Statistical Mesoscopic Hydro-thermodynamics: the Description of Kinetics and Hydrodynamics of Nonequilibrium Processes in Single Liquids

Abstract

Hydrodynamics, a term apparently introduced by Daniel Bernoulli (1700–1783) to comprise hydrostatic and hydraulics, has a long history with several theoretical approaches. Here, after a descriptive introduction, we present so-called mesoscopic hydro-thermodynamics, which is also referred to as higher order generalized hydrodynamics, built within the framework of a mechanical-statistical formalism. It consists of a description of the material and heat motion of fluids in terms of the corresponding densities and their associated fluxes of all orders. In this way, movements are characterized in terms of intermediate to short wavelengths and intermediate to high frequencies. The fluxes have associated Maxwell-like times, which play an important role in determining the appropriate contraction of the description (of the enormous set of fluxes of all orders) necessary to address the characterization of the motion in each experimental setup. This study is an extension of a preliminary article: Physical Review E 91, 063011 (2015).

Appendices

Appendix 1. The Nonequilibrium Statistical Operator

The construction of nonequilibrium statistical ensembles, that is, a nonequilibrium statistical ensemble formalism, NESEF for short, which essentially consists of a derivation of a nonequilibrium statistical operator (probability distribution in the classical case), has been attempted along several lines. In a brief summarized way, we describe the construction of the NESEF within a heuristic approach. First, it should be noticed that for systems away from equilibrium, several important points need to be carefully taken into account in each case under consideration:

• 1. The choice of the basic variables (a wholly different choice from the case of equilibrium when it suffices to take a set of variables that are constants of motion), which is based on an analysis of what sort of macroscopic measurements and processes are actually possible; moreover, one should focus attention not only on what can be observed but also on the character and expectation concerning the equations of evolution for these variables [12, 13, 25].

• 2. The question of irreversibility (or Eddington’s arrow of time).

• 3. Historicity needs be introduced, that is, the idea that the past dynamics of the system (or historicity effects) must be incorporated along the time interval going from an initial description of the macrostate of the sample in a given experiment, say at t₀, to the time t when a measurement is performed.

Concerning the choice of the basic variables, in contrast to the case of equilibrium, immediately after an open system of N particles, in contact with external sources and reservoirs, has been driven out of equilibrium, it becomes necessary to describe its state in terms of all the observables and, eventually, introduce direct and cross-correlation. However, as time elapses, Bogoliubov’s principle of correlation weakening allows us to introduce increasing contractions of the descriptions. Let us say that we can introduce a description based on the observables $\{\hat {P}_{j}\}$, j = 1, 2... , n, on which the nonequilibrium statistical operator depends.

On the question of irreversibility, in the absence of a proper way to introduce such an effect, one needs to resort to the interventionist’s approach, which is based on the ineluctable process of randomization leading to the asymmetric evolution of the macrostate.

The “intervention” consists of introducing into the Liouville equation of the statistical operator, of an otherwise isolated system, a particular source accounting for Krylov’s “jolting” effect [52], in the form (written for the logarithm of the statistical operator)

$$ \frac{\partial}{\partial t} \ln \mathcal{R}_{\varepsilon }(t) + \frac{1}{ i \hbar }[\ln \mathcal{R}_{\varepsilon }(t),\hat{H}] = - \varepsilon \lbrack \ln \mathcal{R}_{\varepsilon}(t) - \ln \overline{\mathcal{R}}(t,0)], $$

(68)

where ε (a kind of reciprocal of a relaxation time) is taken to go to + 0 after the calculations of average values have been performed. Such a mathematically inhomogeneous term, in the otherwise normal Liouville equation, implies a continuous tendency of relaxation of the statistical operators towards a referential distribution, $\overline {\mathcal {R}}$, which represents an instantaneous quasi-equilibrium condition.

