Heat fluctuation distribution from non-equilibrium states

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Abstract
A non-equilibrium state is described by parameters alpha j=(Aj) which are ensemble averages of dynamical phase functions Aj( Gamma ). By using a projection operator technique of Zwanzig (1961), one can derive from the classical Liouville equation an integro-differential equation for g((vj)), the amplitude for the probability that the Aj have respective values vj. These previously derived results are applied to the heat flux AJ in a fluid and an approximate differential equation derived for g(v-J), where J=(AJ). The first moment of this equation yields the Catteneo-Vernotte equation (1958) with a set of auxiliary assumptions which are also employed in solving the equation for g. The author obtains g=g(0)(1+G(v-J)), where g(0)=( kappa / pi mu )3/2 exp(- mu (v-J)2/ kappa ) is the Einstein distribution, and G is a sum of products of scalar combinations of J, v-J, and Del T and Laguerre polynomials Ln(12)/(z2), where z identical to ( mu / kappa )1/2(v-J). The Einstein distribution g(0) gives correctly ( nu 2) to O(J2) when Del T=0 but not in the presence of a temperature gradient.

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