Entropy Minimization for Many-Body Quantum Systems

Abstract

The problem considered here is motivated by a work by Nachtergaele and Yau where the Euler equations of fluid dynamics are derived from many-body quantum mechanics, see (Commun Math Phys 243(3):485–540, 2003). A crucial concept in their work is that of local quantum Gibbs states, which are quantum statistical equilibria with prescribed particle, current, and energy densities at each point of space (here ${\mathbb {R}}^d$, $d \ge 1$). They assume that such local Gibbs states exist, and show that if the quantum system is initially in a local Gibbs state, then the system stays, in an appropriate asymptotic limit, in a Gibbs state with particle, current, and energy densities now solutions to the Euler equations. Our main contribution in this work is to prove that such local quantum Gibbs states can be constructed from prescribed densities under mild hypotheses, in both the fermionic and bosonic cases. The problem consists in minimizing the von Neumann entropy in the quantum grand canonical picture under constraints of local particle, current, and energy densities. The main mathematical difficulty is the lack of compactness of the minimizing sequences to pass to the limit in the constraints. The issue is solved by defining auxiliary constrained optimization problems, and by using some monotonicity properties of equilibrium entropies.

Appendix

1.1 Justification of Definition 2.2

The fact that $\varrho ^{(1)}$ is well-defined is established as follows. Set

$$\begin{aligned} {\mathbb {B}}= 0 \oplus \bigoplus _{n = 1}^{+\infty } {\mathbb {B}}_{(n)}:=0 \oplus \bigoplus _{n = 1}^{+\infty } n^{-1}{\mathbb {A}}_{(n)}, \end{aligned}$$

where ${\mathbb {A}}$ is the second quantization of A and ${\mathbb {A}}_{(n)}$ its component on the n-th sector. Then, we have the estimate

$$\begin{aligned} \Vert {\mathbb {B}}\Vert _{{\mathcal {L}}({\mathfrak {F}}_{b/f})} \le \Vert A\Vert _{{\mathcal {L}}({\mathfrak {h}})}. \end{aligned}$$

(32)

Indeed, for $\psi =\{\psi ^{(n)}\}_{n\in {\mathbb {N}}}\in {\mathfrak {F}}_{b/f}$ and $n \ge 1$, we have

$$\begin{aligned} {\mathbb {B}}_{(n)} \psi ^{(n)}=({\mathbb {B}}\psi )_{(n)}=\frac{1}{n} \left( \sum _{i=1}^n (A)_{i} \psi ^{(n)} \right) , \end{aligned}$$

for $(A)_{i}$ the operator A acting on the variable $x_i$. This yields directly

$$\begin{aligned} \Vert {\mathbb {B}}_{(n)} \psi ^{(n)}\Vert _{n} \le \Vert A\Vert _{{\mathcal {L}}({\mathfrak {h}})} \Vert \psi ^{(n)}\Vert _{n}. \end{aligned}$$

Since $\Vert {\mathbb {B}}\Vert _{{\mathcal {L}}({\mathfrak {F}}_{b/f})}=\sup _{n \in {\mathbb {N}}} \Vert {\mathbb {B}}_{(n)}\Vert _{n}$, this gives (32).

For $\varrho \in {\mathcal {S}}_0$, consider now the linear map

$$\begin{aligned} F: A \in {\mathcal {L}}({\mathfrak {h}}) \mapsto F(A)={{\,\mathrm{Tr}\,}}( {\mathbb {A}}\varrho ). \end{aligned}$$

It is well-defined since, using the above notation for ${\mathbb {B}}$,

$$\begin{aligned} \left| {{\,\mathrm{Tr}\,}}( {\mathbb {A}}\varrho ) \right| =\left| \mathrm {Tr} \left( {\mathbb {B}}{\mathcal {N}}^{1/2} \varrho \, {\mathcal {N}}^{1/2}\right) \right| \le C \Vert A\Vert _{{\mathcal {L}}({\mathfrak {h}})}, \end{aligned}$$

where we used the fact that

$$\begin{aligned} \mathrm {Tr} \left( {\mathcal {N}}^{1/2} \varrho \, {\mathcal {N}}^{1/2}\right) =C<\infty , \end{aligned}$$

since $\varrho \in {\mathcal {S}}_0$. When A is compact, F is therefore a linear continuous map on the space of compact operators on ${\mathfrak {h}}$. By duality, we can then conclude that there exists a unique $\varrho ^{(1)} \in {\mathcal {J}}_1({\mathfrak {h}})$ such that

$$\begin{aligned} F(A)=\mathrm {Tr}_{{\mathfrak {h}}}\left( A \varrho ^{(1)}\right) , \end{aligned}$$

for all compact operators A on ${\mathcal {L}}({\mathfrak {h}})$. The case A bounded follows finally by approximation.

