Gibbs variational principles and Boltzmann irreversible theorem

Abstract

We analyze the Gibbs variational principles associated with the probability distributions of (i) an isolated system and (ii) a system at constant temperature. We give an example of using the Gibbs inequality to obtain the free energy and analyze the phase diagram of an Ising model with competing interactions. We also review the Boltzmann irreversible theorem, and show how it is connected to the Gibbs variational principles. This connection is established by using the Kolmogorov equation for the evolution of the probability distribution, which predicts a monotonic increase of entropy for an isolated system, and a decrease of the free energy for a system in contact with a thermal reservoir.

Keywords:
Gibbs variational principles; Boltzmann irreversible equation; Kolmogorov equation.

1. Introduction

Variational principles have been a useful resource in the formulation of physics theories [¹]. Variational formulations have even been considered as more fundamental, as they conform to a certain theological vision of science, which of course cannot be scientifically validated. In general, variational principles are equivalent to usual formulations and can lead us to a deeper understanding of some theories.

In geometric optics, Fermat introduced a minimum principle associated with the shortest time that it takes for a ray of light to travel between two given points along a certain path. Consider the path integral between two points,

(1) $$ℐ={\int }_1^2\frac{ds}{v},$$

where $ds$ is an element of distance and $v$ is the absolute value of the speed of light. The integral $ℐ$ represents the time it takes for the ray of light to travel along the path between the two given points. Fermat principle states that the actual path traveled by light corresponds to the minimum value of this integral.

In mechanics, in the domain of statics, the principle of virtual work of Johann Bernoulli is a variational principle. It was extended by d’Alembert to dynamics. On the basis of this principle, Lagrange formulated analytical mechanics [²]. Lagrange considers the time integral

(2) $$ℐ={\int }_1^2ℒ\mathit{d}t,$$

between two given instants of time, where $ℒ=K-V$ is the difference between the kinetic energy $K$ and the potential energy $V$ of a mechanical system. Among the various trajectories that we can imagine, the only one that occurs, in agreement with the Newton equations of motion, corresponds to the smallest value of $ℐ$. The Hamilton formulation of mechanics is based on a minimum principle involving a function that is written in terms of the positions and momenta of a mechanical system, which is obtained from $ℒ$ by a Legendre transformation.

In statistical mechanics, Gibbs used the thermodynamic potentials to introduce some variational principles [³, ⁴, ⁵]. According to thermodynamics, the entropy of an isolated system reaches its largest value in the equilibrium state. If the system is kept at constant temperature, it reaches its lowest free energy potential in the equilibrium state. Gibbs variational principles are the translations of these propositions into statistical mechanics, in which case the states of the systems are understood as probability distributions and the equilibrium state is given by the Gibbs distribution.

Another approach to achieving an equilibrium distribution was proposed by Maxwell [⁶, ⁷, ⁸] and by Boltzmann [⁹, ¹⁰, ¹¹, ¹², ¹³]. The historical “Boltzmann method” leads to the well-known transport equation for the molecular probabilities, and to the $H$-theorem for the irreversibility of the physical processes. The Gibbs proposal to obtain the equilibrium state from variational principles does not use a dynamic process; in other words, it does not need the consideration of the time evolution of the state. However, a dynamic process that leads to the equilibrium probability was considered by Maxwell and by Boltzmann, who succeeded in reaching a fundamental transport equation.

We initially discuss the two Gibbs variational principles that lead to the equilibrium probability distributions of an isolated system and of a system in contact with a heat reservoir. The Gibbs inequality for the free energy has been rediscovered and used by several investigators [¹⁴, ¹⁵]. In the quantum mechanical version, it is known as the Bogoliubov inequality. This Gibbs-Bogoliubov inequality has been used in the 1970s to obtain the mean-field behavior of the well-known Blume-Emery-Griffiths (BEG) model [¹⁶]. A few years later, by the suggestion of Luiz Guimarães Ferreira, we have taken advantage of the Gibbs inequality to analyze some temperature and applied field effects in spin models, which are not obtained in the context of simple mean-field approximations, but which were detected in experimental results in our laboratory of low temperature physics [¹⁷]. Nowadays, the use of the Gibbs-Bogoliubov inequality to obtain mean-field results is already incorporated in standard textbooks of statistical mechanics [¹⁸, ¹⁹]. We then present a brief discussion of our early use of this Gibbs-Bogoliubov inequality to analyze the thermodynamic behavior of the axial-next-nearest-neighbor Ising (ANNNI) model [²⁰], which is known to display one of the richest phase diagrams of the literature. Besides establishing the full phase diagram, it has been feasible to use our variational method to go beyond a first mean-field approximation, and check the effects of the introduction of some statistical fluctuations [²¹].