We can see that Eq. (68) consists of a regular Liouville equation but with an infinitesimal source, which provides Bogoliubov’s symmetry breaking of time reversal and is responsible for disregarding the advanced solutions. This is described by a Poisson distribution, and the result at time t is obtained by averaging over all t^′ in the interval (t₀, t), and the solution of (68) is

$$ \mathcal{R}_{\varepsilon }(t) = \exp \left\{ - \hat{S}(t,0) + \int\limits_{t_{0}}^{t}dt^{\prime} e^{\varepsilon (t^{\prime} - t)} \frac{d}{dt^{\prime}} \hat{S}(t^{\prime},t^{\prime} - t) \right\} , $$

(69)

where

$$ \hat{S}(t,0) = - \ln \overline{\mathcal{R}}(t,0) , $$

(70)

$$\begin{array}{@{}rcl@{}} \hat{S}(t^{\prime},t^{\prime} - t) &=& \exp \left\{ -\frac{1}{i \hbar}(t^{\prime} - t) \hat{H} \right\} \hat{S}(t^{\prime},0) \\ && \times\exp \left\{ \frac{1}{i \hbar}(t^{\prime} - t) \hat{H} \right\} , \end{array} $$

(71)

and the initial-time condition at time t₀, when the application of the formalism begins, is

$$ \mathcal{R}_{\varepsilon }(t_{0}) = \overline{\mathcal{R}}(t_{0},0) . $$

(72)

In $\overline {\mathcal {R}}$ and $\hat {S}$, the first time variable in the argument refers to the evolution of the nonequilibrium thermodynamic variables, and the second time variable refers to the time evolution of the dynamical variables, both of which have an effect on the operator. The statistical operator can be written in the form

$$ \mathcal{R}_{\varepsilon }(t) = \overline{\mathcal{R}}(t,0) + \mathcal{R}_{\varepsilon}^{\prime}(t) . $$

(73)

involving the auxiliary probability distribution $\bar {\rho }(t,0)$, with $\rho _{\varepsilon }^{\prime }(t)$, which contains the historicity and irreversibility effects. Moreover, in most cases, we can consider a system that consist of the system of interest (on which we are performing an experiment) in contact with ideal reservoirs. Thus, we can write

$$ \overline{\mathcal{R}}(t,0) = \bar{\rho}(t,0) \times \rho_{R} . $$

(74)

and

$$ \mathcal{R}_{\varepsilon }(t) = \rho_{\varepsilon}(t) \times \rho_{R} , $$

(75)

where ϱ_ε(t) is the statistical operator of the nonequilibrium system, $\bar {\rho }$ is the auxiliary operator, and ϱ_R is the stationary operator of the ideal reservoirs, with ϱ_ε(t) given by

$$ \rho_{\varepsilon}(t) = \exp \left\{ - \hat{S}(t,0) + \int\limits_{- \infty}^{t} dt^{\prime} e^{\varepsilon (t^{\prime} - t)} \frac{d}{dt^{\prime}} \hat{S}(t^{\prime},t^{\prime} - t) \right\} , $$

(76)

with the initial value $\bar {\rho }(t_{0},0)$ (t₀ → −∞), and where

$$ \hat{S}(t,0) = - \ln \bar{\rho}(t,0) . $$

(77)

Finally, the auxiliary statistical operator $\bar {\rho }(t,0)$ must be provided. This operator defines an instantaneous distribution at time t, which describes a “frozen” equilibrium defining the macroscopic state of the system at the given time, and for that reason the operator is sometimes dubbed as the quasi-equilibrium statistical operator. On this basis (or, alternatively, via the variational procedure [12,13,14]) and considering the description of the nonequilibrium state of the system in terms of the basic set of dynamical variables $\hat {P}_{j}$, the reference or instantaneous quasi-equilibrium statistical operator is taken as a canonical-like operator given by

$$ \bar{\rho}(t,0) = \exp \left\{ - \phi (t) - \sum\limits_{j}^{n} F_{j}(t) \hat{P}_{j} \right\} , $$

(78)

with ϕ(t) ensuring the normalization of $\bar {\rho }$ and playing the role of a logarithm of a partition function, say, $\phi (t) = \ln \bar {Z}(t)$. Moreover, in (78), F_j are the nonequilibrium thermodynamic variables associated with each kind of basic dynamical variable $\hat {P}_{j}$. The nonequilibrium thermodynamic space of states consists of the basic variables {Q_j(t)}, which consist of averages of $\{\hat {P}_{j}\}$ over the nonequilibrium ensemble, namely,

$$ Q_{j}(t) = \text{Tr} \{\hat{P}_{j} \rho_{\varepsilon}(t)\} , $$

(79)

which are then functionals of {F_j(t)}, and the equations of state are

$$ Q_{j}(t) = - \frac{\delta \phi (t)}{\delta F_{j}(t)} = - \frac{\delta \ln \bar{Z}(t)}{\delta F_{j}(t)} , $$

(80)

where δ stands for a functional derivative [45].

