The H-Theorem and Entropy

In this section a relationship between the equilibrium value of H and entropy, S, is derived [12] (p. 78). 

Considering a gas at uniform steady state, (2.216) can be reformulated in terms of the absolute Maxwellian distribution (2.224), hence  

Assuming that the kinetic theory definition of temperature and the thermodynamic counterpart are consistent, the entropy variable may be considered a function of p and T as obtained combining the first and second law of thermodynamics [32]. The entropy of the gas is then given by [12] (p. 41)  

This relation connects Hq with the entropy when the gas is in a uniform steady state  

It is commonly argued that the constant reflects the arbitrariness of the zero point of entropy. 

In this section a relationship between the equilibrium value of H and entropy, S, is derived [20] (p. 78). For a gas at uniform steady state, the H(t) quantity defined by (2.235) can be expressed in terms of n and T. By use of the absolute Maxwellian distribution (2.243) and the energy equipartition theorem yields  

Consideringagaswithtotalmass,M,thevolumeoccupiedbythegasis V= y= Integrating H(t) over the gas volume, it follows that  

the small change in entropy caused by the change of state can be expressed as [20] (see p. 41 and p. 81)  

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