In the final sections, we turn to the analysis of the dynamic behavior. However, instead of using the Boltzmann approach, we introduce a more general formalism, which we call Boltzmann-Kolmogorov equation. We point out that the Boltzmann equation governs the evolution of the one-particle probability distribution. In contrast, our proposal of a Boltzmann-Kolmogorov equation refers to the probability associated with all particles of the system.

2. Gibbs First Variational Principle

The Gibbs probability distribution refers to a mechanical system whose states define a phase space $(q,p)$, where $q$ represents the set of position coordinates and $p$ is the set of momenta. For an isolated system, the Gibbs distribution ${\rho }_m$ inphase space $(q,p)$ is given by the microcanonical form [³, ¹²],

(3) $${\rho }_m=\{\begin{array}{cc}\frac{1}{W},& \mathrm{if}E\le \mathscr{H}\le E+\Delta E,\\ 0,& \text{otherwise,}\end{array}$$

where $\mathscr{H}=\mathscr{H}(q,p)$, with $E\le \mathscr{H}\le E+\Delta E$, is the energy function, and $W$ is a normalization constant,

(4) $$W={\int }_R\mathit{d}q\mathit{d}p,$$

which can be understood as the volume of the region $R$ of phase space defined by $E\le \mathscr{H}\le E+\Delta E$. The entropy is given by the Gibbs expression,

(5) $$S=-k\int \rho \ln \rho dqdp,$$

where $k$ is the Boltzmann constant, and $\rho (q,p)$ is any probability distribution that vanishes outside the region $R$ of the phase space $(q,p)$. The function $S$ reaches the largest value when $\rho$ is the microcanonical Gibbs probability distribution, ${\rho }_m$. The largest value of $S$ is the actual value $S_m$ for the system at equilibrium, which is given by

(6) $$S_m=-k\int {\rho }_m\ln {\rho }_{m}dqdp=k\ln W.$$

Therefore, we write

(7) $$S_m\ge S,$$

which is the expression of the Gibbs first variational principle.

We now demonstrate this result according to the arguments of Gibbs in his influential book on the principles of statistical mechanics [³]. We then introduce a quantity $W^{\prime }=W^{\prime }(q,p)$ and write the density in phase space,

(8) $$\rho =\frac{1}{W^{\prime }},E\le \mathscr{H}\le E+\Delta E.$$

The normalization of $\rho$ leads to the condition

(9) $${\int }_R\frac{1}{W^{\prime }}dqdp=1.$$

Therefore, using this notation, we have to show that

(10) $$-{\int }_R\frac{1}{W}\ln \left(\frac{1}{W}\right)\mathrm{}dqdp\ge -{\int }_R\frac{1}{W^{\prime }}\ln \left(\frac{1}{W^{\prime }}\right)\mathrm{}dqdp,$$

which can also be written as

(11) $$\ln W\ge {\int }_R\frac{1}{W^{\prime }}\ln W^{\prime }dqdp,$$

from which we also write

(12) $${\int }_R\frac{1}{W^{\prime }}\ln W^{\prime }dqdp\le {\int }_R\frac{1}{W^{\prime }}\ln Wdqdp,$$

where we have used equation (9) and we have taken into account that $W$ is constant. This inequality can also be written as

(13) $${\int }_R\frac{1}{W^{\prime }}\ln \left(\frac{W}{W^{\prime }}\right)\mathrm{}dqdp\ge 0,$$

since $W$ is a constant related to the constant microcanonical density ${\rho }_m$. Taking into account that $W$ is a constant, we finally write

(14) $${\int }_R\left[\frac{1}{W^{\prime }}\ln \left(\frac{W}{W^{\prime }}\right)+\frac{1}{W}-\frac{1}{W^{\prime }}\right]\mathrm{}dqdp\ge 0.$$

To prove this final relation, it is sufficient to resort to a well-known algebraic property of the convex logarithmic functions. In fact, it easy to see that the positivity of this last equation comes from the inequality

(15) $$a\ln a\ge a-1,\text{for}a=\frac{W}{W^{\prime }}>0.$$

3. Gibbs Second Variational Principle

For a system at constant temperature, the Gibbs equilibrium distribution is called canonical. It is given by [³, ¹²]

