**1. Introduction**

The theoretical bases and the procedures that allow us to describe equilibrium systems are well-established. These procedures can be applied to a wide range of natural systems, including both the macroscopic phenomenological methods (thermodynamics) and the microscopic description (statistical mechanics) (Out-of-equilibrium systems, of course, are still a challenge). For instance, in the kinetic theory of gases, the behavior of a dilute classical gas is described through the Boltzmann transport equation [1], and the timeevolution of a system towards equilibrium is finely accounted for through the Boltzmann *H*-theorem.

However, for quantum out-of-equilibrium systems, the construction of a kinetic framework with the same level of success and universality as the classical version still presents some fundamental challenges. For instance, to obtain a complete correspondence principle between classical mechanics and quantum mechanics, the form of the quantum analogues of both the Boltzmann *H*-theorem and the Boltzmann transport equation is inadequate. In this context, Tolman was one of the earliest physicists to propose a quantum *H*-theorem [2], using a probability transition relationship, the random phases hypothesis, and an *H*-functional defined in terms of a spatially homogeneous distribution function. Tolman also proposed a potential quantum analogue of the transport equation, in terms of the occupation numbers, by applying time perturbation theory. Additional attempts, under quantum operator formalism, have addressed the description of quantum transport phenomena through the Hamiltonian of the system and the master equation (which is, in these works, the analogue of the Boltzmann transport equation) [3–7]. However, these approaches are not consistent with the classical-quantum correspondence principle.

**Citation:** Medel-Portugal, C.; Solano-Altamirano, J.M.; Carrillo-Estrada, J.L.E. Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes. *Entropy* **2021**, *23*, 366. https://doi.org/10.3390/e23030366

Academic Editor: Petr Jizba

Received: 27 January 2021 Accepted: 16 March 2021 Published: 19 March 2021

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Similarly, some authors have proposed *H*-functionals and attempted to proof a quantum *H*-theorem [8–16]. However, whether or not the homogeneous distribution function hypothesis is assumed or if its framework fulfills the correspondence principle is unclear or not discussed. Since the pioneering work of Tolman, at several stages, there has been some discussion regarding the general validity of the quantum *H*-theorem, some possible violations of the second law of thermodynamics, and the interpretation of the quantum entropy [10,12–14,16–21].

Nonetheless, the framework to describe spatially non-homogeneous systems is still under construction, although several approaches have been developed. For instance, the celebrated Onsager formulation (linear thermodynamics) [22,23] has been successful in describing irreversible chemical and physical phenomena. However, some descriptions, such as those the internal behavior of gases [24] and the entropy measurement [25,26], cannot be completely addressed with linear thermodynamics.

In addition, some aspects regarding the classical *H*-theorem and the Boltzmann *H*-functional require revision to improve their mutual consistency. One example is the modification of the *H*-theorem to include phenomena stemming from stochastic trajectories, violations of the second law of thermodynamics, the relationship between Shannon's measure of information and the Boltzmann's entropy, and the calculation of thermodynamical quantities and thermalization of specific systems [8,26–29].

To contribute to the construction of a consistent classical and quantum *H*-theorem, within a formalism that describes out-of-equilibrium non-homogeneous systems, we propose a new theoretical framework. Specifically, for both classical and quantum systems, we include non-homogeneous distribution functions in the *H*-functionals, and consider non-homogeneous systems in the proofs of the resulting *H*-theorems. Our proposed *H*functionals satisfy the correspondence principle, but more importantly, these functionals describe the time-evolution of spatially non-homogeneous systems towards equilibrium.

The organization of this article is as follows. In Section 2, we highlight, for our purposes, the most fundamental assumptions required to proof the Boltzmann *H*-theorem, we provide an alternative method to obtain the Maxwell–Boltzmann distribution using the variational method, propose an alternative *H*-functional for classical systems, and demonstrate the respective *H*-theorem. In Section 3, we review the Tolman proposal for the quantum version of the *H*-theorem (quantum *H*-theorem) and how the Bose–Einstein and Fermi–Dirac distributions are treated within this framework. Subsequently, we present our proposal for a quantum *H*-functional and the proof of the corresponding quantum *H*-theorem. In Section 4, we analyze the classical-quantum correspondence between the quantum and classical *H*-functionals. In Section 5, we explore how relaxation processes occur in a quantum ideal gas and, based on what we call variational entropy, propose a time-evolution equation for the distribution function. Finally, we discuss some key ideas resulting from our approach and close with a summary in Section 6.

#### **2. Classical Scheme**

The Boltzmann kinetic theory of gases represents a fundamental connection between the microscopic nature of matter and the phenomenological macroscopic laws of classical thermodynamics. The stochasticity introduced by the molecular chaos hypothesis in the otherwise deterministic kinetics of the particles allows for the demonstration of the celebrated Boltzmann *H*-theorem. In contrast, in this article, we propose an alternative approach developed using a variational procedure applied to an *H*-functional. We start this section by briefly accounting for the important elements of the standard derivation of the Boltzmann transport equation and demonstrating the *H*-theorem, such as they are presented in classical textbooks [1].

