Energy equipartition relationship

This result implies that the energy equipartition relationship of Eq. (2.S) applies as well as the general definitions of Chapter I. Note that for Af m the variable turns out to be coupled weakly to the thermal bath. This condition generates that time-scale separation which is indispensable for recovering an exponential time decay. To recover the standard Brownian motion we have therefore to assiune that the Brownian particle be given a macroscopic size. In the linear case, when M = w we have no chance of recovering the properties of the standard Brownian motion. In the next two sections we shall show that microscopic nonlinearity, on the contrary, may allow that the Markov characters of the standard Brownian motion be recovered with increasing temperature. 

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 ... 

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