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arXiv:1109.4696

Comment on "Foundation of Statistical Mechanics under Experimentally Realistic Conditions"

Zhexuan Gong, Luming Duan

Published Sep 22, 2011• 3 citations
arXivHTMLPDF

Abstract

Reimann [Phys. Rev. Lett. 101, 190403 (2008)] claimed that generic isolated macroscopic quantum system will equilibrate under experimentally realistic conditions by proving a theorem. We here show that the proof is invalid for most many-body systems and is unable to demonstrate equilibration in realistic experiment.
DOI
10.48550/arxiv.1109.4696

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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.
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.
Understanding how macroscopic systems exhibit irreversible thermal behavior has been a longstanding challenge, first brought to prominence by Boltzmann. Recent advances have established rigorous conditions for isolated quantum systems to equilibrate to a maximum entropy state, contingent upon weak assumptions. These theorems, while powerful, applied for a sudden quench. However, natural processes involve finite-time perturbations or quenches, which raises a crucial question: Can these systems still equilibrate under more realistic, finite-time dynamics? In this work, we extend the established results to account for finite-time quenches, demonstrating that, even under finite-time perturbations, the system will equilibrate provided it populates many significant energy levels. While the mathematical proof is more intricate than in the instantaneous case, the physical conclusion remains the same: sufficient perturbation leads to equilibration. Our results provide a broader and more physically realistic framework for understanding thermalization in isolated quantum systems.