<![CDATA[<b>On the Markovian limit in a kinetic theory for dissipative systems</b>]]>

Abstract

We reconsider a nonlinear quantum kinetic theory which is built within the context of a nonequilibrium statistical ensemble formalism. This is the Nonequilibrium Statistical Operator Method based on a variational principle, namely, the Maximization of the InformationalStatistical Entropy, and referred to as MaxEnt-NESOM. It may be considered as encompassed within the framework of E. T. Jaynes' Predictive Statistical Mechanics. These theory has an ample domain of application covering a large class of experimental conditions. We consider a particular and quite important limiting case, consisting in the Markovian approximation. For illustration we applied it to the study of a spin system in interaction with the lattice. The presentation is an extended and detailed version of a Brief Report published in Phys. Rev. E 57, 3637-3640 1998.

Evolution of dissipative processes via a statistical thermodynamic approach. I. Generalized Mori–Heisenberg–Langevin equations

“…For that purpose we start introducing the densities of particles n (r) and of energy ĥ (r). As described elsewhere, 10 reiterated application of the law of Eq. ͑4͒ requires that, for the closure condition to be satisfied, we must introduce as basic dynamical variables the fluxes of all order of both densities, Î n [r] (r) and Î h [r] (r), with rϭ1,2,3, .…”

“…which is a generating vector for the operators of the fluxes of any order of the dynamical densities. 10 We can see that this generating vector ͑with dimensions of velocity͒ consists of a first term (lϭ0) which is the group velocity of the particle in state ͉k͘ ͑and which is the only one that survives for linear or quadratic dispersion relations, i.e., k ϭs͉k͉ of the soundwave type or ប k ϭប 2 k 2 /2m of the ideal-gas type͒, plus additional terms when contributions higher than the quadratic term are present in k , and which are proportional to even powers of the wave vector Q.…”

“…͑20͒ above is used, represents a generalized nonequilibrium grand-canonical statistical operator. 10 In the asymptotic limit when the system initially perturbed out of equilibrium recovers final equilibrium with the reservoirs, the nonequilibrium statistical operator ⑀ (t) of Eq. ͑6͒, but in terms of of Eq.…”