Moreover,

$$ \bar{S}(t) = \text{Tr} \{ \hat{\bar{S}}(t,0) \bar{\rho}(t,0) \} = - \text{Tr} \{ \bar{\rho}(t,0) \ln \bar{\rho}(t,0)\} , $$

(81)

is the so-called informational entropy characteristic of the distribution $\bar {\rho }$, a functional of the basic variables {Q_j(t)}, and the alternative form of the equations of state is given by

$$ -\frac{\delta \bar{S}(t)}{\delta Q_{j}(t)} = F_{j}(t) . $$

(82)

In the contracted description used for the construction of HOGH of order 2, that is, in terms of the variables of Eq. (36), we have that the auxiliary statistical operator, better written in reciprocal space, is

$$\begin{array}{@{}rcl@{}} \bar{\rho}(t,0) \!&\!=&\! \exp \left\{ - \phi(t) - \! \sum\limits_{\mathbf{Q}}\left[F_{h}({\mathbf{Q}},t)\widehat{h}(\mathbf{Q}) + F_{n}({\mathbf{Q}},t)\widehat{n}({\mathbf{Q}})\right.\right.\\ &&+ \mathbf{F}_{n}(\mathbf{Q},t) \cdot \widehat{\mathbf{I}}_{n}(\mathbf{Q}){+} \mathring{F}_{n}^{[2]}(\mathbf{Q},t) \!\otimes\! \widehat{\mathring{I}}_{n}^{[2]}(\mathbf{Q}). \end{array} $$

(83)

Quantities with a triangular hat are the mechanical operators corresponding to the quantities in Eq. (3). In (83), the conjugated nonequilibrium thermodynamical variables, F’s listed in Eq. (37), are introduced.

Appendix 2. Single-Particle Kinetic Equation

The NESEF-based single-particle kinetic equation was reported in Ref. [34], which contained an interaction with a surrounding thermal bath and assumed that in a dilute solution the interparticle interaction could be treated in the weak-coupling approximation. We reproduce below this result but incorporate the collision integral associated with the two-particle interaction in full. We have

$$ \frac{\partial }{\partial t}f_{1}(\mathbf{r},\mathbf{p};t) = J_{1}^{(0)}(\mathbf{r},\mathbf{p};t) + J_{1}^{(1)}(\mathbf{r},\mathbf{p};t) + J_{1}^{(2)}(\mathbf{r},\mathbf{p};t) , $$

(84)

where the first terms on the right-hand side are

$$\begin{array}{@{}rcl@{}} J_{1}^{(0)}(\mathbf{r},\mathbf{p};t) &=& \text{Tr} \left\{ \{\widehat{n}_{1}(\mathbf{r},\mathbf{p}), \widehat{H}_{0} \} \bar{\rho}(t,0) \rho_{R} \right\} \\ &=& - \frac{\mathbf{p}}{m} \cdot \nabla f_{1}(\mathbf{r},\mathbf{p};t) , \end{array} $$

(85)

which are, as already noticed, the precession term in Moris’s terminology, and

$$\begin{array}{@{}rcl@{}} J_{1}^{(1)}(\mathbf{r},\mathbf{p};t) &=& \text{Tr} \left\{ \{\widehat{n}_{1}(\mathbf{r},\mathbf{p}), \widehat{H}^{\prime} \} \bar{\rho}(t,0) \rho_{R} \right\} \\ &=& [\nabla U_{ex}(\mathbf{r},t) + \nabla U(\mathbf{r},t)] \cdot \nabla_{\mathbf{p}} f_{1}(\mathbf{r},\mathbf{p};t) , \\ \end{array} $$

(86)

is the contribution containing the action of the external and internal forces, where

$$ U(\mathbf{r},t) = \int d^{3}r^{\prime} d^{3}p^{\prime} V(|\mathbf{r} - \mathbf{r}^{\prime}|) f_{1}(\mathbf{r}^{\prime},\mathbf{p}^{\prime};t) $$

(87)

plays the role of a mean-field potential of the interaction between particles or a Vlasov-like potential. Observe that the dot stands for the scalar product of vectors, and we write ⊙ for the full contraction of tensors.