(16) $${\rho }_c=\frac{1}{Z}{e}^{-\mathscr{H}/kT},$$

where $Z$ is a normalization constant,

(17) $$Z=\int e^{-\mathscr{H}/kT}\mathrm{}dqdp.$$

This distribution corresponds to the maximum value of $S$, given by the Gibbs formula for the entropy, (eq.5), among the probability distributions such that

(18) $$U=\int \mathscr{H}\rho \mathrm{}dqdp$$

is equal to a value $E$ given by

(19) $$\int \mathscr{H}{\rho }_c\mathrm{}dqdp=E.$$

We now use the technique of Lagrange multipliers, which is equivalent to finding the minimum of $F=U-TS$, where $T$ is understood as a Lagrange multiplier. The explicit form of $F$ is given by

(20) $$F=\int \mathscr{H}\rho \mathrm{}dqdp+kT\int \rho \ln \rho dqdp,$$

and can be understood as a functional of $\rho$. Denoting by $F_c$ the value of $F$ for which $\rho$ is equal to the canonical distribution ${\rho }_c$, the second Gibbs variational principle may be written as

(21) $$F_c\le F.$$

It should be pointed out that this inequality is valid for any distribution $\rho$ and not just for the particular case in which the integral (18) is equal to $E$. This is a consequence of the introduction of the Lagrange multiplier, which makes the variations in $\rho$ independent and not restricted to the condition that the integral (18) is equal to $E$.

We remark that $F_c$ is given by

(22) $$F_c=\int \mathscr{H}{\rho }_c\mathrm{}dqdp+kT\int {\rho }_c\ln {\rho }_{c}dqdp.$$

Using (16), we obtain

(23) $$F_c=-kT\ln Z.$$

If we write the density in the form

(24) $$\rho =\frac{1}{Z_0}{e}^{-{\mathscr{H}}_0/kT},$$

where $F_0=-kT\ln Z_0$, the function $F$ can be written as

(25) $$F=F_0+⟨\mathscr{H}-{\mathscr{H}}_0⟩,$$

where the average $⟨\dots ⟩$ is calculated by using the distribution $\rho$. The inequality thus can be written in a very convenient form,

(26) $$F_c\le F_0+⟨\mathscr{H}-{\mathscr{H}}_0⟩.$$

The second Gibbs variational principle is usually written in the form of equation (26). It has been adapted to quantum statistical mechanics by Peierls [¹⁴] and by Bogoliubov [¹⁵]. In both versions, the principle was used by Ferreira, Salinas and Oliveira [¹⁷] for the formulation of variational methods for obtaining the properties of cooperative statistical models that exhibit phase transitions and critical phenomena.

To demonstrate the second Gibbs variational principle, we start by writing $Z$ in the form

(27) $$Z=Z_0\int e^{-\beta \left(\mathscr{H}-{\mathscr{H}}_0\right)}\rho \mathrm{}dqdp,$$

which is equivalent to

(28) $$Z=Z_0⟨e^{-\beta \left(\mathscr{H}-{\mathscr{H}}_0\right)}⟩$$

where $\beta =1/kT$, and the average is calculated using the distribution $\rho$. We then use the algebraic inequality [⁵]

(29) $$⟨e^{-x}⟩\ge e^{-⟨x⟩},$$

and write

(30) $$⟨e^{-\beta \left(\mathscr{H}-{\mathscr{H}}_0\right)}⟩\ge e^{-\beta ⟨\mathscr{H}-{\mathscr{H}}_0⟩}.$$

If we replace this result in the previous equation, and take the logarithm of both sides, we finally find that

(31) $$\ln Z\ge \ln Z_0-\beta ⟨\mathscr{H}-{\mathscr{H}}_0⟩.$$

Taking into account that $F_c=-kT\ln Z$ and $F_0=kT\ln Z_0$, we finally obtain the inequality (26).

4. Mean-field Treatment of the ANNNI Model

We now use the second Gibbs inequality to establish the phase diagram of a paradigmatic Ising model on a cubic lattice, with ferromagnetic nearest-neighbor interactions and the addition of competitive antiferromagnetic second-neighbor interactions along an axial direction, which has been called Axial-Next-Nearest-Neighbor Ising or ANNNI model.

