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arXiv:1210.5821 · Physica Scripta

Equilibration of isolated macroscopic quantum systems under experimentally realistic conditions

Peter Reimann

Published Oct 19, 2012• 34 citations
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Abstract

In how far does a non-equilibrium initial ensemble evolve towards a stationary long time behavior for an isolated macroscopic quantum system?We demonstrate that deviations from a steady state indeed become unmeasurably small or exceedingly rare after initial transients have died out under the following conditions: the Hamiltonian does not exhibit exceedingly large degeneracies of energy eigenvalues and energy gaps.A large number of energy levels are significantly populated by the initial ensemble.The system is observed by a measurement device with a reasonably bound working range compared to the resolution limit.The entire experiment, ending in the quantum mechanical measurement process, can only be repeated a 'reasonable' number of times.It is argued that all these prerequisites for equilibration are fulfilled under many, if not all, experimentally realistic conditions.

DOI
10.1088/0031-8949/86/05/058512

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We investigate the equilibration of an isolated macroscopic quantum system in the sense that deviations from a steady state become unmeasurably small for the overwhelming majority of times within any sufficiently large time interval.The main requirements are that the initial state, possibly far from equilibrium, exhibits a macroscopic population of at most one energy level and that degeneracies of energy eigenvalues and of energy gaps (differences of energy eigenvalues) are not of exceedingly large multiplicities.Our approach closely follows and extends recent works by Short and Farrelly (2012 New.J. Phys.14 013063), in particular going beyond the realm of finite-dimensional systems and large effective dimensions.
We investigate the equilibration of an isolated macroscopic quantum system in the sense that deviations from a steady state become unmeasurably small for the overwhelming majority of times within any sufficiently large time interval. The main requirements are that the initial state, possibly far from equilibrium, exhibits a macroscopic population of at most one energy level and that degeneracies of energy eigenvalues and of energy gaps (differences of energy eigenvalues) are not of exceedingly large multiplicities. Our approach closely follows and extends recent works by Short and Farrelly (2012 New. J. Phys. 14 013063), in particular going beyond the realm of finite-dimensional systems and large effective dimensions.
Focusing on isolated macroscopic systems, described either in terms of a quantum mechanical or a classical model, our two key questions are: In how far does an initial ensemble (usually far from equilibrium and largely unknown in detail) evolve towards a stationary long-time behavior ("equilibration")? In how far is this steady state in agreement with the microcanonical ensemble as predicted by Statistical Mechanics ("thermalization")? In the first part of the paper, a recently developed quantum mechanical treatment of the problem is briefly summarized, putting particular emphasis on the realistic modeling of experimental measurements and non-equilibrium initial conditions. Within this framework, equilibration can be proven under very weak assumptions about those measurements and initial conditions, while thermalization still requires quite strong additional hypotheses. In the second part, an analogous approach within the framework of classical mechanics is developed and compared with the quantum case. In particular, the assumptions to guarantee classical equilibration are now rather strong, while thermalization then follows under relatively weak additional conditions.
Focusing on isolated macroscopic systems, described in terms of either a quantum mechanical or a classical model, our two key questions are how far does an initial ensemble (usually far from equilibrium and largely unknown in detail) evolve towards a stationary long-time behavior (equilibration) and how far is this steady state in agreement with the microcanonical ensemble as predicted by statistical mechanics (thermalization). A recently developed quantum mechanical treatment of the problem is briefly summarized, putting particular emphasis on the realistic modeling of experimental measurements and nonequilibrium initial conditions. Within this framework, equilibration can be proven under very weak assumptions about those measurements and initial conditions, while thermalization still requires quite strong additional hypotheses. An analogous approach within the framework of classical mechanics is developed and compared with the quantum case. In particular, the assumptions to guarantee classical equilibration are now rather strong, while thermalization then follows under relatively weak additional conditions.
We discuss the approach toward equilibrium of an isolated quantum system. For a wide class of systems, in particular, for chaotic systems, we argue that the expectation value of a local operator averaged over a time interval of length TTT is bounded by the so-called deviation function, which characterizes maximal deviation from the equilibrium of all states with a given value of energy fluctuations. This result applies to any initial state with a well-defined effective temperature. We provide numerical evidence that the bound is approximately saturated by the initial configurations with spatial inhomogeneities at a macroscopic level. In this way the deviation function establishes an explicit connection between the macroscopically observed timescales associated with the transport and the properties of microscopic matrix elements. The form of the deviation function indicates that among the ``slowest'' states which saturate the bound there are also those with arbitrarily long equilibration times.