2.3.1. Properties of H for Systems in Equilibrium

If the complete system is in equilibrium without external forces applied to the gas, from classical thermodynamics of systems in equilibrium, we ascertain that the local number of particles and the local energy do not depend on the cell number. In a statistical sense, this is

$$\mathcal{N}'\_{M} = \mathcal{N}' \equiv \mathcal{N}' \tag{14}$$

and

$$
\mathcal{E}'\_M = \mathcal{E}' \equiv \bar{\mathcal{E}}'.\tag{15}
$$

In Equations (14) and (15) the bar implies averaged properties over the complete system. Moreover, the global distribution function is homogeneous, hence *f <sup>M</sup>* does not depend on the cell number *M* (i.e., *f <sup>M</sup>*(*v*, *t*) = *f* (*v*, *t*), ∀*M*). Several properties arise directly from this, *e.g.,* from Equations (14) and (15) *E* = ∑*<sup>M</sup>* E *<sup>M</sup>* <sup>=</sup> *<sup>K</sup>*E¯ , *N* = ∑*<sup>M</sup>* N *<sup>M</sup>* <sup>=</sup> *<sup>K</sup>*N¯ . Here we have used Equations (8) and (9). Equation (5), in terms of Equation (3), can be rewritten as:

$$\mathcal{H}'(t) = \int \sum\_{M=1}^{K} [f'(\vec{v}, t) \ln f'(\vec{v}, t)] \mathrm{d}^3 v = K \int f'(\vec{v}, t) \ln f'(\vec{v}, t) \mathrm{d}^3 v = K H\_\mathbb{B}(t). \tag{16}$$

To identify H with the entropy, we need to show that H (*t*) is extensive, with respect to *K f* (*v*, *t*). This is shown by analyzing the following expression:

$$\begin{aligned} \int \left[ \mathbb{K} f'(\vec{v}, t) \right] \ln \left[ \mathbb{K} f'(\vec{v}, t) \right] \mathrm{d}^3 v &= \int \left[ (\mathbb{K} \ln \mathcal{K}) f'(\vec{v}, t) + \mathbb{K} f'(\vec{v}, t) \ln f'(\vec{v}, t) \right] \mathrm{d}^3 v \\ &= \ \mathrm{K} \int \left\{ f'(\vec{v}, t) \left[ \ln \mathcal{K} + \ln f'(\vec{v}, t) \right] \right\} \mathrm{d}^3 v. \end{aligned} \tag{17}$$

We observe that if the number of particles in the *μ*-space, *f* (*v*, *t*), is much larger than the number of cells, *K*, then the first term of Equation (17) is negligible, and consequently:

$$\int \left[ \mathbb{K} f'(\vec{v}, t) \right] \ln \left[ \mathbb{K} f'(\vec{v}, t) \right] \mathrm{d}^3 v \approx \mathrm{K} \int f'(\vec{v}, t) \ln f'(\vec{v}, t) \mathrm{d}^3 v,\tag{18}$$

i.e., H *<sup>M</sup>* is extensive, and the sum ∑*<sup>M</sup>* H *<sup>M</sup>* is the *H*-functional of the complete system, which reduces to the Boltzmann *H*-functional. Therefore, H can be identified with the entropy density of the system.

Furthermore, in equilibrium, the Lagrange multipliers are position- and time-independent, thus *f <sup>M</sup>*(*v*) reduces to

$$f\_M'(\vec{v}) = \mathbb{C} \exp(\alpha + \beta \epsilon(\vec{v})) \equiv f'(\vec{v}), \quad M = 1, \ldots, K. \tag{19}$$

The constant *C* can be omitted, which is shown by defining the following H functional:

$$\mathcal{H}''(t) = \sum\_{M=1}^{K} \int \left[ f\_M'(\vec{v}, t) \ln f\_M'(\vec{v}, t) - f\_M'(\vec{v}, t) \right] \mathrm{d}^3 v. \tag{20}$$

Since ∑*<sup>K</sup> M*=1 *f <sup>M</sup>*(*v*, *<sup>t</sup>*)d3*<sup>v</sup>* = *<sup>N</sup>* (a constant), and because we are mainly interested in the time-derivative of H, *C* can be conveniently omitted. In other words, H leads to the Maxwell–Boltzmann distribution function of systems in equilibrium.