The fact that $\varrho ^{(1)}$ is nonnegative is established as follows. Let $\varphi \in {\mathfrak {h}}$ with $\Vert \varphi \Vert _{{\mathfrak {h}}}=1$, and consider the rank one projector $P={|}\varphi {\rangle }{\langle }\varphi {|}$. Then

$$\begin{aligned} (\varphi ,\varrho ^{(1)} \varphi )_{\mathfrak {h}}=\mathrm {Tr}_{{\mathfrak {h}}}\left( P \varrho ^{(1)}\right) =\mathrm {Tr} \left( d\Gamma (P) \varrho \right) . \end{aligned}$$

Denoting by $\{\rho _p\}_{p \in {\mathbb {N}}}$ and $\{\psi _p\}_{p \in {\mathbb {N}}}$ the (nonnegative) eigenvalues and eigenfunctions of $\varrho \in {\mathcal {E}}_0$, the last term is equal to

$$\begin{aligned} \sum _{p \in {\mathbb {N}}} \sum _{n \in {\mathbb {N}}^*} \sum _{j=1}^n \rho _p \left( \psi _p^{(n)}, P_{(j)}\psi _p^{(n)}\right) _{n}= \sum _{p \in {\mathbb {N}}} \sum _{n \in {\mathbb {N}}^*} \sum _{j=1}^n \rho _p \left( P_{(j)} \psi _p^{(n)}, P_{(j)}\psi _p^{(n)}\right) _{n} \ge 0. \end{aligned}$$

Above $P_{(j)}$ is the operator P acting on $x_j$, and we used the fact that $P^2_{(j)}=P_{(j)}$. This yields the positivity and ends the justification of Definition 2.2.

1.2 Proof of Lemma 2.4

Denote by $\{\rho _p\}_{p \in {\mathbb {N}}}$ and $\{\psi _p\}_{p \in {\mathbb {N}}}$ the eigenvalues and eigenfunctions of $\varrho \in {\mathcal {E}}_0$. A direct calculation shows first that

$$\begin{aligned} {{\,\mathrm{Tr}\,}}\left( {\mathbb {H}}_0^{1/2} \varrho \,{\mathbb {H}}_0^{1/2} \right) =\sum _{p \in {\mathbb {N}}} \sum _{n \in {\mathbb {N}}^*} \rho _p \left\| d\Gamma (h_0)^{1/2}_{(n)} \psi _p^{(n)}\right\| ^2_{n}, \end{aligned}$$

where $\psi _p^{(n)}$ is the component of $\psi _p$ on the n-th sector and

$$\begin{aligned} d\Gamma (h_0)^{1/2}_{(n)} = \left( \sum _{j=1}^n -\Delta _{x_j}\right) ^{1/2}. \end{aligned}$$

Since $h_0$ is not bounded, we proceed by regularization in order to use (2) for the definition of the one-body density matrix. For $\varepsilon >0$, set then $h_\varepsilon =h_0(\mathrm {Id}_{{\mathfrak {h}}}+\varepsilon h_0)^{-1} \in {\mathcal {L}}({\mathfrak {h}})$. According to (2), we have

$$\begin{aligned} \mathrm {Tr}_{\mathfrak {h}}\left( h_\varepsilon ^{1/2} \varrho ^{(1)} h_\varepsilon ^{1/2} \right) =\mathrm {Tr}_{\mathfrak {h}}\left( h_\varepsilon \varrho ^{(1)} \right) = {{\,\mathrm{Tr}\,}}\left( d\Gamma (h_\varepsilon ) \varrho \right) . \end{aligned}$$

(33)

