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Physical Review Letters

Uncertainty, Entropy, and the Statistical Mechanics of Microscopic Systems

R. Blankenbecler, M. Hossein Partovi

Published Feb 4, 1985• 45 citations
Publisher

Abstract

Noting that quantum measurements are in general incomplete, we develop, starting from a recent entropic formulation of uncertainty, a maximum uncertainty principle to define the statistical mechanics of microscopic systems. The resulting ensemble entropy coincides with the expression of von Neumann, thus providing a unified, quantum basis for statistical physics of all systems. Examples involving momentum and position measurements are discussed.

DOI
10.1103/physrevlett.54.373

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Noting that quantum measurements are in general incomplete, we develop, starting from a recent entropic formulation of uncertainty, a maximum uncertainty principle to define the statistical mechanics of microscopic systems. The resulting ensemble entropy coincides with the expression of von Neumann, thus providing a unified, quantum basis for statistical physics of all systems. Examples involving momentum and position measurements are discussed.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
In thermodynamics and statistical mechanics entropy is the fundamental physical quantity to describe the evolution of a statistical ensemble. Its microscopic definition was provided by Boltzmann through the celebrated expression S = kB ln Γ, where kB is the Boltzmann constant and Γ is the number of microstates exploiting the given macroscopic properties. In this context, it is well known that in classical mechanics the entropy: i) allows the violation of the uncertainty principle [1], [2]; ii) can be considered as a special case of the so-called Boltzmann-Gibbs-Shannon entropy that enables one to apply results of information theory to physics [1], [3]. In particular, by introducing the principle of maximum entropy (MEP) it was found possible to derive rigorous hydrodynamic (HD) models based on the moments of the distribution function to all orders of a power expansion and including appropriate closure conditions [4], [5]. Accordingly, making use of the Lagrange multipliers technique, it was found possible to construct the set of evolution equations for the macro-variables of interest. Apart from some partial attempts [3], [6], this is no longer the case in quantum mechanics. Here, the main difficulties concern with: i) the definition of a proper quantum entropy that includes particle indistinguishability; ii) the formulation of a global quantum MEP (QMEP) that allows one to obtain a quantum distribution function both for thermodynamic equilibrium and nonequilibrium configurations. From one hand, in the framework of a nonlocal quantum theory, the generalization of the corresponding Lagrange multipliers is also an open problem. Fron another hand, a rigorous formulation of quantum HD (QHD) closed models is a demanding issue for many kinds of problems in quantum systems like, interacting fermionic and bosonic gases, quantum turbulence, quantum fluids, quantized vortices, nuclear physics, confined carrier transport in semiconductor heterostrucures, phonon and electron transport in nanostructures, nanowires and thin layers. The aim of this talk is to address the above drawbacks by defining a quantum entropy in terms of the reduced density matrix, thus explicitly incorporating quantum statistics into problems involving a system of identical particles. As a further step, with respect to the uncertainty principle, starting from the Wigner representation we formulate a quantum maximum entropy principle which, in the framework of information theory, allows us to obtain a non-local theory for the system under study. As a final step, we determine a closed quantum hydrodynamic model for the macroscopic variables used as constraints in the QMEP approach.
Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.
In the Entropic Dynamics (ED) framework quantum theory is derived as an application of the method of maximum entropy. The particles have well-defined positions but since they follow non differentiable Brownian trajectories they cannot be assigned an instantaneous momentum. Nevertheless, four different notions of momentum can be usefully introduced. We derive relations among them and the corresponding uncertainty relations. The main conclusion is that in ED momentum is not to be interpreted as an attribute of the particles but as a statistical concept associated to the probability distributions.
This paper attempts to explain both the basic concepts and the mathematical formalism of von Neumann's operator method in quantum statistics, in terms that should be understandable by anyone who has mastered first courses in quantum mechanics and statistical mechanics. A simple model is used throughout the paper to illustrate the various theorems and techniques. The discussion begins with the statistical interpretation of the Heisenberg uncertainty principle, the definition of mixed ensembles, and the meaning of measurement in quantum mechanics. The von Neumann projection operator is explained, and is used to set up the ensemble density matrix and the von Neumann entropy operator. In the second part of the paper, to be published later, the quantum-mechanical equation of change with time is applied to the von Neumann operators.