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Journal of Modern Physics

About Statistical Mechanics in the Configuration Space (x, v)

Gustavo V. López

Published Jan 1, 2025• 0 citations
Publisher

Abstract

The classical and quantum Statistical Mechanics is straightforwardly formulated in the configuration space (x, v), and an example of a system is given to make the comparison with the formulation of the Statistical Mechanics in the phase space (x, p). It is shown that the advantage of using configuration space for system with their interaction force depends explicitly on the velocity of the bodies. In addition, for these types of systems it is shown that their partition function can be well defined in the configuration space but it is not defined in the phase space, or they can bring about different results, bringing the possibility of experimental verification of the formulation of Statistical Mechanics in the configuration space.

DOI
10.4236/jmp.2025.1610076

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The classical and quantum Statistical Mechanics is straightforwardly formulated in the configuration space (x, v), and an example of a system is given to make the comparison with the formulation of the Statistical Mechanics in the phase space (x, p). It is shown that the advantage of using configuration space for system with their interaction force depends explicitly on the velocity of the bodies. In addition, for these types of systems it is shown that their partition function can be well defined in the configuration space but it is not defined in the phase space, or they can bring about different results, bringing the possibility of experimental verification of the formulation of Statistical Mechanics in the configuration space.
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We review some connections between quantum information and statistical mechanics. We focus on three sets of results for classical spin models. First, we show that the partition function of all classical spin models (including models in different dimensions, different types of many-body interactions, different symmetries, etc) can be mapped to the partition function of a single model. Second, we give efficient quantum algorithms to estimate the partition function of various classical spin models, such as the Ising or the Potts model. The proofs of these two results are based on a mapping from partition functions to quantum states and to quantum circuits, respectively. Finally, we show how classical spin models can be used to describe certain fluctuating lattices appearing in models of discrete quantum gravity.