#### *2.1. The Boltzmann Transport Equation*

The first step in the Boltzmann kinetic theory of gases is defining the distribution function, *f*(*r*,*v*, *t*), as the average number of molecules that, at time *t*, have position*r* and velocity*v*, and are contained in a *μ*-space volume element *d*3*rd*3*v*. Assuming a deterministic Newtonian description of molecular motion, as well as the invariance of the *μ*-space volume measure, one arrives at the Boltzmann rate equation:

$$f(\vec{r} + \vec{v}\delta t, \vec{v} + \vec{F}\delta t, t + \delta t) = f(\vec{r}, \vec{v}, t) + \left(\frac{\partial f}{\partial t}\right)\_{\text{coll}}\delta t. \tag{1}$$

Here, the term (*∂ f* /*∂t*)coll describes the in and out fluxes from and towards the volume element, due to the collisions. Subsequently, from the previous equation, the integrodifferential Boltzmann transport equation is obtained:

$$\int \left( \frac{\partial}{\partial t} + \vec{v}\_1 \cdot \nabla\_{\vec{r}} + \frac{\vec{F}}{m} \cdot \nabla\_{\vec{v}\_1} \right) f\_1 = \int \mathbf{d} \Omega \int \mathbf{d}^3 v\_2 r(\Omega) |\vec{v}\_1 - \vec{v}\_2| (\vec{f}\_2 \vec{f}\_1 - f\_2 f\_1). \tag{2}$$

In Equation (2), Ω is the solid angle, *σ* is the scattering cross section, *F* the external force applied to the system, and *f*<sup>1</sup> and *f*<sup>2</sup> ( ˜ *f*<sup>1</sup> and ˜ *f*2) are the distribution functions of particles 1 and 2, respectively, before (and after) the collision.

Particle dynamics and the effects of external forces are described by the left-hand side of Equation (2). The right-hand side is derived by considering binary collisions between particles and accepting the *molecular chaos hypothesis*, i.e., it is assumed that the positions and velocities of the particles are not time-correlated.

### *2.2. A Summary of the H-Theorem and the Maxwell—Boltzmann Distribution*

The evolution of a dilute gas towards thermodynamic equilibrium is frequently addressed by first defining the *H*-functional [1,2]:

$$H\_{\rm B} = \int f(\vec{v}, t) \ln f(\vec{v}, t) \,\mathrm{d}^3 v. \tag{3}$$

Notice that *fB*(*v*, *t*) is a spatially homogeneous distribution function. The functional *HB*, originally introduced by Boltzmann in 1872, describes a dilute gas occupying a volume *V*, at temperature *T*, with total energy *E*, and total number of free classical particles *N*. To clearly distinguish the Boltzmann functional *HB*, we denote hereafter the Maxwell–Boltzmann distribution function as *fB*.

The physically correct spontaneous time-evolution of an out of equilibrium dilute gas is corroborated by the *H*-theorem. This theorem establishes that if (a) the homogeneous function *f*(*v*, *t*) satisfies the Boltzmann transport equation and (b) the molecular chaos hypothesis is valid, then the system evolves in such a manner that *dHB*/*dt* ≤ 0, and if *dHB*/*dt* = 0, then the system is in the equilibrium state. The *H*-theorem is straightforward to prove using Equation (2) [1], and it assures the consistency between our microscopic approach to describe the system's spontaneous time-evolution and the phenomenological observations established by the second law of classical thermodynamics; in fact, *HB* can be associated with an entropy density.

On the other hand, considering a dilute gas in equilibrium with no applied external forces, *i.e.,* (*∂ f* /*∂t*) = 0 and *f* is independent of*r*, we can directly prove that the equilibrium distribution function obtained from Equation (2) is precisely the Maxwell–Boltzmann distribution function. The proof of the above first requires identification of the sufficient condition for *f* to render a null r.h.s. of Equation (2). Such an *f* , which we denote here as *f*0, must satisfy

$$f\_0(\vec{v}\_2')f\_0(\vec{v}\_1') - f\_0(\vec{v}\_2)f\_0(\vec{v}\_1) = 0. \tag{4}$$

Subsequently, the Maxwell–Boltzmann distribution function can be obtained by taking the logarithm of Equation (4) and conserved mechanical quantities (see ([1], ch. 4.2)).

Before introducing our proposed *H*-functional, we must state that in defining *HB*, Equation (3), it is assumed that the distribution function *f* is spatially homogeneous. This assumption simplifies the demonstration of the *H* theorem. However, it also introduces

a limited conception of the out-of-equilibrium condition of the gas. Given the relatively simple nature of a dilute gas, one of the salient features of an out-of-equilibrium condition is the existence of inhomogeneities in the system, which is not considered in the above.

#### *2.3. Non-Homogeneous Classical H-Functional*

As we saw in the previous section, the validity of the *H*-theorem relies significantly on assuming that the distribution function is homogeneous and the molecular chaos hypothesis is fulfilled. To extend the previous procedure to systems with non-homogeneous distribution functions, which might allow for the study of systems in a more general outof-equilibrium condition, we introduce a modified *H*-functional. For the sake of clarity and simplicity, we use primed functions and quantities to denote the classical case to differentiate them from the quantum analogues.