Taking all the interactions in Eq. (26), denoted by H^′, that is, pair interactions with applied external sources, lengthy but straightforward calculations give the final expression of the collision integral $J_{1}^{(2)}$. Taking this final expression, Eqs. (85) and (86) into Eq. (84), leads to the kinetic equation:

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

(88)

where

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

(89)

plays the role of a generalized momentum,

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

(90)

is a generalized force, in which

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

(91)

and the coefficients are as follows:

$$ \mathbf{A}_{1}(\mathbf{p}) = - \frac{n_{B}}{\mathcal{V}} \frac{M \beta_{0}}{m} \sum\limits_{\mathbf{Q}}\mathbf{Q} \frac{\left\vert \psi (Q)\right\vert^{2}}{Q^{2}} \mathbf{Q} \cdot \nabla_{\mathbf{p}} F(\mathbf{p,Q}) , $$

(92)

where

$$ F(\mathbf{p,Q}) = 1 + \sum\limits_{n = 1}^{\infty}\frac{(-1)^{n}}{(2n-1)!!} \left( \frac{M \beta_{0}}{Q^{2}m^{2}} \right)^{n} \left( \mathbf{Q} \cdot \mathbf{p} \right)^{2n} , $$

(93)

$$\begin{array}{@{}rcl@{}} \mathbf{B}_{1}(\mathbf{p}) &=& \frac{n_{B}}{\mathcal{V}} \frac{\pi \sqrt{(M \beta_{0})^{3}}}{m\sqrt{2\pi}} \\ &\times& \sum\limits_{\mathbf{Q}}\mathbf{Q} \frac{\left\vert \psi (Q)\right\vert^{2}}{Q} \left( \frac{1}{M} - \frac{1}{m}\right) \left( \mathbf{Q} \cdot \mathbf{p} \right) e^{- \frac{M \beta_{0} \left( \mathbf{Q} \cdot \mathbf{p} \right)^{2}}{2Q^{2}m^{2}}} , \end{array} $$

(94)

$$\begin{array}{@{}rcl@{}} \mathbf{G}_{NL}(\mathbf{r}^{\prime} - \mathbf{r},\mathbf{p}^{\prime}) &=& \frac{n_{R} \beta_{0}}{\mathcal{V}} \sum\limits_{\mathbf{Q}}\mathbf{Q} \left\vert \psi (Q) \right\vert^{2}\left\{\vphantom{\frac{M\beta_{0}}{2\pi}}i F(\mathbf{Q},\mathbf{p}^{\prime})\right.\\ &&\left.+\!\left( \frac{M\beta_{0}}{2\pi} \right)^{1/2} \frac{\pi}{m}\frac{\mathbf{Q}\cdot\mathbf{p}^{\prime}}{Q}e^{- \alpha \left( \frac{\mathbf{Q}}{Q} \cdot \mathbf{p}^{\prime }\right)^{2}}\right\} e^{i \mathbf{Q} \cdot (\mathbf{r}^{\prime} - \mathbf{r})},\\ \end{array} $$

(95)

$$ A_{2}^{[2]}(\mathbf{p}) = \frac{n_{B}}{\mathcal{V}} \frac{M \beta_{0}}{m} \sum\limits_{\mathbf{Q}} \frac{\left\vert \psi (Q) \right\vert^{2}}{Q^{2}} F(\mathbf{p,Q}) [\mathbf{QQ}] , $$

(96)

$$ B_{2}^{[2]}(\mathbf{p}) = \frac{n_{B}}{\mathcal{V}} \sqrt{ \frac{M \beta_{0}}{2 \pi}} \pi \sum\limits_{\mathbf{Q}} \frac{ \left\vert \psi (Q) \right\vert^{2}}{Q} e^{- \frac{M\beta_{0}\left( \mathbf{Q} \cdot \mathbf{p}\right)^{2}}{2Q^{2}m^{2}}} [\mathbf{QQ}] , $$

