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Thermal fluctuations in quantized chaotic systems

Mark Srednicki

Published Jan 1, 1996• 92 citations
OpenAlex

Abstract

We consider a quantum system with N degrees of freedom which is classically chaotic. When N is large, and both ¯h and the quantum energy uncertainty ∆E are small, quantum chaos theory can be used to demonstrate the following results: (1) given a generic observable A, the infinite time average A of the quantum expectation value 〈A(t) 〉 is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of A; (2) the time variations of 〈A(t)〉−A are too small to represent thermal fluctuations; (3) however, the time variations of 〈A 2 (t) 〉 − 〈A(t) 〉 2 can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when N is small. Typeset using REVTEX

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We consider a quantum system with N degrees of freedom which is classically chaotic. When N is large, and both ¯h and the quantum energy uncertainty ∆E are small, quantum chaos theory can be used to demonstrate the following results: (1) given a generic observable A, the infinite time average A of the quantum expectation value 〈A(t) 〉 is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of A; (2) the time variations of 〈A(t)〉−A are too small to represent thermal fluctuations; (3) however, the time variations of 〈A 2 (t) 〉 − 〈A(t) 〉 2 can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when N is small. Typeset using REVTEX
We consider a quantum system with N degrees of freedom which is classically chaotic. When N is large, and both and the quantum energy uncertainty are small, quantum chaos theory can be used to demonstrate the following results: (i) given a generic observable A, the infinite time average of the quantum expectation value is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of A; (ii) the time variations of are too small to represent thermal fluctuations; (iii) however, the time variations of can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when N is small.
If a many-body quantum system approaches thermal equilibrium from a generic initial state, then the expectation value ⟨ψ(t)∣Ai∣ψ(t)⟩\langleψ(t)|A_i|ψ(t)\rangle⟨ψ(t)∣Ai​∣ψ(t)⟩, where ∣ψ(t)⟩|ψ(t)\rangle∣ψ(t)⟩ is the system's state vector and AiA_iAi​ is an experimentally accessible observable, should approach a constant value which is independent of the initial state, and equal to a thermal average of AiA_iAi​ at an appropriate temperature. We show that this is the case for all simple observables whenever the system is classically chaotic.
We consider many-body quantum systems that exhibit quantum chaos, in the sense that the observables of interest act on energy eigenstates like banded random matrices. We study the time-dependent expectation values of these observables, assuming that the system is in a definite (but arbitrary) pure quantum state. We induce a probability distribution for the expectation values by treating the zero of time as a uniformly distributed random variable. We show explicitly that if an observable has a non-equilibrium expectation value at some particular moment, then it is overwhelmingly likely to move towards equilibrium, both forwards and backwards in time. For deviations from equilibrium that are not much larger than a typical quantum or thermal fluctuation, we find that the time dependence of the move towards equilibrium is given by the Kubo correlation function, in agreement with Onsager's postulate. These results are independent of the details of the system's quantum state.
We consider many-body quantum systems that exhibit quantum chaos, in the sense that the observables of interest act on energy eigenstates like banded random matrices. We study the time-dependent expectation values of these observables, assuming that the system is in a definite (but arbitrary) pure quantum state. We induce a probability distribution for the expectation values by treating the zero of time as a uniformly distributed random variable. We show explicitly that if an observable has a nonequilibrium expectation value at some particular moment, then it is overwhelmingly likely to move towards equilibrium, both forwards and backwards in time. For deviations from equilibrium that are not much larger than a typical quantum or thermal fluctuation, we find that the time dependence of the move towards equilibrium is given by the Kubo correlation function, in agreement with Onsager’s postulate. These results are independent of the details of the system’s quantum state.
We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey Berry's conjecture. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were Gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell-Boltzmann, Bose-Einstein, or Fermi-Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon eigenstate thermalization. We show that a generic initial state will approach thermal equilibrium at least as fast as O(\ensuremath{\Elzxh}/\ensuremath{\Delta})${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\Delta} is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a sound foundation for quantum statistical mechanics.