Irreversible processes, equilibrium Boltzmann

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

Boltzmann connected his ideas with those of Rudolf Clausius, who had introduced the concept of entropy in 1865. Somehow related to heat, entropy was known to increase during irreversible processes, but its exact nature was unknown. From the distribution of gas atoms, Boltzmann described a quantity—later symbolized by the letter H—which is a minimum when atoms assume a Maxwell-Boltzmann distribution. He recognized his H function as the negative of entropy, which is a maximum when the atoms reach thermal equilibrium. Thus Boltzmann offered a kinetic explanation for entropy and, more generally, a connection between the behavior of atoms and thermodynamics. 

The constant growth of entropy postulated in the second law is closely related to the following principle By the irreversible process the system is spontaneously directed towards the most frequently occurring set of micro states which corresponds to the same macro state of the system. The macroscopic state of equilibrium - where entropy is at its maximum - is the state which is linked to the largest number of different micro states. This basic concept is the basis of the Boltzmann relation that defines the entropy function S in statistical thermodynamics. 

In the 19th century the variational principles of mechanics that allow one to determine the extreme equilibrium (passing through the continuous sequence of equilibrium states) trajectories, as was noted in the introduction, were extended to the description of nonconservative systems (Polak, 1960), i.e., the systems in which irreversibility of the processes occurs. However, the analysis of interrelations between the notions of "equilibrium" and "reversibility," "equilibrium processes" and "reversible processes" started only during the period when the classical equilibrium thermodynamics was created by Clausius, Helmholtz, Maxwell, Boltzmann, and Gibbs. Boltzmann (1878) and Gibbs (1876, 1878, 1902) started to use the terms of equilibria to describe the processes that satisfy the entropy increase principle and follow the "time arrow."... 

Such an attitude to equilibrium thermodynamics - the science which revealed irreversibility of the evolution of isolated systems and asymmetry of natural processes with respect to time - is related to some circumstances that require a thorough analysis. Here we will emphasize only one of them which is the most important for imderstanding further text. It lies in the fact that the most important notion of thermodynamics, i.e. equilibrium, became interpreted exclusively as the state of rest (absence of any forces and flows in the thermodynamic system) and equilibrium processes - as those identical to reversible ones. These one-sided interpretations ignored the Galileo principle of relativity, the third law of Newton and the Boltzmann probabilistic interpretations of entropy that allow dynamic interpretations of equilibria and irreversible interpretations of equilibrium processes. 

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