(97)

$$\begin{array}{@{}rcl@{}} B(\mathbf{p}) &=& \frac{n_{B}}{\mathcal{V}} \frac{\pi \sqrt{(M \beta_{0})^{3}}}{m M \sqrt{2\pi}} \\ &\times& \sum\limits_{\mathbf{Q}}\frac{\left\vert \psi (Q) \right\vert^{2}}{Q} \left( Q^{2} - \frac{M \beta_{0}}{m^{2}} \left( \mathbf{Q} \cdot \mathbf{p} \right)^{2} \right) e^{- \frac{M \beta_{0} \left( \mathbf{Q} \cdot \mathbf{p} \right)^{2}}{2Q^{2}m^{2}}}.\\ \end{array} $$

(98)

In Eqs. (96) and (97), [QQ] stands for a tensor product of vectors, i.e., a rank-2 tensor. Moreover, in these equations, ψ(Q) is the Fourier transform of the potential energy w(|(r_j −R_μ)|, $\mathcal {V}$ is the volume, and $n_{B}=N_{B}/\mathcal {V}$.

Appendix 3. Summary of Heims-Jaynes Procedure

Given a statistical operator of the form

$$ \rho = \frac{1}{Z} e^{\widehat{A} + \widehat{B}} , $$

(99)

where

$$ Z = \text{Tr} \left\{ e^{\widehat{A} + \widehat{B}} \right\} , $$

(100)

ensures its normalization and introducing

$$ \rho_{0} = \frac{e^{\widehat{A}}}{\text{Tr} \left\{ e^{\widehat{A}} \right\} } , $$

(101)

according to Heims-Jaynes, given an operator $\widehat {\Theta }$, it follows that

$$ \text{Tr}\left\{\widehat{{\Theta} }\rho \right\} = \langle \widehat{\Theta} \rangle_{0} + \sum\limits_{n = 1}^{\infty} \langle \widehat{Q}_{n}(\widehat{\Theta} - \langle \widehat{\Theta} \rangle_{0})\rangle , $$

(102)

where

$$ \langle \widehat{\Theta} \rangle_{0} = \text{Tr} \left\{ \widehat{\Theta} \rho_{0} \right\} , $$

(103)

with

$$ \widehat{Q}_{n} = \widehat{S}_{n} - \sum\limits_{k = 1}^{n-1} \langle \widehat{Q}_{n} \rangle_{0}\widehat{S}_{n-k} , $$

(104)

for n ≥ 2, and $\widehat {Q}_{0}=\widehat {1}$ and $\widehat {Q}_{1} = \widehat {S}_{1}$,

$$ \widehat{S}_{n} = \frac{B^{n}}{n!} , \quad \widehat{S}_{0} = \widehat{1} . $$

(105)

Equation (102) consists of the average value of $\widehat {\Theta }$ with ρ₀ (that is, only depending on A) plus a contribution in the form of a series expansion in powers of B. In a first-order approximation, we have that

$$ \text{Tr} \left\{ \widehat{\Theta} \rho \right\} \simeq \langle \widehat{\Theta} \rangle_{0} + \text{Tr} \left\{ \widehat{B}(\widehat{\Theta} - \langle \widehat{\Theta} \rangle_{0}) \rho_{0}\right\} . $$

(106)

In Section 3, we have used

$$ \overline{\rho}(t,0) = \frac{1}{\overline{Z}(t)} e^{\widehat{A} + \widehat{B}} , $$

(107)

where

$$ \widehat{A}(t) = - \sum\limits_{\mathbf{Q}} F_{h}(\mathbf{Q},t) \widehat{h}(\mathbf{Q}) , $$

(108)

$$\begin{array}{@{}rcl@{}} \widehat{B}(t) &=& - \sum\limits_{\mathbf{Q}} [F_{n}(\mathbf{Q},t) \widehat{n}(\mathbf{Q}) + \mathbf{F}_{n}(\mathbf{Q},t) \cdot \widehat{\mathbf{I}}_{n}(\mathbf{Q}) \\ && + \mathring{F}_{n}^{[2]}(\mathbf{Q},t) \otimes \widehat{I}_{n}^{[2]}(\mathbf{Q})] , \end{array} $$

(109)

and

$$ \overline{Z}(t)=e^{\phi (t)} , $$

(110)

with the calculations performed in the first-order (linear) Heims-Jaynes expansion.