Recent Applications of Boltzmann’s Theory

Walter Greiner, Ludwig Neise, Horst Stöcker B

Kai Zhang, Sora Sakai

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.

Eleftherios N. Economou

Sauro Succi A

Abstract The Lattice Boltzmann concepts and applications described so far refer to classical, i.e., non-quantum physics. However, the LB formalism is not restricted to classical Newtonian mechanics and indeed an LB formulation of quantum mechanics, going by the name of quantum LB (QLB) has been in existence for more than two decades. Even though it would far-fetched to say that QLB represents a mainstream, in the recent years it has captured some revived interest, mostly on account of recent developments in quantum-computing research. This chapter provides an account of the QLB formulation: stay tuned, LBE goes quantum!

Mehmet Kadiroḡlu, Jochen Gemmer B

We investigate the transport behavior of finite modular quantum systems. Such systems have recently been analyzed by different methods. These approaches indicate diffusive behavior even, and especially, for finite systems. Inspired by these results we analytically and numerically analyze if and in which sense the dynamics of those systems are in agreement with an appropriate Boltzmann equation. We find that the transport behavior of a certain type of finite modular quantum systems may indeed be described in terms of a Boltzmann equation. However, the applicability of the Boltzmann equation appears to be rather limited to a very specific type of model.

L. Lanz, O. Melsheimer, Erhard Wacker