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Mathematics of the USSR-Izvestiya

CONVERGENCE TO AN EQUILIBRIUM STATE FOR A ONE-DIMENSIONAL QUANTUM SYSTEM OF HARD RODS

Ju M Sukhov

Published Jun 30, 1983• 1 citation
Publisher

Abstract

In this paper a theorem is proved on the convergence to an equilibrium state during the course of the time evolution of an infinite, one-dimensional system of quantum particles interacting by means of a hard-rod potential. The basic idea consists in the reduction of the time evolution of the system of hard rods to the time evolution of free particles. The case of free motion was studied earlier by O. Lanford and D. Robinson. Bibliography: 23 titles.

DOI
10.1070/im1983v021n03abeh001804

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In this paper a theorem is proved on the convergence to an equilibrium state during the course of the time evolution of an infinite, one-dimensional system of quantum particles interacting by means of a hard-rod potential. The basic idea consists in the reduction of the time evolution of the system of hard rods to the time evolution of free particles. The case of free motion was studied earlier by O. Lanford and D. Robinson. Bibliography: 23 titles.
Abstract. We study the out-of-equilibrium evolution of a strongly interacting quantum spin chain which is mapped on a system of hard rods that are coherently deposited on and removed from a lattice. We show that this closed quantum system approaches an equilibrium steady state which strongly resembles a microcanonical ensemble of classical hard rods. Starting from the fully coherent evolution equation we derive a Master equation for the evolution of the number of hard rods on the lattice. This equation does not only capture properties of the equilibrium state but also describes the dynamical non-equilibrium evolution into it for the majority of initial conditions. We analyze this in detail for hard rods of varying size. ar X iv
We study the out-of-equilibrium evolution of a strongly interacting quantum spin chain which is mapped on a system of hard rods that are coherently deposited on and removed from a lattice. We show that this closed quantum system approaches an equilibrium steady state which strongly resembles a microcanonical ensemble of classical hard rods. Starting from the fully coherent evolution equation we derive a master equation for the evolution of the number of hard rods on the lattice. This equation not only captures properties of the equilibrium state but also describes the dynamical non-equilibrium evolution into it for the majority of initial conditions. We analyze this in detail for hard rods of varying size.
We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our theorems allow to describe the dynamics of both pure states (zero temperature states) and mixed states (positive temperature states). Our results hold for all times, and give effective estimates on the rate of convergence towards the Hartree-Fock evolution. The results on pure states are based on joint works with N. Benedikter and B. Schlein, [5, 6]; while those on mixed states are based on a joint work with N. Benedikter, V. Jaksic, C. Saffirio and B. Schlein, [7].