In its simplest form, the Hamiltonian of this ANNNI model is given by

(32) $$\mathscr{H}=\sum _{x,y,z}[-J_1{S}_{x,y,z}(S_{x+1,y,z}+S_{x,y+1,z})-J_1{S}_{x,y,z}{S}_{x,y,z+1}-J_2{S}_{x,y,z}{S}_{x,y,z+2}],$$

where $S_{x,y,z}=\pm 1$ is an Ising spin on the $(x,y,z)$ site of a cubic lattice with unit lattice parameter. Interactions between first neighbors are ferromagnetic ($J_1>0$), and interactions between second neighbors along the $z$ direction favor an antiferromagnetic alignment ($J_2<0$). It is known that this axial competition between ferro and antiferro alignments gives rise to a Lifshitz point and a succession of modulated phases in a phase diagram in terms of temperature $T$ and the parameter of competition $p=-J_2/J_1>0$[²⁰].

Mean-field solutions for this problem are based on the Gibbs inequality, given by equation (26), with the trial Hamiltonian

(33) $${\mathscr{H}}_0=-\sum _z{h}_1\left(z\right)=-\sum _{x,y}\sum _z{\eta }_z^{\left(1\right)}{S}_{x,y,z},$$

where $\left\{{\eta }_z^{\left(1\right)}\right\}$ is a set of trial fields to be determined. The solutions of this mean-field problem have been published in a pioneer paper by Yokoi, Coutinho-Filho and Salinas [²⁰], which also contains a complete analysis of the $T-p$ phase diagram, including connections with other calculations and with a Landau expansion in the vicinity of the paramagnetic border.

We now use the method proposed by Ferreira, Salinas and Oliveira [¹⁷] to go beyond the mean-field results. In the pair approximation, which corresponds to the first stage of the method, we consider a certain number of pair interactions in a modified trial Hamiltonian,

(34) $${\mathscr{H}}_0=-\sum _z\left[\sum _{x,y}{h}_1\left(z\right)+\sum _{x,y}^{\left(1\right)}{h}_2\left(z\right)+\sum _{x,y}^{\left(2\right)}{h}_2^{\prime }\left(z\right)\right],$$

where $h_1\left(z\right)$ is the usual one-site term, and the independent two-site terms are given by

(35) $$h_2\left(z\right)=J_1{S}_{x,y,z}{S}_{x+1,y,z}+{\eta }_z^{\left(2\right)}\left[S_{x,y,z}+S_{x+1,y,z}\right]$$

and

(36) $$h_2^{\prime }\left(z\right)=J_1{S}_{x,y,z}{S}_{x,y+1,z}+{\eta }_z^{\left(2\right)}\left[S_{x,y,z}+S_{x,y+1,z}\right],$$

with the interactions restricted to a certain number of nearest-neighbor sites belonging to the same $x-y$ plane.

Now it is straightforward to write

(37) $$G_0=-kT{n}_1\sum _z\ln \left[2\cosh \beta {\eta }_z^{\left(1\right)}\right]-kT{n}_2\sum _z\ln \left[2\exp \left(\beta J_0\right)\cosh \beta {\eta }_z^{\left(2\right)}+2\exp \left(-\beta J_0\right)\right],$$

where $n_1=N^2-2n_2$ is the number of independent sites, $n_2$ is the number of isolated pairs, and $N^2$ is the number of independent sites in each $x,y$ plane. We also write

(38) $$\begin{array}{l}⟨\mathscr{H}-{\mathscr{H}}_0⟩=-\left(2N^2-n_2\right)J_0\sum _z{m}_z^2\\ -\frac{1}{2}{N}^2{J}_1\sum _z{m}_z\left(m_{z-1}+m_{z+1}\right)\\ -\frac{1}{2}{N}^2{J}_{21}\sum _z{m}_z\left(m_{z-2}+m_{z+2}\right)\\ +\sum _z\left[N^2{\eta }_z^{\left(1\right)}+2n_2\left({\eta }_z^{\left(2\right)}-{\eta }_z^{\left(1\right)}\right)\right]m_z.\end{array}$$

From the minimization of this trial free energy, it is straightforward to write equations of state and discuss all of the delicate features of the phase diagrams. At this point the reader is invited to check the detailed results in the paper by Tomé and Salinas [²¹]. However, there is a delicate question about the choice of the number $n_2$ of isolated pairs. According to our previous work, Tomé and Salinas resort to the first few and relatively easy to obtain terms of the exact high-temperature expansion of the free energy. According to this choice, we should take $n_2=2N^2$, which corresponds to an analytic continuation of our expressions, and which considerably simplifies the equations. Incidentally, this kind of choice also leads to the well known Bethe-Peierls equation of state for the simple Ising ferromagnet.

