The Boltzmann H-Theorem

In this section the elementary definitions and results deduced from the H- 

In summary, the statistical tf-theorem of kinetic theory relates to the Maxwellian velocity distribution function and thermodynamics. Most important, the Boltzmann s //-theorem provides a mechanistic or probabilistic prove for the second law of thermodynamics. In this manner, the //-theorem also relates the thermodynamic entropy quantity to probability concepts. Further details can be found in the standard references [97] [39] [12] [100] [47] [28] [61] [85]. 

In practice, during the process of developing novel models for the collision term, the //-theorem merely serves as a requirement for the constitutive relations in order to fulfill the second law of thermodynamics (in a similar manner as for the continuum models). 

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Since 1876 numerous papers have called attention to these foundations. In these papers the Boltzmann H-theorem, a central theorem of the kinetic theory of gases, was attacked. Without exception all studies so far published dealing with the connection of mechanics with probability theory grew out of the synthesis of these polemics and of Boltzmann s replies. These discussions will therefore be referred to frequently in our report. 

In other words, in approximate accordance with the original paper by Boltzmann [6], we assume that in a given volume element the expected number of collisions between molecules that belong to different velocity ranges can be computed statistically. This assumption is referred to as the Boltzmann Stosszahlansatz (German for Collision number assumption). A result of the Boltzmann H-theorem analysis is that the latter statistical assumption makes Boltzmann s equation irreversible in time (e.g., [28], sect. 4.2). 

The Boltzmann H-Theorem act upon the molecules. Thus (2.185) reduces to ... 

Shear, D. (1967). An analog of the Boltzmann H-theorem (a Liapunov function) for systems of coupled chemical reactions. J. Theor. Biol., 16, 212-28. 

A result of the Boltzmann H-theorem analysis is that the latter statistical assumption makes Boltzmann s equation irreversible in time (e.g., [39], Sect. 4.2). 

H-property function in the Boltzmann H-theorem Liquid hight in standard turbulent stirred tank (m)... 

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. 

One may also show that MPC dynamics satisfies an H theorem and that any initial velocity distribution will relax to the Maxwell-Boltzmann distribution [11]. Figure 2 shows simulation results for the velocity distribution function that confirm this result. In the simulation, the particles were initially uniformly distributed in the volume and had the same speed v = 1 but different random directions. After a relatively short transient the distribution function adopts the Maxwell-Boltzmann form shown in the figure. 

The Boltzmann //-theorem generalizes the condition that with a state ol a system represented by its distribution function /. a quantity H. defined as the statistical average of In /, approaches a minimum when equilibrium is reached. This conforms lo the Boltzmann hypothesis of distribution in the above in that S = —kH accounts for equilibrium as a consequence of collisions which change the distribution toward that of equilibrium conditions. 

The year 1872 was the year of the formulation of the famous Boltzmann equation (BE), which is one of the most important equations of statistical physics. One of the remarkable consequences of the BE is the H-theorem. Furthermore, the BE is the basic equation for transport processes in macroscopic systems. 

The subsequent attempts of Maxwell and Boltzmann to obtain a derivation reached their first conclusion in the H-theorem. Before reaching this they had irone through the following stages of development ... 

Boltzmann (1872, as one of the corollaries of the H-theorem)47—The Maxwell-Boltzmann distribution is the only distribution which can maintain its invariance,48 and any other distribution under the influence of collisions finally goes over into the Maxwellian one. 

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. 

Equations 28-35 and 28-36 are known as Newton s equations of motion. MD. simulations apply these two cquatioas to all the atoms in a molecular structure. According to the kinetic-molecular theorem, the kinetic energy is proportional to the temperature. This remarkable relationship is shown in Equation 28-37 without derivation, where M is the number of molecules, h is the Boltzmann constant, and Tis the abso-... 

Finally, the expression (78) for q(z) can be used to modify the nonlinear Poisson-Boltzmann theory in order to consider a highly charged surface [59, 60]. In this case, for the profile (h z) the new term appears which exactly reproduces the last electrostatic term in the contact theorem (73). 

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... 

Boltzmann showed that one can give an affirmative answer to the first question without actually having to construct a solution to the equation, if one assumes that a solution exists. To do this, he proved a theorem, the Boltzmann //-theorem, which states that the function H(t) defined by... 

It is very satisfactory from a macroscopic point of view that the Boltzmann equation, through the H-theorem, predicts the approach to equilibrium of an initial nonequilibrium state of the gas. However, one can raise serious objections to the //-theorem, and to the Boltzmann equation, from a microscopic point of view. The fundamental difficulty is that the Boltzmann equation is inconsistent with the laws of mechanics. The laws of mechanics require that any equation of motion describing the gas be invariant under time reversal if the particles make specular collisions with the walls. Otherwise any dynamical processes that do not involve collisions with the walls must be time reversal invariant. That is, the form of the equations of motion must be invariant if v-> —V and t —t. It is clear from an inspection of the Boltzmann equation for points far from the walls. 

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