#### 2.3.2. Proof of the *H*-Theorem for Non-Homogeneous Distributions

Throughout this section, we consider a classical gas with an initial condition close to the equilibrium, which ensures that the local equilibrium hypothesis remains valid during the time-evolution of the system. Also, we use the following definitions for the deviations of concentration and energy, relative to the equilibrium values:

$$\mathcal{N}\_M'(t) = \int f\_M'(\vec{v}, t) \mathbf{d}^3 v = \bar{\mathcal{N}}' + \Delta\_M'(t) \tag{21}$$

and

$$\mathcal{E}'\_M(t) = \int f'\_M(\vec{v}, t) \epsilon(\vec{v}) \mathbf{d}^3 v = \mathcal{E}' + \delta'\_M(t). \tag{22}$$

Here <sup>N</sup>¯ <sup>=</sup> *<sup>N</sup>*/*<sup>K</sup>* and <sup>E</sup>¯ <sup>=</sup> *<sup>E</sup>*/*<sup>K</sup>* are the cell particle number and the cell energy in equilibrium, respectively, which are given by

$$
\bar{\mathcal{N}}' = \int \bar{f}'(\vec{v}) \mathbf{d}^3 v \quad \text{and} \quad \bar{\mathcal{E}}' = \int \bar{f}'(\vec{v}) \epsilon(\vec{v}) \mathbf{d}^3 v,\tag{23}
$$

where we have used ¯ *f* (*v*) as defined in Equation (19). In Equations (21) and (22), Δ *M* and *δ <sup>M</sup>* are considered deviations relative to <sup>N</sup>¯ and <sup>E</sup>¯ , respectively. For systems that are sufficiently close to equilibrium, it is reasonable to expect first that Δ *<sup>M</sup>*(*t*) <sup>N</sup>¯ and *δ <sup>M</sup>*(*t*) <sup>E</sup>¯ , and second that Δ *<sup>M</sup>* and *δ <sup>M</sup>* are sufficiently large compared to the fluctuations of <sup>N</sup>¯ and <sup>E</sup>¯ . Similarly, we can assume that every local distribution function can be written as

$$f\_M'(\vec{v}, t) = \vec{f}'(\vec{v}) (1 + \mathcal{g}\_M'(\vec{v}, t)), \quad 1 \gg |\mathcal{g}\_M'(\vec{v}, t)|. \tag{24}$$

With the previous considerations, in the following, we proof an alternative *H*-theorem, considering the *H*-functional, H , defined by Equation (5).

We commence by differentiating Equation (5) with respect to time:

$$\frac{d\mathcal{H}'}{dt} = \sum\_{M=1}^{K} \int \left[1 + \ln f\_M'(\vec{v}, t)\right] \dot{f}\_M'(\vec{v}, t) \,\mathrm{d}^3 v. \tag{25}$$

(Starting here, we use the standard notation ˙ *h* ≡ (*dh*/*dt*)). Substituting Equation (24) into Equation (25) yields

$$\frac{d\mathcal{H}'}{dt} = \sum\_{M=1}^{K} \int \vec{f}'(\vec{v}) \left[1 + \ln\left\{\vec{f}'(\vec{v}) + \vec{f}'(\vec{v}) \mathbf{g}'\_{M}(\vec{v}, t)\right\}\right] \mathbf{g}'\_{M}(\vec{v}, t) \mathbf{d}^{3}v. \tag{26}$$