In conclusion, according to Tomé and Salinas [²¹], the general features of the variational layer-by-layer mean-field phase diagram of the ANNNI model are not changed by the inclusion of some pair spin interactions. The paramagnetic lines are depressed, in agreement with analyses on the basis of high-temperature series expansions. In the phase diagram in terms of $p$ and $T$, the paramagnetic-modulated and the paramagnetic-ferromagnetic critical lines still meet smoothly at a Lifshitz point with the first-order ferromagnetic-modulated transition line. The more refined mean-field calculations show that the Lifshitz point, as in the experiments on the magnetic compound MnP, is and inflection point of the paramagnetic border. Numerical calculations show that the main commensurate phase are stable under the introduction of some spin fluctuations.

5. Liouville Equation

The search for an evolution equation that leads to the final equilibrium state of a system was carried out by Boltzmann within the context of the classical kinetic theory of gases. Boltzmann proposed a nonlinear equation that governs the time evolution of the one-particle probability distribution.

Let us call $x_i$ the collection of the Cartesian components of the position and the momentum of particle $i$. The probability distribution $\rho \left(x\right)$ is then a function of all variables $x_i$ which we are denoting by $x$. The one-particle distribution function $f_i\left(x_i\right)$ considered by Boltzmann can be understood as the marginal distribution given by

(39) $$f_i\left(x_i\right)=\int \rho \left(x\right)\prod _{j\left(\ne i\right)}d{x}_j,$$

where the integration is over all variables $x_j$ except $x_i$. One of the most relevant results of Boltzmann refers to the behavior of a function $S_B$, given by

(40) $$S_B=-k\int f_i\left(x_i\right)\ln f_i\left(x_i\right)d{x}_i,$$

which has been shown to be an increasing function of time and to reach its maximum value at equilibrium. This has been known as the $H$-theorem. As $S_B$ refers to the distribution of one particle, we are tempted to say that the entropy of the system is $N{S}_B$ where $N$ is the number of particles. However, for a system consisting of interacting particles, the entropy $S$ defined by the Gibbs formula (5) cannot be $N{S}_B$.

An alternative to the Boltzmann equation is the Liouville equation [¹²],

(41) $$\frac{\partial \rho }{\partial t}=\{\mathscr{H},\rho \}$$

where the right-hand side are the Poisson brackets,

(42) $$\{\mathscr{H},\rho \}=\sum _i\left(\frac{\partial \mathscr{H}}{\partial q_i}\frac{\partial \rho }{\partial p_i}-\frac{\partial \mathscr{H}}{\partial p_i}\frac{\partial \rho }{\partial q_i}\right),$$

which govern the time evolution of $\rho$. The Liouville equation is understood as describing an isolated system because the energy function $\mathscr{H}(q,p)$ is strictly conserved along a path in phase space. However, using the Liouville equation one can show that the entropy $S$ given by the Gibbs formula (5) does not increase in time. In fact, it is constant in time. It is easy to show this result by writing the Gibbs entropy,

(43) $$S=-k\int \rho \ln \rho dx,$$

and taking the derivative with respect to time,

(44) $$\frac{dS}{dt}=-k\int \frac{\partial \rho \left(x\right)}{\partial t}\ln \rho \left(x\right)dx.$$

It should be remarked that there is a second term, but it vanishes because $\rho$ is normalized. Using the Liouville equation, we find

(45) $$\frac{dS}{dt}=-k\int \{\mathscr{H},\rho \}\ln \rho \left(x\right)dx.$$

If we integrate by parts, it is not difficult to show that

(46) $$\frac{dS}{dt}=0.$$

Therefore, in this formulation $S$ is constant in time. We then have to abandon the Liouville equation and try to look for another formalism that should be capable of predicting an increase of entropy.

6. Boltzmann-Kolmogorov Equation

The assumptions contained in the Boltzmann reasoning to reach his transport equation can be retrospectively understood if we consider a Markov stochastic process [²², ²³]. The Boltzmann assumption was called collision number hypothesis (Stosszahlansatz) by the Ehrenfests, in their famous review of the Boltzmann method [²⁴]. The crucial point of the Boltzmann derivation is the introduction of the conditional probability that two colliding particles have certain velocities given the velocities of the particles at an earlier time. Referring to two particles $i$ and $j$, this quantity is proportional to the rate $w_{ij}(x_i^{\prime },x_j^{\prime };x_i,x_j)$ of the transition $(x_i,x_j)\to (x_i^{\prime },x_j^{\prime })$ connecting the states of two particles. The total rate $w(x^{\prime },x)$ of a transition $x\to x^{\prime }$ is the sum of the transition rates associated with two particles, that is,

(47) $$w(x^{\prime },x)=\sum _{ij}{w}_{ij}(x_i^{\prime },x_j^{\prime };x_i,x_j).$$

Given two states $x$ and $x^{\prime }$ we assume that just one of the several terms of the right-hand side of this equation is nonzero.

