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arXiv:cond-mat/0005128

From Quantum Dynamics to the Second Law of Thermodynamics

Hal Tasaki

Published May 8, 2000• 0 citations
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Abstract

In quantum systems which satisfy the hypothesis of equal weights for eigenstates [4], the maximum work principle (for extremely slow and relatively fast operation) is derived by using quantum dynamics alone. This may be a crucial step in establishing a firm connection between macroscopic thermodynamics and microscopic quantum dynamics. For special models introduced in [4,5], the derivation of the maximum work principle can be executed without introducing any unproved assumptions.

DOI
10.48550/arxiv.cond-mat/0005128

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In quantum systems which satisfy the hypothesis of equal weights for eigenstates [4], the maximum work principle (for extremely slow and relatively fast operation) is derived by using quantum dynamics alone. This may be a crucial step in establishing a firm connection between macroscopic thermodynamics and microscopic quantum dynamics. For special models introduced in [4,5], the derivation of the maximum work principle can be executed without introducing any unproved assumptions.
We treat a quantum mechanical system with certain general properties which are expected to be common in macroscopic quantum systems. Starting from a PURE initial state (which may not describe an equilibrium) in which energy is mildly concentrated at a single value, we consider a time evolution determined by a time-dependent Hamiltonian as a model of an adiabatic operation in thermodynamics. We take a family of operations with the same procedure and various ``waiting times.'' Then the Minimum Work Principle is rigorously proved for almost all choices of the waiting time.
Quantum dynamics controlled by a time-dependent coupling constant are studied. It is proven that an energy eigenstate expectation value of work done by the system in a quench process cannot exceed the work in the corresponding quasi-static process, if and only if the energy eigenvalue is a concave function of the coupling constant. We propose this concavity of energy eigenvalues as a new universal criterion for quantum dynamical systems to satisfy the second law of thermodynamics. We argue simple universal conditions on quantum systems for the concavity, and show that every energy eigenvalue is indeed concave in some specific quantum systems. These results agree with the maximal work principle for adiabatic quench and quasi-static processes as an expression of the second law of thermodynamics. Our result gives a simple example of an integrable system satisfying an analogue to the strong eigenstate thermalization hypothesis (ETH) with respect to the principle of maximum work.
Quantum dynamics controlled by a time-dependent coupling constant are studied. It is proven that an energy eigenstate expectation value of work done by the system in a quench process cannot exceed the work in the corresponding quasi-static process, if and only if the energy eigenvalue is a concave function of the coupling constant. We propose this concavity of energy eigenvalues as a new universal criterion for quantum dynamical systems to satisfy the second law of thermodynamics. We argue simple universal conditions on quantum systems for the concavity, and show that every energy eigenvalue is indeed concave in some specific quantum systems. These results agree with the maximal work principle for adiabatic quench and quasi-static processes as an expression of the second law of thermodynamics. Our result gives a simple example of an integrable system satisfying an analogue to the strong eigenstate thermalization hypothesis (ETH) with respect to the principle of maximum work.