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arXiv:1211.3265 · Europhysics Letters (EPL)

Onset of Fokker-Planck dynamics within a closed finite spin system

Hendrik Niemeyer, Daniel Schmidtke, Jochen Gemmer B

Published Jan 1, 2013• 13 citations
arXivHTMLPDF

Abstract

Relaxation according to Fokker-Planck equations is a standard scenario in classical statistical mechanics. It is, however, not obvious how such a Fokker-Planck dynamics may emerge within a closed, finite quantum system. We present an analysis of a system comprising sixteen spins in which spatial inhomogeneities of the magnetization relax approximately in accord with a standard Fokker-Planck equation for a Brownian particle in a parabolic potential. While a single, specfic example is addressed in detail, arguments which indicate that the above findings may hold for a wider range of systems are given.

DOI
10.1209/0295-5075/101/10010

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Relaxation according to Fokker-Planck equations is a standard scenario in classical statistical mechanics. It is, however, not obvious how such a Fokker-Planck dynamics may emerge within a closed, finite quantum system. We present an analysis of a system comprising sixteen spins in which spatial inhomogeneities of the magnetization relax approximately in accord with a standard Fokker-Planck equation for a Brownian particle in a parabolic potential. While a single, specfic example is addressed in detail, arguments which indicate that the above findings may hold for a wider range of systems are given.
Thermalization has been shown to occur in a number of closed quantum many-body systems, but the description of the actual thermalization dynamics is prohibitively complex. Here, we present a model-in one and two dimensions-for which we can analytically show that the evolution into thermal equilibrium is governed by a Fokker-Planck equation derived from the underlying quantum dynamics. Our approach does not rely on a formal distinction of weakly coupled bath and system degrees of freedom. The results show that transitions within narrow energy shells lead to a dynamics which is dominated by entropy and establishes detailed balance conditions that determine both the eventual equilibrium state and the nonequilibrium relaxation to it.
The dynamics of quantum systems subject to dissipation is a subject of fundamental importance which has recently been of great interest due to its relation to macroscopic quantum phenomena, In particular the behaviour of the magnetic flux trapped in a SQUID is an example of this. The authors derive the quantum mechanical analogue of the Fokker-Planck equation which describes such systems. The major difference between the quantum and classical Fokker-Planck equations arises from the presence of a quantum mechanical memory term.
Starting from the master equation of Glauber (1963), for the Ising system of N spins, and working in the Bragg-Williams approximation, a Fokker-Planck equation is derived for the probability that the system has an overall magnetization. The expansion of the master equation is discussed in powers of N -1/2 and it is shown that a certain choice of transition probability has to be made to get the correct width of the probability distribution. This is in contrast to the recent work of Goldstein and Scully (1973), which though starting from a Liouville equation effectively works only to the molecular field approximation which is not as satisfactory as the Bragg-Williams approximation. The dynamic magnetic susceptibility is also calculated.
The dynamics of linear quantum systems is classical in the Wigner representation, yet linear problems are often analyzed using such general techniques as influence functionals and Bogoliubov transformations. In fact, the classical equations of motion provide a simpler and more intuitive formalism for these systems. As an important example, we show that quantum Brownian dynamics in the independent oscillator model is described directly and completely by a c-number Langevin equation. The corresponding Fokker-Planck equation is always local in time, regardless of the environmental spectrum.