Instead of proceeding according to Boltzmann, we may use the transition rates to establish a Kolmogorov equation [²⁵, ²⁶], which has been proposed to govern the evolution of the probability distributions of a Markovian stochastic process. We then write

(48) $$\frac{\partial \rho }{\partial t}=\int \left[w(x,x^{\prime })\rho \left(x^{\prime }\right)-w(x^{\prime },x)\rho \right]d{x}^{\prime }.$$

However, this equation includes only transitions due to a stochastic force. To be more precise, it is restricted to forces that we assume to be stochastic, as in the collisions of two hard spheres. To include deterministic transitions coming from conservative forces, we should add the Poisson term of the Liouville equation. The evolution equation is then written as

(49) $$\frac{\partial \rho }{\partial t}=\{\mathscr{H},\rho \}+\int \left[w(x,x^{\prime })\rho \left(x^{\prime }\right)-w(x^{\prime },x)\rho \left(x\right)\right]d{x}^{\prime }.$$

Boltzmann did not obtain either equation (49) or equation (48). He obtained an equation which can retrospectively be understood as an approximation to equation (49), which consists in writing $\rho$ as a product of the probability distribution associated with each particle. This assumption, which is implicit in Boltzmann writings, was called hypothesis of molecular disorder (molekularen Unordnung) in the well-known review of the Ehrenfests [²⁴].

To go beyond the Boltzmann proposal, we keep using the essential properties of the transition rates which were already used by Boltzmann and by Maxwell. The rates $w(x^{\prime },x)$ are in accordance with the conservation of energy and momentum, that is, $w(x^{\prime },x)$ is nonzero only when the energy $\mathscr{H}\left(x^{\prime }\right)$ is equal to $\mathscr{H}\left(x\right)$. The momentum associated with state $x^{\prime }$ is equal to the momentum of state $x$. Another essential property that is crucial for the development of the present approach is the reversal property, $w(x,x^{\prime })=w(x^{\prime },x)$. This last property leads to the following form of Boltzmann-Kolmogorov equation,

(50) $$\frac{\partial \rho }{\partial t}=\{\mathscr{H},\rho \}+\int w(x,x^{\prime })\left[\rho \left(x^{\prime }\right)-\rho \left(x\right)\right]d{x}^{\prime }.$$

We remark that in 1956 Kac [²⁷] introduced a similar equation, without the Poisson term, and using transition rates restricted to the original Boltzmann proposal.

From the property of energy conservation we conclude that the trajectory of a representative point in phase space obeys the relation $\mathscr{H}\left(x\right)=E$, which is a constant for all points $x$ of this trajectory. Therefore, $\rho \left(x\right)$ will be nonzero only when $\mathscr{H}\left(x\right)=E$. Taking into account that in the stationary state $\rho \left(x\right)$ does not depend on time, we concludee that the probability distribution at the stationary state is the microcanonical Gibbs distribution, given by (3).

We now determine the time variation of the entropy. If we replace equation (50) in equation (44), we have

(51) $$\frac{dS}{dt}=-k\int w(x,x^{\prime })\left[\rho \left(x^{\prime }\right)-\rho \left(x\right)\right]\ln \rho \left(x\right)dxd{x}^{\prime }.$$

As we have shown, the term associated with the Poisson brackets vanishes. We then write this expression in the equivalent form

(52) $$\frac{dS}{dt}=\frac{k}{2}\int w(x,x^{\prime })\left[\rho \left(x^{\prime }\right)-\rho \left(x\right)\right]\ln \frac{\rho \left(x^{\prime }\right)}{\rho \left(x\right)}dxd{x}^{\prime }.$$

Using the inequality $\left(a-b\right)\ln \left(a/b\right)\ge 0$, we conclude that the integrand is nonnegative because $w(x,x^{\prime })\ge 0$. Therefore,

(53) $$\frac{dS}{dt}\ge 0$$

which we call the first Boltzmann irreversible theorem.

If $S_v$ is the entropy associated with an initial distribution ${\rho }_v$ and if $S_m$ is the entropy of the equilibrium state, which is the microcanonical Gibbs distribution ${\rho }_m$, then from (53) it follows that $S_m\ge S_v$. We remark that this is the representation of the first Gibbs variational principle, given by equation (7).