The logarithmic term of Equation (26) expanded up to the first-order term of its Taylor series, around *g <sup>M</sup>*(*v*, *t*) = 0, is

$$
\ln[\bar{f}'(\vec{v}) + \bar{f}'(\vec{v})g\_M'(\vec{v}, t)] \approx \ln[\bar{f}'(\vec{v})] + \mathcal{g}\_M'(\vec{v}, t),
\tag{27}
$$

and substituting this into Equation (26) gives

$$\frac{d\mathcal{H}'}{dt} \quad = \sum\_{M=1}^{K} \int \vec{f}'(\vec{v}) \left[1 + \ln \vec{f}'(\vec{v}) + \mathcal{g}'\_M(\vec{v}, t)\right] \mathcal{g}'\_M(\vec{v}, t) \mathrm{d}^3 v. \tag{28}$$

Substituting ln ¯ *f* (*v*) = exp(*α* + *β*(*v*)), see Equation (19) and subsequent text, and omitting *C* we obtain

$$\frac{d\mathcal{H}'}{dt} = \sum\_{M=1}^{K} \int f(\vec{v}) [\alpha + \beta \epsilon(\vec{v})] \dot{\xi}\_M(\vec{v}, t) \mathbf{d}^3 v + \sum\_{M=1}^{K} \int f(\vec{v}) \xi\_M(\vec{v}, t) \dot{\xi}\_M(\vec{v}, t) \mathbf{d}^3 v. \tag{29}$$

From the definitions of N *<sup>M</sup>* and E *<sup>M</sup>* —Equations (21) and (22)—and *f <sup>M</sup>*(*v*, *t*) —Equation (24) it is straightforward to show that

$$
\int \vec{f}(\vec{v}) \mathbf{\hat{g}} M(\vec{v}, t) \mathbf{d}^3 v = \Delta\_M(t) \quad \Rightarrow \quad \int \vec{f}(\vec{v}) \dot{\mathbf{\hat{g}}} M(\vec{v}, t) \mathbf{d}^3 v = \dot{\Delta}\_M(t), \tag{30}
$$

$$
\int \mathbf{f}(\vec{v}) \mathbf{g}\_M(\vec{v}, t) \mathbf{c}(\vec{v}) \mathbf{d}^3 v = \delta\_M(t) \quad \Rightarrow \quad \int \mathbf{f}(\vec{v}) \mathbf{g}\_M(\vec{v}, t) \mathbf{c}(\vec{v}) \mathbf{d}^3 v = \delta\_M(t), \tag{31}
$$

and as a consequence of ∑*<sup>K</sup> <sup>M</sup>*=<sup>1</sup> <sup>Δ</sup>*M*(*t*) = <sup>∑</sup>*<sup>K</sup> <sup>M</sup>*=<sup>1</sup> *δM*(*t*) = 0, we find

$$\sum\_{M=1}^{K} \dot{\Delta}\_M(t) = \sum\_{M=1}^{K} \dot{\delta}\_M(t) = 0. \tag{32}$$

Therefore, due to Equations (30)–(32), Equation (29) simplifies to

$$\frac{d\mathcal{H}'}{dt} = \sum\_{M=1}^{K} \int f(\vec{v}) \underline{\mathcal{G}}\_{M}(\vec{v}, t) \dot{\underline{\mathcal{G}}}\_{M}(\vec{v}, t) \,\mathrm{d}^3 v. \tag{33}$$

To clearly determine the time-evolution of Equation (33), we split the summation over *M* into two terms:

$$\frac{d\mathcal{H}'}{dt} = \sum\_{l}^{L} \int \vec{f}'(\vec{v}) \boldsymbol{g}\_{l}'^+(\vec{v}, t) \boldsymbol{\check{g}}\_{l}'^+(\vec{v}, t) \,\mathrm{d}^3 v + \sum\_{l}^{P} \int \vec{f}'(\vec{v}) \boldsymbol{g}\_{l}'^-(\vec{v}, t) \boldsymbol{\check{g}}\_{l}'^-(\vec{v}, t) \mathrm{d}^3 v \tag{34}$$

where *L* + *P* = *K*. The above split is made based on the assumption that for any given initial state of the system, at *t*0, some cells will have either a *g <sup>I</sup>*(*v*, *t*0) ≥ 0 or a *g <sup>J</sup>*(*v*, *t*0) < 0, which we denote as *g*˙ *I* +(*v*, *t*) or *g*˙ *J* −(*v*, *t*), respectively.

