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Mathematical Notes

Approximate nonsymmetric solutions of many-particle equations and some identities for the density matrix

Виктор Павлович Маслов, O. Yu. Shvedov

Published Sep 1, 1996• 0 citations
Publisher
DOI
10.1007/bf02320379

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We consider an open quantum many-particle system in which there are dissipative processes. The evolution of this system is described by a kinetic equation for the density matrix. From the equation describing a random Markov process in this system, we obtain an equation for the single-particle statistical operator. This equation describes the evolution of a system of identical particles in a mean-field approximation. The equation for interacting particles in thermodynamic equilibrium was obtained. The distribution function of a system of interacting electrons in metals has multivalence in a certain region of wave vectors. Among many solutions one is isotropic. Other solutions have the anisotropy of the electron distribution over the wave vectors. The anisotropy arises as a result of repulsion and attraction between electrons.
We use the mean-field approximation to simplify the master equation for sympathetic cooling of bosons. For the mean single-particle occupation numbers, this approach yields the same equations as the factorization assumption introduced in an early paper. The stationary or equilibrium solution of the resulting master equation for the one-body density matrix shows that the mean-field approximation breaks down whenever the fraction of condensate bosons exceeds 10% or so of the total. Using group-theoretical methods, we also solve the time-dependent master equation for the one-body density matrix. Given the time dependence of the mean single-particle occupation numbers, this solution is obtained by quadratures. It tends asymptotically towards the equilibrium solution.