Local Boltzmann entropy

Perhaps one of the most important consequences of the Boltzmann equation, discovered by Boltzmann himself, was that it led to a kinetic molecular theory basis for the law of entropy increase. This law is contained within the so-called Boltzmann -theorem. Here we will work directly with the Boltzmann entropy Sb rather than the i/-function used in Boltzmann s original work.1 We define a local Boltzmann entropy Ssfri, t) as... 

It is easiest to start with the configurational entropy, Sc- Suppose that the number of defects, which is equal to the number of mobile (localized) holes or electrons, is nd and moreover that only one type of mobile carrier, either holes or electrons, is present. The configurational entropy Sc is given by using the Boltzmann formula ... 

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. 

Keywords Planck system of units, L-regions (local regions), O-regions (observable regions), comoving frame, Second Law of Thermodynamics, heat death, Planck power versus heat death, low-entropy boundary conditions versus heat death, kinetic versus thermodynamic control, kinetic control versus heat death, minimal Boltzmann brains, extraordinary observers. 

Whereas the MC method selects configurations correctly with the Boltzmann PD, it does not provide the value of this PD, and therefore the absolute entropy cannot be obtained in a direct manner with Eq. [29]. The difficulty stems from the fact that if the simulation starts from configuration i and after t MC steps reaches /, one knows the probability density of the specific avenue i j that was chosen in the simulation. However, to obtain the PD of / one has to sum up the probability densities of all the large number of possible avenues / in t MC steps. Yet as mentioned in the Introduction, P (x) [like E(x)j is also written on the configurations x and can be deciphered approximately using the local states method or the hypothetical scanning method (discussed later). A direct estimation of S (hence F) by Eq. [29] becomes possible with these methods. 

The first two summations represent the flow of entropy and the last summation corresponds to the local entropy production. If we substitute dfyjdt from the Boltzmann equation 66 and rewrite with use of the relation dVydv = dv ydv f, we find... 

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