Velocity-dependent forces and non-hydrodynamic initial conditions in quantum and classical fluids

Abstract

We consider a fermionic fluid in a non-equilibrium steady state where the fluctuation–dissipation theorem is not valid and fields conjugate to the hydrodynamic variables are explicitly required to determine response functions. We identify velocity-dependent forces in the kinetic equation that are equivalent to such fields. They lead to driving terms in the hydrodynamic equations and to corrections to the hydrodynamic initial conditions.

Appendix 1: Derivation of Eq. (15)

Here, we show how to derive Eq. (15) from Eq. (14). We define a scalar product in the space of modes

$$\begin{aligned} \langle A_{\varvec{p}}\vert B_{\varvec{p}}\rangle = (1/V)\sum _{\varvec{p}} w({\varvec{p}}) A_{\varvec{p}} B_{\varvec{p}}, \end{aligned}$$

(16)

with $w({\varvec{p}})$ as given after Eq. (6b) as the weight function. Let ${{\mathcal {L}}}_0$ be the hydrodynamic subspace spanned by the hydrodynamic modes $a_{\rho }$, $a_{\perp }$, and $a_s$, and in addition the longitudinal component of the velocity and the second transverse component, and define a projector ${{\mathcal{P}}}$ that projects onto ${{\mathcal{L}}}_0$

$$\begin{aligned} {{\mathcal {P}}} = \sum _{\alpha } \frac{\vert a_{\alpha }({\varvec{p}})\rangle \langle a_{\alpha }({\varvec{p}})\vert }{\langle a_{\alpha }({\varvec{p}})\vert a_{\alpha }({\varvec{p}})\rangle }. \end{aligned}$$

(17)

The projector onto the complementary space is ${{\mathcal {P}}}_{\perp } = \mathbb{1} - {{\mathcal {P}}}$, with $\mathbb{1}$ the unit operator.

Now, consider Eq. (14), let ${\varvec{k}} = (k,0,0)$, and impose the initial conditions in the form of an initial shear velocity

$$\begin{aligned} u_{\perp }^{(0)}({\varvec{k}}) = \frac{1}{V} \sum _{\varvec{p}} v_{\varvec{p}}^y \delta f_{\varvec{p}}({\varvec{k}},t=0), \end{aligned}$$

(18a)

and an initial shear stress

$$\begin{aligned} \sigma _{\perp }^{(0)}({\varvec{k}}) = \frac{1}{V} \sum _{\varvec{p}} p_y v_{\varvec{p}}^x\, \delta f_{\varvec{p}}({\varvec{k}},t=0). \end{aligned}$$

(18b)

Operating on Eq. (14) from the left, we obtain

$$\begin{aligned} {{\mathcal {P}}} \delta f_{\varvec{p}}({\varvec{k}},t=0)= & {} \left( -iz + {{\mathcal {P}}} i{\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}}\right) {{\mathcal {P}}} \delta f_{\varvec{p}}({\varvec{k}},z)\nonumber \\{} & {} + {{\mathcal {P}}} i{\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}} {{\mathcal {P}}}_{\perp } \delta f_{\varvec{p}}({\varvec{k}},z). \end{aligned}$$

(19)

Here, we have used the projector properties $\mathbb {1} = {{\mathcal {P}}} + {{\mathcal {P}}}_{\perp }$ and ${{\mathcal {P}}}\Lambda ({\varvec{p}}) = 0$. Performing the same operation with ${{\mathcal {P}}}_{\perp }$ instead of ${{\mathcal {P}}}$, we find

$$\begin{aligned} {{\mathcal {P}}}_{\perp } \delta f_{\varvec{p}}({\varvec{k}},t=0)= & {} -\left[ \Lambda _{\perp }({\varvec{p}}) + O(z,k)\right] {{\mathcal {P}}}_{\perp } \delta f_{\varvec{p}}({\varvec{k}},z)\nonumber \\{} & {} + {{\mathcal {P}}}_{\perp } i{\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}} {{\mathcal {P}}} \delta f_{\varvec{p}}({\varvec{k}},z), \end{aligned}$$

