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arXiv:cond-mat/0602625

On the Basis of Quantum Statistical Mechanics

Ayumu Sugita

Published Feb 27, 2006• 16 citations
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Abstract

We propose a new approach to justify the use of the microcanonical ensemble for isolated macroscopic quantum systems. Since there are huge number of independent observables in a macroscopic system, we cannot see all of them. Actually what we observe can be written in a rather simple combination of local observables. Considering this limitation, we show that almost all states in a energy shell are practically indistinguishable from one another, and hence from the microcanonical ensemble. In particular, the expectation value of a macroscopic observable is very close to its microcanonical ensemble average for almost all states.

DOI
10.48550/arxiv.cond-mat/0602625

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•Jan 1, 2010•0 citations
We propose a new approach to justify the use of the microcanonical ensemble for isolated macroscopic quantum systems. Since there are huge number of independent observables in a macroscopic system, we cannot see all of them. Actually what we observe can be written in a rather simple combination of local observables. Considering this limitation, we show that almost all states in a energy shell are practically indistinguishable from one another, and hence from the microcanonical ensemble. In particular, the expectation value of a macroscopic observable is very close to its microcanonical ensemble average for almost all states.
A quantum mechanical explanation of the relaxation to equilibrium is shown for macroscopic systems for nonintegrable cases and numerically verified. The macroscopic system is initially in an equilibrium state, subsequently externally perturbed during a finite time, and then isolated for a sufficiently long time. We show a quantitative explanation that the initial microcanonical state typically reaches a state whose expectation values are well approximated by the average over another microcanonical ensemble.
For a macroscopic, isolated quantum system in an unknown pure state, the expectation value of any given observable is shown to hardly deviate from the ensemble average with extremely high probability under generic equilibrium and nonequilibrium conditions. Special care is devoted to the uncontrollable microscopic details of the system state. For a subsystem weakly coupled to a large heat bath, the canonical ensemble is recovered under much more general and realistic assumptions than those implicit in the usual microcanonical description of the composite system at equilibrium.
We show insurmountable contradictions which arise if statistical ensembles are considered a consequence of the influence of the environment of the physical systems. We regard the multiplicity of states with a definite energy value as a result of internal probabilistic processes in the macrosystem; these internal probabilistic processes are not taken into account by quantum mechanics. On a simple example it is demonstrated, that canonical distribution is not independent and equal to microcanonical one, but is a result of averaging by microcanonical ensemble.
We discuss a generalized quantum microcanonical ensemble. It describes isolated systems that are not necessarily in an eigenstate of the Hamilton operator. Statistical averages are obtained by combining a time average and a maximum entropy argument to resolve the lack of knowledge about initial conditions. As a result, statistical averages of linear observables coincide with values obtained in the canonical ensemble. Non-canonical averages can be obtained by taking into account conserved quantities which are non-linear functions of the microstate.
The capabilities of a new approach towards the foundations of Statistical Mechanics are explored. The approach is genuine quantum in the sense that statistical behavior is a consequence of objective quantum uncertainties due to entanglement and uncertainty relations. No additional randomness is added by hand and no assumptions about a priori probabilities are made, instead measure concentration results are used to justify the methods of Statistical Physics. The approach explains the applicability of the microcanonical and canonical ensemble and the tendency to equilibrate in a natural way. This work contains a pedagogical review of the existing literature and some new results. The most important of which are: i) A measure theoretic justification for the microcanonical ensemble. ii) Bounds on the subsystem equilibration time. iii) A proof that a generic weak interaction causes decoherence in the energy eigenbasis. iv) A proof of a quantum H-Theorem. v) New estimates of the average effective dimension for initial product states and states from the mean energy ensemble. vi) A proof that time and ensemble averages of observables are typically close to each other. vii) A bound on the fluctuations of the purity of a system coupled to a bath.
The capabilities of a new approach towards the foundations of Statistical Mechanics are explored. The approach is genuine quantum in the sense that statistical behavior is a consequence of objective quantum uncertainties due to entanglement and uncertainty relations. No additional randomness is added by hand and no assumptions about a priori probabilities are made, instead measure concentration results are used to justify the methods of Statistical Physics. The approach explains the applicability of the microcanonical and canonical ensemble and the tendency to equilibrate in a natural way. This work contains a pedagogical review of the existing literature and some new results. The most important of which are: i) A measure theoretic justification for the microcanonical ensemble. ii) Bounds on the subsystem equilibration time. iii) A proof that a generic weak interaction causes decoherence in the energy eigenbasis. iv) A proof of a quantum H-Theorem. v) New estimates of the average effective dimension for initial product states and states from the mean energy ensemble. vi) A proof that time and ensemble averages of observables are typically close to each other. vii) A bound on the fluctuations of the purity of a system coupled to a bath.