Boltzmann’s H-theorem

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

Internally equilibrated subsystems, which act as free energy reservoirs, are already as random as possible given their boundary conditions, even if they are not in equilibrium with one another because of some bottleneck. Tlius, the only kinds of perturbation that can arise and be stabilized when they are coupled are those that make the joint system less constrained than the subsystems originally were. (This is Boltzmann s H-theorem [9] only a less constrained joint system has a liigher maximal entropy than die sum of entropies from the subsystems independently and can stably adopt a different form.) The flows that relax reservoir constraints are thermochemical relaxation processes toward the equilibrium state for tlte joint ensemble. The processes by wliich such equilibration takes place are by assumption not reachable within the equilibrium distribution of either subsystem. As the nature of the relaxation phenomenon often depends on aspects of the crosssystem coupling that are much more specific than the constraints that define either reservoir, they are often correspondingly more complex than the typical processes... 

Moreover, since gaAi, g)dQg dcdci is a positive measure, the integral on the right hand side of (2.238) is either negative or zero, so H can never increase. The resulting relation is known as Boltzmann s H-theorem ... 

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