7. Contact with a Thermal Reservoir

The transition rates that we have considered in the last section describe the stochastic interaction of molecules. They are in agreement with the conservation of energy and momentum in addition to the reversal property, which from now on we denote by $w^b\left(x^{\prime }x\right)$. The conservation of energy of the transition rate $w^b\left(x^{\prime }x\right)$ allows us to say that the system described by the Boltzmann-Kolmogorov equation (49) is an isolated system. Now we wish to describe an open system in contact with a heat reservoir at a temperature $T$. In this case, at long times, the system will be found in the equilibrium state with the Gibbs canonical probability distribution.

The transition rate $w^c(x^{\prime },x)$ that describes the contact with the heat reservoir is assumed to obey the property

(54) $$\frac{w^c(x^{\prime },x)}{w^c(x,x^{\prime })}=\frac{\exp \left(-\mathscr{H}\left(x^{\prime }\right)/kT\right)}{\exp \left(-\mathscr{H}\left(x\right)/kT\right)}.$$

The Boltzmann-Kolmogorov equation has now two stochastic terms corresponding to the internal stochastic interactions described by the rate $w^b\left(x^{\prime }x\right)$ and the stochastic interaction due to the contact with the heat reservoir described by the rate $w^c(x^{\prime },x).$We then write

(55) $$\frac{\partial \rho }{\partial t}=\{\mathscr{H},\rho \}+\int w^b(x,x^{\prime })\left[\rho \left(x^{\prime }\right)-\rho \left(x\right)\right]d{x}^{\prime }+\int \left[w^c(x,x^{\prime })\rho \left(x^{\prime }\right)-w^c(x^{\prime },x)\rho \left(x\right)\right]d{x}^{\prime }.$$

We remark that this equation, without the first integral, was introduced by Bergmann and Lebowitz [²⁸] in 1955 to study nonequilibrium processes. They used transition rates with the property (54) and also considered the case where the system is in contact with several reservoirs at distinct temperatures. However, at this point we wish to make connections with the Gibbs canonical distribution, which occurs when there is just one reservoir.

It is straightforward to show by a simple substitution that the Gibbs canonical distribution (16) is the equilibrium distribution of equation (55). Initially, we observe that the Poisson brackets vanishes because ${\rho }_c$ is a function of $\mathscr{H}$. To see that the second integral in (55) vanishes, it is sufficient to write (54) as

(56) $$w^c(x^{\prime },x){\rho }_c\left(x\right)=w^c(x,x^{\prime }){\rho }_c\left(x^{\prime }\right).$$

Now, taking into account that $w^b(x,x^{\prime })$ vanishes when $\mathscr{H}\left(x^{\prime }\right)$ is not equal to $\mathscr{H}\left(x\right)$, we obtain the identity

(57) $$w^b(x^{\prime },x)\left[{\rho }_c\left(x^{\prime }\right)-{\rho }_c\left(x\right)\right]=0.$$

Using this result we see that the first integral in (55) vanishes.

In the previous section, we have seen that the energy was strictly conserved. In the present case, this no longer happens due to the contact with the reservoir with which the system exchanges energy. In fact, the energy exchanged is called heat because we are not considering external work done on the system. Let $U=⟨H\left(x\right)⟩$ be the average energy,

(58) $$U=\int H\left(x\right)\rho \left(x\right)dx.$$

Deriving this equation with respect to time we have

(59) $$\frac{dU}{dt}=\int H\left(x\right)\frac{\partial \rho \left(x\right)}{\partial t}dx$$

Using the evolution equation (55), we obtain

(60) $$\frac{dU}{dt}=\int [H\left(x^{\prime }\right)-H\left(x\right)]w^c(x^{\prime },x)\rho \left(x\right)]dx{}^{\prime }dx,$$

and we remark that the terms associated with the Poisson brackets and with the internal stochastic processes vanish. Using the property (54), we may write

(61) $$\frac{dU}{dt}=-kT\int w^c(x^{\prime },x)\rho \left(x\right)\ln \frac{w^c(x^{\prime },x)}{w^c(x,x^{\prime })}d{x}^{\prime }dx.$$

This expression means that the change in the energy of the system is entirely due to the contact with the heat reservoirs, as it should be expected.