The last term is equal to

$$\begin{aligned} \sum _{p \in {\mathbb {N}}} \sum _{n \in {\mathbb {N}}^*} \rho _p \left\| {\mathbb {A}}^\varepsilon _{(n)} \psi _p^{(n)}\right\| ^2_{n}, \end{aligned}$$

(34)

with

$$\begin{aligned} {\mathbb {A}}^\varepsilon _{(n)}=\left( \sum _{j=1}^n -\Delta _{x_j} (\mathrm {Id}_{{\mathfrak {h}}}- \varepsilon \Delta _{x_j})^{-1}\right) ^{1/2}. \end{aligned}$$

For $\psi \in {\mathfrak {F}}_{b/f}^{(n)}$, we find by a Fourier transform

$$\begin{aligned} \Vert {\mathbb {A}}^\varepsilon _{(n)} \psi \Vert ^2_{(n)} =&(2 \pi )^{-n} \int _{({\mathbb {R}}^d)^n} \left( \sum _{j=1}^n \frac{|k_j|^2}{1+\varepsilon |k_j|^2}\right) ^{1/2} |{{\widehat{\psi }}}(k_1, \ldots ,k_d)|^2 d k_1 \ldots d k_d\nonumber \\ \le&\Vert d\Gamma (h_0)^{1/2}_{(n)} \psi \Vert ^2_{(n)} . \end{aligned}$$

(35)

Hence,

$$\begin{aligned} \mathrm {Tr}_{\mathfrak {h}}\left( h_\varepsilon ^{1/2} \varrho ^{(1)} h_\varepsilon ^{1/2} \right) \le {{\,\mathrm{Tr}\,}}\left( {\mathbb {H}}_0^{1/2} \varrho \,{\mathbb {H}}_0^{1/2} \right) , \end{aligned}$$

and there exists therefore $\alpha \in {\mathcal {J}}_1({\mathfrak {h}})$ and a subsequence such that $h_{\varepsilon _\ell }^{1/2} \varrho ^{(1)} h_{\varepsilon _\ell }^{1/2} \rightarrow \alpha$ in ${\mathcal {J}}_1({\mathfrak {h}})$ weak-$*$ as $\ell \rightarrow \infty$ with

$$\begin{aligned} \mathrm {Tr}_{\mathfrak {h}}\left( \alpha \right) \le {{\,\mathrm{Tr}\,}}\left( {\mathbb {H}}_0^{1/2} \varrho \,{\mathbb {H}}_0^{1/2} \right) . \end{aligned}$$

We now identify $\alpha$ with $h_0^{1/2} \varrho ^{(1)} h_0^{1/2}$. Let K be a compact operator on ${\mathfrak {h}}$. Then,

$$\begin{aligned} \mathrm {Tr}_{\mathfrak {h}}\left( (\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} K (\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} h_\varepsilon ^{1/2} \varrho ^{(1)} h_\varepsilon ^{1/2} \right) = \mathrm {Tr}_{\mathfrak {h}}(K_\varepsilon \varrho ^{(1)}), \end{aligned}$$

with

$$\begin{aligned} K_\varepsilon =h_\varepsilon ^{1/2} (\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} K (\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} h_\varepsilon ^{1/2}. \end{aligned}$$

As $\varepsilon \rightarrow 0$, the operator $K_\varepsilon$ converges strongly to $h_0^{1/2} (\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} K (\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} h_0^{1/2}$ in ${\mathcal {L}}({\mathfrak {h}})$. As a consequence

$$\begin{aligned} \mathrm {Tr}_{\mathfrak {h}}\big ( (\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} K&(\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} \alpha \big )\\&= \mathrm {Tr}_{\mathfrak {h}}\left( K h_0^{1/2} (\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} \varrho ^{(1)} (\mathrm {Id}_{{\mathfrak {h}}}+h_0)^{-1} h_0^{1/2} \right) , \end{aligned}$$

which allows us to identify $\alpha$ with $h_0^{1/2} \varrho ^{(1)} h_0^{1/2}$. The relation (4) is obtained by passing to the limit in (33): in the l.h.s, we use the fact that $h_0^{1/2} \varrho ^{(1)} h_0^{1/2} \in {\mathcal {J}}_1({\mathfrak {h}})$, and in the r.h.s., we use (34), (35), and monotone convergence. This ends the proof.