Our proposed classical *H*-functional, denoted as H , describes a dilute classical gas occupying a volume *V*. In our theoretical treatment, we divide this volume into *K* cells, which, without loss of generality, have identical volumes, *δVM* = *V*/*K*, *M* = 1, ... , *K*. Each cell of index *M* has the following local functions, properties, and variables: an *H*functional, H *<sup>M</sup>*, a homogeneous distribution function, *f <sup>M</sup>*(*v*, *t*), number of particles, N *M*, temperature, *T <sup>M</sup>*, and energy, E *<sup>M</sup>*. Taken as a whole, the system has an energy *E*, and a global number of free classical particles *N*. We also assume that the system is perfectly isolated, and that the number of particles in each cell is sufficiently large, so as to obtain accurate averages. We start our analysis by proposing the following inhomogeneous *H*-functional:

$$\mathcal{H}'(t) = \sum\_{M=1}^{K} \int\_{\delta V\_M} f'\_M(\vec{v}, t) \ln f'\_M(\vec{v}, t) \mathrm{d}^3 v. \tag{5}$$

The distribution functions, { *f <sup>M</sup>*(*v*, *t*)}, depend implicitly on the position of the cells, relative to the global system, and on the velocity *v* and time *t*. Notice that each *f <sup>M</sup>* can be formally extended to the complete coordinate space by defining each *f <sup>M</sup>* to be zero outside the *M*-th cell, in such a manner that the distribution function of the complete system is a piece-wise sum of { *f <sup>M</sup>*(*v*, *t*)}:

$$f'(\vec{r}, \vec{v}, t) = \sum\_{M=1}^{K} f'(\vec{r}\_{M\prime} \vec{v}, t) = \sum\_{M=1}^{K} f'\_{M}(\vec{v}, t). \tag{6}$$

Here, *rM* is the center of the cell of index *M*, and *f <sup>M</sup>*(*v*, *t*) = *f <sup>N</sup>*(*v*, *t*) for *M* = *N*. This extended definition allows us to omit the symbol *δVM* in all integrals performed over the cell volume. In terms of *f <sup>M</sup>*(*v*, *t*) and a local variable of energy, (*v*), we have

$$\mathcal{H}\_{M\_{\perp}}^{\prime} = -\int f\_{M}^{\prime}(\vec{v}\_{\prime}t) \ln f\_{M}^{\prime}(\vec{v}\_{\prime}t) \mathrm{d}^{3}v\_{\prime} \tag{7}$$

$$\mathcal{N}'\_{M} \quad = \int f'\_{M}(\vec{v}, t) \mathbf{d}^{3} v\_{\prime} \tag{8}$$

$$\mathcal{E}'\_M = -\int f'\_M(\vec{v}, t) \epsilon(\vec{v}) \mathrm{d}^3 v. \tag{9}$$

Notice that assuming every *f <sup>M</sup>* to be homogeneous implies that we are accepting the validity of the local equilibrium hypothesis. In addition, the set { *f <sup>M</sup>*(*v*, *t*)} must satisfy the following restrictions:

$$\sum\_{M=1}^{K} \int f\_M'(\vec{v}\_\prime t) \mathbf{d}^3 v = N \tag{10}$$

and *<sup>K</sup>*

$$\sum\_{M=1}^{K} \int f\_M'(\vec{v}, t) \epsilon(\vec{v}) \mathbf{d}^3 v = E. \tag{11}$$

We now use the variational method to find the extremal of H , consistent with restrictions (7)–(11) together with the corresponding Lagrange multipliers {*αM*} and {*βM*}. This yields

$$\begin{split} \frac{\delta \mathcal{H}'}{\delta f'\_f(\vec{v}\vec{r})} &= \sum\_{M=1}^{K} \int \frac{\delta}{\delta f'\_f(\vec{v}\vec{r})} \left[ f'\_M(\vec{v}) \ln f'\_M(\vec{v}) \right] \mathbf{d}^3 v - \sum\_{M=1}^{K} a\_M \int \frac{\delta f'\_M(\vec{v})}{\delta f'\_f(\vec{v}')} \mathbf{d}^3 v \\ &- \sum\_{M=1}^{K} \beta\_M \int \varepsilon(\vec{v}) \frac{\delta f'\_M(\vec{v})}{\delta f'\_f(\vec{v}')} \mathbf{d}^3 v \\ &= \ln f'\_f(\vec{v}') + 1 - a\_f - \beta\_f \varepsilon(\vec{v}') = 0. \end{split} \tag{12}$$

Solving the last line for *f J*(*v* ) renders

$$f\_I'(\vec{v}') = \mathbb{C} \exp\left(\mathfrak{a}\_I + \beta\_I \varepsilon(\vec{v}')\right) \tag{13}$$

where *C* is a constant. We notice that by applying the variational procedure on H', we predict that when equilibrium is reached, the distribution function of each cell has the form of the Maxwell–Boltzmann distribution function, which is consistent with the local equilibrium assumption.