Let us determine the time variation of the entropy. Replacing (55) in the expression (44) we have

(62) $$\frac{dS}{dt}=k\int w^b(x^{\prime },x)\rho \left(x\right)\ln \frac{\rho \left(x\right)}{\rho \left(x^{\prime }\right)}dxd{x}^{\prime }+k\int w^c(x^{\prime },x)\rho \left(x\right)\ln \frac{\rho \left(x\right)}{\rho \left(x^{\prime }\right)}dxd{x}^{\prime },$$

and we remark that the term containing the Poisson brackets vanishes.

We now define the free energy $F$ by $F=U-TS$, and consider the time derivative of $F$,

(63) $$\frac{dF}{dt}=\frac{dU}{dt}-T\frac{dS}{dt}.$$

We demonstrate that

(64) $$\frac{dF}{dt}\le 0,$$

which we call the second Boltzmann irreversible theorem. This result was also demonstrated by Bergmann and Lebowitz in his paper of 1955 using the equation they introduced [²⁸], which corresponds to equation (55) without the first integral.

Replacing (61) and (62) in (63), we find

(65) $$\begin{array}{l}\frac{dF}{dt}=-kT\int w^b(x^{\prime },x)\rho \left(x\right)\ln \frac{\rho \left(x\right)}{\rho \left(x^{\prime }\right)}dxd{x}^{\prime }\\ -kT\int w^c(x^{\prime },x)\rho \left(x\right)\ln \frac{w^c(x^{\prime },x)\rho \left(x\right)}{w^c(x,x^{\prime })\rho \left(x^{\prime }\right)}d{x}^{\prime }dx,\end{array}$$

which can be written as

(66) $$\begin{array}{l}\frac{dF}{dt}=-\frac{kT}{2}\int w^b(x^{\prime },x)\left[\rho \left(x\right)-\rho \left(x^{\prime }\right)\right]\ln \frac{\rho \left(x\right)}{\rho \left(x^{\prime }\right)}dxd{x}^{\prime }\\ -\frac{kT}{2}\int [w^c\left(x^{\prime c}(x,x^{\prime })\rho \left(x^{\prime }\right)\right]\ln \frac{w^c(x^{\prime },x)\rho \left(x\right)}{w^c(x,x^{\prime })\rho \left(x^{\prime }\right)}d{x}^{\prime }dx.\end{array}$$

Using again the inequality $\left(a-b\right)\ln \left(a/b\right)\ge 0$ we see that both integrals are nonnegative, and we conclude that the right hand side of this equation is negative or zero, leading us to the inequality (64). We remark that $dF/dt$ vanishes when $\rho$ equals ${\rho }_c$, that is, in equilibrium. Therefore, $F$ decreases monotonically in time and reaches its minimum values at equilibrium. Denoting by $F_c$ the value of $F$ when $\rho$ equals ${\rho }_c$, then $F_c\le F$, which is the expression of the second Gibbs variational principle, given by equation (21).

8. Conclusions

We have shown that the Gibbs variational principles are direct related to the Boltzmann irreversible theorems. They involve a thermodynamic potential, which is the entropy $S$ in the case of an isolated system. According to the Boltzmann theorem, $S$ increases monotonically. Since $S$ is bounded by the Gibbs principle, then $S$ reaches the equilibrium value in the long run. In the case of a system at constant temperature, the associated thermodynamic potential is the free energy $F$. In this case, the Boltzmann theorem states that $F$ decreases monotonically; since $F$ is bounded by the Gibbs principle, then $F$ reaches the equilibrium value in the long run.

If we define $\psi =F-F_c$ then the second Gibbs principle and the Boltzmann theorem are written as

(67) $$\Psi \le 0,\frac{d\Psi }{dt}\le 0.$$

and the explicit form of $\psi$ is given by

(68) $$\Psi =kT\int \rho \ln \frac{\rho }{{\rho }_c}dx.$$

The inequalities (67) are the properties that define a Lyapunov function. In accordance with the Lyapunov theorem, the stability of the stationary solution of a set of ordinary equations of first order in time is proved if we construct a function that obeys the inequalities of equation (67). In this case, the set of equations is the Boltzmann-Kolmogorov equation and the Lyapunov theorem is nothing more than the combination of Gibbs variational principle and Boltzmann irreversible theorem.

Data Availability

This work is purely theoretical and does not rely on any datasets. No data were generated or analyzed in this study.

Acknowledgments

This paper is dedicated to Luiz Guimarães Ferreira, who extensively used variational methods, including applications to statistical mechanics in collaboration with the authors [¹⁷].

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Edited by

• Editor-in-Chief:
  Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  10 Apr 2026
• Date of issue
  2026

History

• Received
  09 Feb 2026
• Accepted
  07 Mar 2026