Related papers: The Canonical Ensemble and the Central Limit Theor…
A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain conditional $\alpha$-mixing…
We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained hermitean operators. Thanks to the hermicity constraints, we obtain positive-definite…
Breaking of ensemble equivalence between the microcanonical ensemble and the canonical ensemble may occur for random graphs whose size tends to infinity, and is signaled by a non-zero specific relative entropy of the two ensembles. In [3]…
We investigate a model system consisting of a Morse oscillator strongly coupled to a doubly-degenerate bending degree of freedom and show that Canonical Perturbation Theory is able to provide a fairly precise, though not exact,…
In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields…
Classical multidimensional scaling is a widely used method in dimensionality reduction and manifold learning. The method takes in a dissimilarity matrix and outputs a low-dimensional configuration matrix based on a spectral decomposition.…
The applicability conditions of a recently reported Central Limit Theorem-based approximation method in statistical physics are investigated and rigorously determined. The failure of this method at low and intermediate temperature is proved…
In this paper we study counting functions representing the number of solutions of systems of linear inequalities which arise in the theory of Diophantine approximation. We develop a method that allows us to explain the random-like behavior…
We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical…
Regularized coherent-state functional integrals are derived for ensembles of identical bosons on a lattice, the regularization being a discretization of Euclidian time. Convergence of the time-continuum limit is shown for various…
We give a simple proof of a central limit theorem for linear statistics of the Circular beta-ensembles which is valid at almost arbitrary mesoscopic scale and for functions of class C^3. As a consequence, using a coupling introduced by…
The microscopic foundation of the generalized equilibrium statistical mechanics based on the Tsallis entropy is given by using the Gibbs idea of statistical ensembles of the classical and quantum mechanics. The equilibrium distribution…
Ensembling has a long history in statistical data analysis, with many impactful applications. However, in many modern machine learning settings, the benefits of ensembling are less ubiquitous and less obvious. We study, both theoretically…
A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully…
It is shown that statistical mechanics is applicable to quantum systems with finite numbers of particles, such as complex atoms, atomic clusters, etc., where the residual two-body interaction is sufficiently strong. This interaction mixes…
Equilibrium statistics of finite Hamiltonian systems is fundamentally described by the microcanonical ensemble (ME). Canonical, or grand-canonical partition functions are deduced from this by Laplace transform. Only in the thermodynamic…
We show how canonical ensemble expectation values can be extracted from quantum Monte Carlo simulations in the grand canonical ensemble. In order to obtain results for all particle sectors, a modest number of grand canonical simulations…
Theory of Random Matrix Ensembles have proven to be a useful tool in the study of the statistical distribution of energy or transmission levels of a wide variety of physical systems. We give an overview of certain q-generalizations of the…
The canonical tensor model (CTM) is a tensor model proposing a classically and quantum mechanically consistent model of gravity, formulated as a first-class constraint system with structural similarities to the ADM formalism of general…
While the canonical ensemble has been tremendously successful in capturing thermal statistics of macroscopic systems, deviations from canonical behavior exhibited by small systems are not well understood. Here, using a small two dimensional…