If the system's initial state is sufficiently close to equilibrium, it is physically appropriate to assume that *g <sup>M</sup>*(*v*, *t*0) → 0 as *t* → ∞ in a monotonous manner, thus *g*˙ *I* +(*v*, *<sup>t</sup>*) <sup>≤</sup> <sup>0</sup> and *g*˙ *J* <sup>−</sup>(*v*, *t*) > 0, for *t* ≥ *t*0. Consequently, Equation (34) can be re-written as

$$\frac{d\mathcal{H}'}{dt} = -\left[\sum\_{I}^{L} \int \vec{f}(\vec{v}) |\mathcal{g}\_{I}^{+}(\vec{v},t)| |\mathcal{g}\_{I}^{+}(\vec{v},t)| \mathrm{d}^{3}v + \sum\_{I}^{P} \int \vec{f}(\vec{v}) |\mathcal{g}\_{I}^{-}(\vec{v},t)| |\mathcal{g}\_{I}^{-}(\vec{v},t)| \mathrm{d}^{3}v\right].\tag{35}$$

Since every integrand in Equation (35) is positive, for all *t* and *v*, and *g <sup>M</sup>*(*v*, *t*0) → 0 as *t* → ∞, it follows that

$$\frac{d\mathcal{H}'}{dt} \le 0.\qquad\qquad\text{QED.}\tag{36}$$

In summary, considering a gas occupying a volume V (which is divided into *K* small cells), with a total energy *E* and *N* classical free particles, whose initial state is not in equilibrium, but sufficiently close to equilibrium, then the functional

$$\mathcal{H}'(t) = \sum\_{M=1}^{K} \int f\_M'(\vec{v}, t) \ln f\_M'(\vec{v}, t) \,\mathrm{d}^3 v \tag{37}$$

where *f <sup>M</sup>*(*v*, *t*) is the cell distribution function, which satisfies *d*H /*dt* ≤ 0, and the equality relation is attained at *t* → ∞. In Equation (37), *f <sup>M</sup>*(*v*, *t*) is the Maxwell–Boltzmann distribution function, which in general is different for different cells—i.e., the complete system can be non-homogeneous—and each *f <sup>M</sup>*(*v*, *t*) is compatible with the cell properties, such as number of particles, N *<sup>M</sup>*, energy, E *<sup>M</sup>*, temperature, *TM*, and Legendre multipliers *α<sup>M</sup>* and *βM*.

#### **3. Quantum Scheme**

The classical *H*-theorem is still considered one pillar on which classical statistical physics is founded. Unfortunately, despite multiple attempts [3,10,11,14–16], the generality of the classical *H*-theorem has no equally robust quantum match. In this section, we propose and analyze an alternative quantum *H*-functional using the variational method. We start by briefly outlining a typical textbook demonstration of the quantum *H*-theorem [2], and subsequently present the analysis of our proposed *H*-functional.

