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Physica A Statistical Mechanics and its Applications

On the Boltzmann observable approach in quantum statistical mechanics

O. Melsheimer

Published Aug 1, 1989• 0 citations
Publisher
DOI
10.1016/0378-4371(89)90152-0

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Through the H theorem, Bolzmann attempted to validate the foundations of statistical mechanics. However, it is incompatible with the fundamental laws of mechanics because its deduction requires the introduction of probability. In this paper we attempt a justification of statistical mechanics without deviating from the existing framework of quantum mechanics. We point out that the principle of equal a priori probabilities is easily proven in the dual space. The dual of the space of the quantum states is the space of the observations. We then prove that time evolution of the operators of observations obeys Boltzmann equation. This result implies that the difference of the states from equal probability becomes unobservable as time elapses.
Abstract A restricted Boltzmann machine (RBM) is a two-layer neural-network machine-learning model comprising a visible layer that represents observed data and a hidden layer that captures correlations among the visible units. Rooted in ideas from statistical mechanics, RBMs provide a probabilistic framework for modeling complex, high-dimensional distributions through an energy function and a corresponding Boltzmann distribution. This review offers an accessible introduction to RBMs from a physics-oriented perspective, focusing on their core concepts and training principles. We survey key theoretical developments that interpret RBMs as many-body systems, highlighting their equilibrium/non-equilibrium thermodynamics, learning dynamics, mean-field properties, phase diagrams, and free energy calculation. We further discuss recent applications of RBMs to a range of physics problems, including quantum-mechanical wavefunction representation, statistical-mechanical Boltzmann distribution modeling, and biological protein sequence analysis.