Nonequilibrium ensemble derivation of hydrodynamic heat transport and higher-order generalizations

Abstract

Thermal transport in classical fluids is analyzed through higher-order generalized hydrodynamics (or mesoscopic hydrothermodynamics) depending on the evolution of the energy density and its fluxes of all orders. It is derived by a kinetic theory based on the nonequilibrium statistical ensemble formalism. A general system of coupled evolution equations is derived. Maxwell times, which are of significance to determine the character of the motion, are derived. They also have an important role in the choice of the contraction of description (limitation in the number of fluxes to be retained) in the studies on hydrodynamic motions. In a description of order 1, an analysis of the technological process of thermal prototyping is presented.

Appendices

Appendix 1: NESEF kinetic equation

A kinetic equation of evolution for the single-particle distribution function $f_{1}({\mathbf{r}},{\mathbf{p}};t)$ follows from the NESEF-based kinetic theory [67, 76, 78],

$$\begin{aligned} \frac{\partial }{\partial t}f_{1}({\mathbf{r}},{\mathbf{p}};t) = {\mathrm{Tr}} \left\{ \{{\widehat{n}}_{1}({\mathbf{r}},{\mathbf{p}}),{\widehat{H}}\} \varrho _{\varepsilon }(t) \right\} , \end{aligned}$$

(57)

that is, the average over the nonequilibrium ensemble characterized by the statistical operator $\varrho _{\varepsilon }(t)$. The method was used successfully in semiconductors far from equilibrium [92,93,94,95,96].

Using the Hamiltonian of Eq. (7), with ${\widehat{H}}_{0}$ being the kinetic energy and ${\widehat{H}}^{\prime }$ containing all the interactions, that is, pair interactions between the system particles, of the system particles with those of the thermal bath, and the interactions with applied external sources. Resorting to the approximation of keeping only the contribution from the irreducible two particle collisions, lengthy but straightforward calculations lead to the kinetic equation

$$\begin{aligned}&\frac{\partial }{\partial t} f_{1}({\mathbf{r}},{\mathbf{p}};t) + \frac{{\mathbf{P}}({\mathbf{r}},{\mathbf{p}};t)}{m} \cdot \nabla f_{1}({\mathbf{r}},{\mathbf{p}};t) \\&\quad +{\mathbf{F}}({\mathbf{r}},{\mathbf{p}};t) \cdot \nabla _{{\mathbf{p}}} f_{1}({\mathbf{r}},{\mathbf{p}};t) - B({\mathbf{p}}) f_{1}({\mathbf{r}},{\mathbf{p}};t) \\&\quad -A_{2}^{[2]}({\mathbf{p}}) \odot [\nabla _{{\mathbf{p}}} \nabla ] f_{1}({\mathbf{r}},{\mathbf{p}};t) \\&\quad - B_{2}^{[2]}({\mathbf{p}}) \odot [\nabla _{{\mathbf{p}}} \nabla _{{\mathbf{p}}}] f_{1}({\mathbf{r}},{\mathbf{p}};t) = J_{S}^{(2)}({\mathbf{r}},{\mathbf{p}};t) , \end{aligned}$$

(58)

where

$$\begin{aligned} {\mathbf{P}}({\mathbf{r}},{\mathbf{p}};t) = {\mathbf{p}} - m{\mathbf{A}}_{1}({\mathbf{p}}) \end{aligned}$$

(59)

plays the role of a generalized nonlinear momentum,

$$\begin{aligned} {\mathbf{F}}({\mathbf{r}},{\mathbf{p}};t) = - \nabla U_{ext}({\mathbf{r}},t) - {\mathbf{B}}_{1}({\mathbf{p}}) - {\mathbf{F}}_{NL}({\mathbf{r}};t) - \nabla U({\mathbf{r}};t) , \end{aligned}$$

(60)

is a generalized force, in which

$$\begin{aligned} {\mathbf{F}}_{NL}({\mathbf{r}};t) = \int d^{3}r^{\prime } \int d^{3}p^{\prime } {\mathbf{G}}_{NL}({\mathbf{r}}^{\prime } - {\mathbf{r}},{\mathbf{p}}^{\prime }) f_{1}( {\mathbf{r}}^{\prime },{\mathbf{p}}^{\prime };t) , \end{aligned}$$

(61)

with full details given in Ref. [61]. The symbol $\odot$ stands for contracted product of tensors.

When Eq. (58) is compared with, say, standard Boltzmann kinetic equation, it contains several additional contributions. First, ${\mathbf{P}}({\mathbf{p}};t)$, Eq. (59), can be interpreted as a modified momentum, composed of the linear one, ${\mathbf{p}}$, plus a contribution arising out of the interaction with the thermal bath, $m {\mathbf{A}}_{1}({\mathbf{p}})$. The contribution $m {\mathbf{A}}_{1}({\mathbf{p}})$ can be considered as implying in a transfer of momentum between system and bath.

