rho _{\alpha }(\mathbf{u}_{\alpha }-\mathbf{u})=\rho \mathbf{u}-\rho \mathbf{u}=\mathbf{0}, \end{aligned}$$

(G23)

we are left with

$$\begin{aligned}&\frac{\partial }{\partial t}(\rho \mathbf{u})+\nabla (\rho \mathbf{u}\otimes \mathbf{u})=-\sum _{\alpha =1}^N p_{\alpha }\nabla (\rho _{\alpha }{} \mathbf{w}_{\alpha }\otimes \mathbf{w}_{\alpha })\nonumber \\&\quad -\nabla P_\mathrm{Slater}-\rho \nabla \Phi -\sum _{\alpha =1}^N p_{\alpha }\frac{\rho _{\alpha }}{m}\nabla Q_{\alpha }. \end{aligned}$$

(G24)

Using the continuity equation (G21), we obtain the identity

$$\begin{aligned} \rho \left[\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}\right]=\frac{\partial }{\partial t}(\rho \mathbf{u})+\nabla (\rho \mathbf{u}\otimes \mathbf{u}). \end{aligned}$$

(G25)

The Euler equation (G24) can then be rewritten as

$$\begin{aligned}&\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\partial _j P_{ij}-\frac{1}{\rho }\nabla P_\mathrm{Slater}\nonumber \\&\quad -\frac{1}{\rho m}\sum _{\alpha =1}^N p_{\alpha }\rho _{\alpha } \nabla Q_{\alpha }-\nabla \Phi , \end{aligned}$$

(G26)

where

$$\begin{aligned} P_{ij}=\sum _{\alpha =1}^N p_{\alpha }\rho _{\alpha } \mathbf{w}_{\alpha }\otimes \mathbf{w}_{\alpha } \end{aligned}$$

(G27)

is the pressure tensor arising from the difference between the multistate velocities $\mathbf{u}_{\alpha }$ and the total velocity $\mathbf{u}$. It can also be written as

$$\begin{aligned} P_{ij}=\sum _{\alpha =1}^N p_{\alpha }\rho _{\alpha } \mathbf{u}_{\alpha }\otimes \mathbf{u}_{\alpha }-\rho \mathbf{u}\otimes \mathbf{u}. \end{aligned}$$

(G28)

If we make the approximation

$$\begin{aligned} -\frac{1}{\rho m}\sum _{\alpha =1}^N p_{\alpha }\rho _{\alpha } \nabla Q_{\alpha }\simeq -\frac{1}{m}\nabla Q, \end{aligned}$$

(G29)

where Q is the quantum potential defined by Eq. (21), we obtain the hydrodynamic equations

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{u})=0, \end{aligned}$$

(G30)

$$\begin{aligned}&\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\partial _j P_{ij}-\frac{1}{\rho }\nabla P_\mathrm{Slater}-\frac{1}{m}\nabla Q-\nabla \Phi ,\nonumber \\ \end{aligned}$$

(G31)

$$\begin{aligned}&\Delta \Phi =4\pi G\rho . \end{aligned}$$

(G32)

These equations are not closed since the pressure tensor $P_{ij}$ depends on the multistate variables. We note that the pressure tensor defined by Eq. (G27) is similar to the pressure tensor defined by Eq. (60) if we identify $p_{\alpha }\rho _{\alpha }$ with the coarse-grained DF $\overline{f}(\mathbf{r},\mathbf{v})$. We shall consider successively the case of bosons and fermions.

1.3 3. Bosons

In the case of bosons at $T=0$ forming a BEC, we just have one pure state ($\alpha =p_\alpha =1$), and we trivially find that the pressure tensor defined by Eq. (G27) vanishes

$$\begin{aligned} P_{ij}=0. \end{aligned}$$

(G33)

We also have $P_\mathrm{Slater}=0$ in that case ($C_S=0$). As a result, the hydrodynamic equations (G30)–(G32) reduce to Eqs. (17)–(20) as it should. However, if we write $\psi$ as a superposition of modes [see Eq. (41)], and use Eqs. (G3) and (G4) with this interpretation (see the Remark at the end of Appendix G1), we can close Eqs. (G30)–(G32) by using the Lynden-Bell pressure from Eq. (65). This leads to Eqs. (72)–(74) and, finally, to Eq. (100). Note, however, that the friction term, which represents a form of nonlinear Landau damping, does not explicitly appear in the present formalism.

1.4 4. Fermions

In the case of fermions, the pressure tensor $P_{ij}$ takes into account the Pauli exclusion principle but, without further assumption, it cannot be explicitly evaluated. Now, if we view $\sum _\alpha$ as a sum over the modes with $p_\alpha$ being the probability of mode $\alpha$ (see the Remark at the end of Appendix G1), we can close Eqs. (G30)–(G32) by using the Lynden-Bell pressure from Eq. (65) which, in the case of fermions, coincides with the Fermi–Dirac pressure. This leads to the hydrodynamic equations

$$\begin{aligned} \frac{\partial \rho }{\partial t}&+\nabla \cdot (\rho \mathbf{u})=0, \end{aligned}$$

(G34)

$$\begin{aligned} \frac{\partial \mathbf{u}}{\partial t}&+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\nabla P_\mathrm{LB/FD}\nonumber \\&-\frac{1}{\rho }\nabla P_\mathrm{Slater}-\frac{1}{m}\nabla Q-\nabla \Phi , \end{aligned}$$

(G35)

$$\begin{aligned}&\Delta \Phi =4\pi G\rho , \end{aligned}$$

(G36)

and, finally, to Eq. (158) with the same comment as above concerning the absence of friction term. We note that the Lynden-Bell (or Fermi–Dirac) pressure has a statistical origin (it depends on the specification of $p_\alpha$) while the Slater pressure has a purely quantum nature.