#### *3.1. H-Theorem and the Fermi–Dirac and Bose–Einstein Distribution Functions*

Consider a dilute gas of *N* non-interacting quantum particles (either bosons or fermions), contained by a vessel of volume *V*, temperature *T*, and total energy E. Starting from the Boltzmann definition of entropy, the quantum *H* functional is

$$H\_T = -\ln G\_\prime \tag{38}$$

where *G* describes the total number of accessible quantum states of the gas that satisfy the above conditions [2]. The quantum *H*-theorem can be demonstrated as follows. *G* can be divided into groups of neighboring states, *gk*, and certain occupation numbers, *nk*, can be associated with each of these groups. Thus, the above functional takes the form

$$H\_T = \sum\_i n\_i \ln n\_i - (n\_i \pm g\_i) \ln(g\_i \pm n\_i) \pm g\_i \ln g\_{i\prime} \tag{39}$$

where the upper and lower signs are for bosons and fermions, respectively. Thus the time derivative of Equation (39) is

$$\frac{dH\_T}{dt} = \sum\_{\mathbf{x}} \left[ \ln n\_{\mathbf{x}} - \ln(g\_{\mathbf{x}} \pm n\_{\mathbf{x}}) \right] \frac{dn\_{\mathbf{x}}}{dt}.\tag{40}$$

Assuming that the energy exchange between particles is produced by interparticle collisions, and using perturbation theory, the rate of change in the number of particles in a group *κ* is

$$\begin{split} \frac{dn\_{\mathbb{K}}}{dt} &= \ -\sum\_{\lambda, (\mu\nu)} A\_{\mathbb{K}\lambda, \mu\nu} n\_{\mathbb{K}} n\_{\mathbb{K}} (\mathcal{g}\_{\mu} \pm n\_{\mu}) (\mathcal{g}\_{\nu} \pm n\_{\nu}) \\ &+ \sum\_{\lambda, (\mu\nu)} A\_{\mu\nu, \varkappa\lambda} n\_{\mu} n\_{\nu} (\mathcal{g}\_{\mathbb{K}} \pm n\_{\kappa}) (\mathcal{g}\_{\lambda} \pm n\_{\lambda}). \end{split} \tag{41}$$

Here *Aκλ*,*μνnκnλ*(*g<sup>μ</sup>* ± *nμ*)(*g<sup>ν</sup>* ± *nν*) is the expected number of collisions per unit time, in which two particles will be moved from groups (*κ*, *λ*) to (*μ*, *ν*), and the tensor *Aκλ*,*μν* is given by

$$A\_{\kappa\lambda,\mu\nu} = \frac{4\pi^2}{h} \frac{|I\_1 \pm I\_2|^2}{\Delta\varepsilon}. \tag{42}$$

In Equation (42), Δ is the net energy change occurring during the collision and |*I*<sup>1</sup> − *I*2| <sup>2</sup> = |*Vmn*,*kl*| 2, where *Vmn*,*kl* is the element of the transition matrix of a binary collision. It is important to remark that in deriving Equation (41), the equal a priori probabilities and the random a priori phase hypotheses were assumed valid. The random a priori phase hypothesis can be considered an analogue of the molecular chaos hypothesis [15], as it is the mechanism by which stochasticity is introduced into the system.

Substituting Equation (41) into Equation (40), it is straightforward to prove that

$$\frac{dH\_T}{dt} \le 0.\tag{43}$$

At equilibrium (at *t* → ∞), *dnκ*/*dt* = 0, hence from Equation (41)

$$
\ln \frac{n\_{\mathbf{k}}}{g\_{\mathbf{k}} \pm n\_{\mathbf{k}}} + \ln \frac{n\_{\lambda}}{g\_{\lambda} \pm n\_{\lambda}} = \ln \frac{n\_{\mu}}{g\_{\mu} \pm n\_{\nu}} + \ln \frac{n\_{\nu}}{g\_{\nu} \pm n\_{\nu}}.\tag{44}
$$

Considering that energy is conserved during the collision, the Bose–Einstein or Fermi–Dirac distribution functions can be recovered from Equation (44):

$$m\_{\mathbf{k}} = \frac{\mathcal{G}\kappa}{\exp(\alpha + \beta \epsilon\_{\mathbf{k}}) \mp 1}. \tag{45}$$

*I*n other words, at equilibrium *dHT*/*dt* = 0, the distribution function obtained from Equation (39) is the expected distribution function.