The force ${\mathbf{F}}({\mathbf{r}},{\mathbf{p}};t)$, in Eq. (60), is composed of four contributions: The first one, $-\nabla U_{ext}$, is the external applied force; next, ${\mathbf{B}}_{1}({\mathbf{p}})$, arising out of the interaction with the thermal bath, is a contribution that, together with the fourth one in Eq. (58), namely ${\mathbf{B}}({\mathbf{p}}) f_{1}$, constitutes a generalization of the so-called effective friction force. In the limit of a Brownian system ($m\gg M$), it is recovered the known expression [97] $\gamma \nabla _{{\mathbf{p}}}[({\mathbf{p}}/m) f_{1}({\mathbf{r}},{\mathbf{p}};t)]$ with the friction coefficient given by $\gamma = m/\tau$, where $\tau$ is the momentum relaxation time. In the limit of a Lorentz system, ($m\ll M$) ${\mathbf{B}}_{1}({\mathbf{p}})$ and ${\mathbf{B}}({\mathbf{p}})$ tend to zero and then this friction force is practically null.

The third contribution to the force ${\mathbf{F}}({\mathbf{r}},{\mathbf{p}};t)$, i.e., ${\mathbf{F}}_{NL}({\mathbf{r}};t)$, is an interesting one, which is an effective force between pairs of particles generated through the interaction of each of the pair with the thermal bath. A similar presence of an induced effective coupling of this type has been evidenced in the case of two Brownian particles embedded in a thermal bath [98], and also, it can be noticed the similarity with the one that leads to the formation of Cooper pairs in type-I superconductivity. Such contribution is of a nonlinear (bilinear in $f_{1}$) character and therefore eventually may lead to the emergence of complex behavior in the system, e. g., the cases of the so-called nonequilibrium Bose–Einstein-like condensation [99,100,101,102,103,104]. The last contribution, $- \nabla U$, is usually called the self-consistent or mean field force, containing the proper single-particle distribution and then providing a nonlinear contribution to the kinetic equation.

The fifth term on the left of Eq. (58) consists of a cross double differentiation, namely the rank-2 tensor $[\nabla _{{\mathbf{p}}} \nabla ] f_{1}$, which takes into account effects of anisotropy caused by nonuniformity. The sixth one consists of a double differentiation in the momentum variable, the rank-2 tensor $[\nabla _{{\mathbf{p}}} \nabla _{{\mathbf{p}}}] f_{1}$, which is related to the so-called diffusion in momentum space. Both contributions have their origins in the interaction of the particles with the thermal bath, that is, in those terms of the collision integral $J_{1}^{(2)}$ containing the potential w (see Eq. (11)).

Finally, on the right, $J_{S}^{(2)}$ is the collision integral resulting from the interaction between the particles taken in the weak coupling limit, i.e., the so-called weakly coupled gas collision integral [105].

To go beyond the weak-coupling approximation, one needs to go back to the general kinetic equation to include memory effects and higher-order collision integrals, so that to include vertex renormalization [100].

Appendix 2: The kinetic coefficients in Eq. (32)

We do have that

$$\begin{aligned} a_{\tau 0} = \frac{\mathcal {V}}{(2 \pi )^{3}} \frac{4 \pi }{3} \int dQ \, Q^{4} f_{\tau 0}(Q) , \end{aligned}$$

(62)

with

$$\begin{aligned} f_{\tau 0}(Q) = - \frac{n_{R}}{\mathcal {V}} \frac{\left( M \beta _{0} \right) ^{3/2} \pi }{\sqrt{2 \pi } m^{2}} \frac{\left| \psi (Q)\right| ^{2}}{Q} \left( \frac{m}{M} + 1 \right), \end{aligned}$$

(63)

where $\psi (Q)$ is the Fourier transform of the potential energy $w(\left| {\mathbf{r}}_{j}-{\mathbf{R}}_{\mu }\right| )$, $n_{R}$ is the density of particles in the thermal bath, $\mathcal {V}$ is the volume, and $\beta _{0}^{-1} = k_{B} T_{0}$. Moreover,

$$\begin{aligned} a_{\tau 1}= & {} - \frac{M \beta _{0}}{10} a_{\tau 0} , \end{aligned}$$

(64)

$$\begin{aligned} a_{L0}= & {} \sqrt{\frac{2}{M \beta _{0} \pi }} \frac{1}{\kappa } a_{\tau 0} , \end{aligned}$$

(65)

$$\begin{aligned} \frac{1}{\kappa }= & {} \frac{\int dQ \, Q^{2}\left| \psi (Q) \right| ^{2} }{\int dQ \, Q^{3}\left| \psi (Q) \right| ^{2} \left( 1 + \frac{m}{M} \right) } , \end{aligned}$$

(66)

$$\begin{aligned} b_{\tau 0}= & {} - \frac{2}{M \beta _{0}} a_{\tau 0} \left( 1 + \frac{m}{M} \right) ^{-1} , \end{aligned}$$

(67)

$$\begin{aligned} b_{\tau 1}= & {} \frac{a_{\tau 0}}{5} \left( 1 + \frac{m}{M} \right) ^{-1} , \end{aligned}$$

(68)

$$\begin{aligned} a_{L 1}= & {} - \frac{1}{5\kappa } \sqrt{\frac{2M \beta _{0}}{\pi }} a_{\tau 0} . \end{aligned}$$

(69)