(20)

where $\Lambda _{\perp }({\varvec{p}}) = {{\mathcal {P}}}_{\perp } \Lambda ({\varvec{p}}) {{\mathcal {P}}}_{\perp }$. Note that $\Lambda _{\perp }({\varvec{p}})$ has an inverse, since the zero eigenvalues of $\Lambda ({\varvec{p}})$ have been projected out. Solving Eq. (20) for ${{\mathcal {P}}}_{\perp } \delta f_{\varvec{p}}({\varvec{k}},z)$, and inserting the result in Eq. (19), we find

$$\begin{aligned} & \left( -iz + {{\mathcal {P}}} i{\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}} + {{\mathcal {P}}} i{\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}} \Lambda _{\perp }^{-1}({\varvec{p}}) i{\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}}\right) {{\mathcal {P}}} \delta f_{\varvec{p}}({\varvec{k}},z)\nonumber \\ & \quad = {{\mathcal {P}}} \delta f_{\varvec{p}}({\varvec{k}},t=0) \nonumber \\ & \qquad + {{\mathcal {P}}} i{\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}} \Lambda _{\perp }^{-1}({\varvec{p}}) \delta f_{\varvec{p}}({\varvec{k}},t=0). \end{aligned}$$

(21)

Of the two initial-condition terms on the right-hand side of Eq. (21), the first one is in the hydrodynamic subspace, but the second one is not. Furthermore, the non-hydrodynamic initial condition involves a collision operator and hence is dissipative in nature, and Eq. (21) is a closed equation for ${{\mathcal {P}}} \delta f_{\varvec{p}}({\varvec{k}},z)$ only if one neglects the non-hydrodynamic initial condition.

By operating on Eq. (21) from the left with the hydrodynamic modes $\langle a_{\alpha }({\varvec{p}})\vert$, we obtain the hydrodynamic equations as an initial-value problem. The only unusual aspect is the non-hydrodynamic initial condition. The latter does not contribute to the density equation, obtained by multiplying from the left with $\langle 1\vert$, on account of the identity

$$\begin{aligned} \langle i{\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}} \vert \Lambda _{\perp }^{-1}({\varvec{p}}) = 0, \end{aligned}$$

(22)

which holds, since ${\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}} \in {{\mathcal {L}}}_0$. If the initial shear stress, Eq. (18b), is the only non-hydrodynamic initial condition, it does not contribute to the heat equation either, since the relevant matrix element vanishes by symmetry. It does, however, contribute to the equation for the shear velocity. The relevant matrix element is

$$\begin{aligned} M = \left\langle p_y\vert {{\mathcal {P}}} i{\varvec{k}}\cdot {\varvec{v}}_{\varvec{p}}^x \vert \Lambda _{\perp }^{-1}({\varvec{p}})\vert \delta f_{\varvec{p}}({\varvec{k}},t=0)\right\rangle . \end{aligned}$$

(23)

If we again take the initial shear stress to be the only non-hydrodynamic initial condition, we can project onto $\sigma _{\perp }$ and have

$$\begin{aligned} M &= \left\langle p_y v_{\varvec{p}}^x \vert \Lambda _{\perp }^{-1}({\varvec{p}}) \vert p_y v_{\varvec{p}}^x \right\rangle \,\frac{ik}{\left\langle p_y v_{\varvec{p}}^x \vert p_y v_{\varvec{p}}^x \right\rangle }\,\\ &\quad\quad\left\langle p_y v_{\varvec{p}}^x \vert \delta f_{\varvec{p}}({\varvec{k}},t=0)\right\rangle . \end{aligned}$$

(24)

The first factor on the right-hand side of Eq. (24) we recognize as minus the shear viscosity $\eta$, and if we remember that the mode $a_{\perp }({\varvec{p}}) = p_y$ yields $\rho$ times the transverse velocity $u_{\perp }$, see Eq. (3), we obtain the transverse velocity equation in the form

$$\begin{aligned} \left( -iz + \nu {\varvec{k}}^2\right) \, u_{\perp }({\varvec{k}},z) = u_{\perp }^{(0)}({\varvec{k}}) - \frac{\nu i k}{\langle p_y v_{\varvec{p}}^x \vert p_y v_{\varvec{p}}^x \rangle }\,\sigma _{\perp }^{(0)}({\varvec{k}}), \end{aligned}$$

(25)

which is Eq. (15). Finally, the matrix element in the denominator is $\langle p_y v_{\varvec{p}}^x \vert p_y v_{\varvec{p}}^x \rangle = 8(n\mu + Ts)/3$ by Eq. (A.23) in Ref. [14]. Ignoring the factor of 8/3 and the Ts correction to $n\mu$, and using Eq. (12), we obtain the transverse velocity equation in the form of Eq. (15').