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H-theorem in quantum systems interacting with reservoir

Gordey B. Lesovik, Ivan Sadovskyy, A. V. Lebedev, M. V. Suslov, V. M. Vinokur, L.D. LANDAU

Published Jul 16, 2014• 0 citations

Abstract

Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny, 141700, Moscow Region, Russia(Dated: July 17, 2014)Boltzmann’s H-theorem stating that entropy of a physical system would tend to increase overtime remains a foundation of thermodynamics. Original Boltzmann’s formulation given for classicalgases was resting on the molecular chaos hypothesis. Von Neumann’s quantum H-theorem claimedquantum origin of irreversibility but included as a part of the proof macroscopic measurement pro-cedure going thus beyond the strictly quantum consideration. Recent developments related entropygrowth to quantum entanglement hinting the solely quantum origin of macroscopic irreversibility.In particular, progress in quantum informatics brought in several rigorous results on the conditionsfor the non-negative entropy gain, however, many questions of how to adapt them to the realm ofphysics remain. Utilizing the mathematical formalism, we show that entanglement of electrons witheither electromagnetic environment or phonon bath results in the evolution of the electron densitymatrix that belong in the classes of evolution described in quantum informatics. Using exemplaryphysical systems we formulate and prove the H-theorem for a wide class of physical phenomena.1. INTRODUCTION

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Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny, 141700, Moscow Region, Russia(Dated: July 17, 2014)Boltzmann’s H-theorem stating that entropy of a physical system would tend to increase overtime remains a foundation of thermodynamics. Original Boltzmann’s formulation given for classicalgases was resting on the molecular chaos hypothesis. Von Neumann’s quantum H-theorem claimedquantum origin of irreversibility but included as a part of the proof macroscopic measurement pro-cedure going thus beyond the strictly quantum consideration. Recent developments related entropygrowth to quantum entanglement hinting the solely quantum origin of macroscopic irreversibility.In particular, progress in quantum informatics brought in several rigorous results on the conditionsfor the non-negative entropy gain, however, many questions of how to adapt them to the realm ofphysics remain. Utilizing the mathematical formalism, we show that entanglement of electrons witheither electromagnetic environment or phonon bath results in the evolution of the electron densitymatrix that belong in the classes of evolution described in quantum informatics. Using exemplaryphysical systems we formulate and prove the H-theorem for a wide class of physical phenomena.1. INTRODUCTION

Remarkable progress of quantum information theory (QIT) allowed to formulate mathematical theorems for conditions that data-transmitting or data-processing occurs with a non-negative entropy gain. However, relation of these results formulated in terms of entropy gain in quantum channels to temporal evolution of real physical systems is not thoroughly understood. Here we build on the mathematical formalism provided by QIT to formulate the quantum H-theorem in terms of physical observables. We discuss the manifestation of the second law of thermodynamics in quantum physics and uncover special situations where the second law can be violated. We further demonstrate that the typical evolution of energy-isolated quantum systems occurs with non-diminishing entropy.

We test Boltzmann's H-theorem for several models of particle random walk. We study the influence of interaction between the particle and reservoir/detectors on entropy and find entropy increasing in time for some models and behaving non-monotonically for others. The key mechanism affecting the entropy growth is the quantum entanglement between the system and the reservoir. We discuss the details of the system-reservoir interaction, such as presence of the interference in the system and number of interactions with detector parts, and their impact on the monotonicity of entropy.

The quantum entropy is usually defined using von Neumann's formula, which measures lack of information and vanishes for pure states. In contrast, we obtain a formula for the entropy of a pure state as it is measured from thermodynamic experiments, solely from the self-entanglement of the wave function, and find strong numerical evidence that the two are in agreement for nonintegrable systems, both for energy eigenstates and for states that are obtained at long times under the evolution of more general initial conditions. This is an extension of Boltzmann's hypothesis for classical systems, relating microscopic motion to thermodynamics.

The irreversible entropy increase described by the second law of thermodynamics is fundamentally tied to thermalization and the emergence of equilibrium. In the first part of our work (Ref: arXiv.2503.04152), we constructed an isolated gas system model and numerically demonstrated irreversible growth of entanglement entropy caused by erasure of spread non-equilibrium state information. Here, we mathematically prove that for a typical macroscopic system in a non-equilibrium state

|ϕ_0\rangle

, the quantum state

|ϕ'_0\rangle = \hat{O}(t)|ϕ_0\rangle

will inevitably evolve toward equilibrium. Our work demonstrates that the second law of thermodynamics, and consequently the ergodic hypothesis in statistical physics, can be understood and proven from a quantum information perspective. From this perspective, the second law can be stated as: In typical macroscopic physical systems, the spreading and erasure of non-equilibrium information is inevitable.

Abstract In this paper the thermodynamical properties of any system whatsoever are deduced from quantum mechanics. Two fundamental irreversible processes are considered: the conversion of other forms of energy into heat, and the flow of heat from one temperature to another. By the proof of a generalized H -theorem, it is shown that in each case the entropy, correctly defined, must increase, and the system tend towards a state of equilibrium. A simple but rigorous proof of Boltzmann’s law is given from which the thermodynamics of reversible processes may be inferred. An appendix includes the exact solution of the most general time-dependent perturbation problem of quantum mechanics.

Since the days of Boltzmann the foundation of thermodynamics has been subject to an ongoing discussion. Various problems like the precise definition of entropy, the origin of the second law, etc. have always been issues. For a long time the discussion has not been in the center of interest, for statistical physics work extremely well from a practical point of view. But now, as research deals with single small objects like molecules or atoms, the question arises whether or not thermodynamics are applicable to them, which leads back to the old question of the foundation of thermodynamics. We examine whether thermodynamics can be viewed as emerging from Schrödinger-type quantum dynamics. Therefore we partially transfer concepts of thermodynamical behavior in Gamma-space to the Hilbertspace of compound systems. Central roles play the local Von Neumann entropy of a system and the inevitability of entanglement with its environment, causing it to increase.