#### *3.2. Out-of-Equilibrium, Non-Homogeneous Quantum Systems*

Consider a dilute gas enclosed by a perfectly isolated vessel of volume *V*, with total energy *E*, and total number of quantum particles *N*, which can be free fermions or bosons. For our purposes, the volume *V* is divided into *K* small cells, each of which has constant volume *δVM* = *V*/*K* (*M* = 1, ... , *K*), temperature *TM*, energy *M*, number of particles N*M*, and distribution function, { *fMn*(*t*)}. Hereafter we use the following short-hand notation:

$$f\_{Mn}(t) \equiv f(\vec{r}\_{M\prime} \epsilon\_{n\prime} t),\tag{46}$$

where*rM* is the radius vector pointing at the center of the *M*-th cell. *fMn*(*t*) represents the number of particles contained in the *M*-th cell that occupies the energy level *<sup>n</sup>* at time *t*. Since the particles are considered to be free, the energy levels should not depend on the cell properties, *i.e.,* the energy spectrum, {*n*}, is the same for all cells; thus, there is no need to label *<sup>n</sup>* with an index *M*.

We propose the following functional as an alternative *H*-functional for quantum non-homogeneous dilute gases:

$$\mathcal{H}(t) \quad = \sum\_{M=1}^{K} \sum\_{n} \left[ f\_{Mn}(t) \ln f\_{Mn}(t) \right]$$

$$\pm \left( 1 \mp f\_{Mn}(t) \right) \ln(1 \mp f\_{Mn}(t)) \Big| \delta V\_M. \tag{47}$$

Here, the upper and lower signs refer to fermions and bosons, respectively. In addition, when needed, each cell has an associated local chemical potential, *αM*, and a local *H*functional, which is defined by

$$\mathcal{H}\_{M}(t) = \sum\_{n} \left[ f\_{Mn}(t) \ln f\_{Mn}(t) \pm \left( 1 \mp f\_{Mn}(t) \right) \ln \left( 1 \mp f\_{Mn}(t) \right) \right] \delta V\_{M}.\tag{48}$$

Therefore, N*<sup>M</sup>* and E*<sup>M</sup>* as functions of time are given by

$$
\mathcal{N}\_M(t) = \sum\_n f\_{Mn}(t)\delta V\_M \tag{49}
$$

and

$$\mathcal{E}\_M(t) = \sum\_n f\_{Mn}(t) \varepsilon\_n \delta V\_{M\nu} \tag{50}$$

which are, for the whole system, constrained by the micro-canonical restrictions

$$\sum\_{M=1}^{K} \left[ \sum\_{n} f\_{Mn}(t) \right] \delta V\_M = \sum\_{M=1}^{K} \mathcal{N}\_M(t) \delta V\_M = N,\tag{51}$$

and *<sup>K</sup>*

$$\sum\_{M=1}^{K} \left[ \sum\_{n} f\_{Mn}(t) \varepsilon\_{n} \right] \delta V\_{M} = \sum\_{M=1}^{K} \mathcal{E}\_{M}(t) \delta V\_{M} = E. \tag{52}$$

Applying the variational method to H, and using the Lagrange multipliers {*αM*} and {*βM*}, we readily obtain (see also the discussion related to Equation (12)):

$$\ln\left(\frac{1\mp f\_{Mn}(t)}{f\_{Mn}(t)}\right) = -\mathfrak{a}\_M(t) - \beta\_M(t)\varepsilon\_{n\prime} \tag{53}$$

and solving for *fMn*(*t*) yields

$$f\_{Mn}(t) = \frac{1}{\exp\left(-\kappa\_M(t) - \beta\_M(t)\epsilon\_n\right) \pm 1}.\tag{54}$$

Thus, in this zero-order approximation, the form of equilibrium distribution functions is conserved.
