Related papers: What mathematical billiards teach us about statist…
A critical examination of some basic conceptual issues in classical statistical mechanics is attempted, with a view to understanding the origins, structure and statuts of that discipline. Due attention is given to the interplay between…
We prove some partial results on the periodicity of billiard systems on graphs. The results specialize to the case of $n$ billiards with equal mass on the unit interval or circle traveling at the same speed.
Statistical physics has proven to be a very fruitful framework to describe phenomena outside the realm of traditional physics. The last years have witnessed the attempt by physicists to study collective phenomena emerging from the…
These lecture notes introduce some topics of classical statistical physics, particularly those that are relevant for neural networks and deep learning. Statistical physics is treated as a branch of probability theory or statistics, with the…
A short overview of the billiard approach for cosmological-type models with n Einstein factor-spaces is presented. We start with the billiard representation for pseudo-Euclidean Toda-like systems of cosmological origin. Then we consider…
Quantum ergodicity of classically chaotic systems has been studied extensively both theoretically and experimentally, in mathematics, and in physics. Despite this long tradition we are able to present a new rigorous result using only…
An elementary application of Algorithmic Complexity Theory to the polygonal approximations of curved billiards-integrable and chaotic-unveils the equivalence of this problem to the procedure of quantization of classical systems: the scaling…
We discuss the key role that Hamiltonian notions play in physics. Five examples are given that illustrate the versatility and generality of Hamiltonian notions. The given examples concern the interconnection between quantum mechanics,…
The aim of this paper is twofold: First, we give a formal introduction to the basics of the mathematical framework of classical mechanics. Along the way, we prove a Hamiltonian and a Lagrangian version of Noether's Theorem, an important…
The classical and quantum behavior of a particle inside a square box under the influence of the gravitational field is studied. Detailed calculations on periodic orbits, probability densities as well as expectation values and uncertainties…
The representations of Clifford algebras and their involutions and anti-involutions are fully investigated since decades. However, these representations do sometimes not comply with usual conventions within physics. A few simple examples…
Hamiltonian theory of hybrid quantum-classical systems is used to study dynamics of the classical subsystem coupled to different types of quantum systems. It is shown that the qualitative properties of orbits of the classical subsystem…
We study the statistical mechanics of classical and quantum systems in non-equilibrium steady states. Emphasis is placed on systems in strong thermal gradients. Various measures and functional forms of observables are presented. The quantum…
In this paper we show that, under certain generic conditions, billiards on ovals have only a finite number of periodic orbits, for each period, all non-degenerate and at least one of them is hyperbolic. Moreover, the invariant curves of two…
We illustrate the theory of one-dimensional pluri-Lagrangian systems with the example of commuting billiard maps in confocal quadrics.
A new class of probability distributions closely connected to generalized hyperbolic distributions is introduced. It is more adapted to study the distributions of sums of random number of random variables. The properties of these…
This is an updated and expanded version of our earlier survey article \cite{Gut5}. Section $\S 1$ introduces the subject matter. Sections $\S 2 - \S 4$ expose the basic material following the paradigm of elliptic, hyperbolic and parabolic…
We attempt to make superdeterminism more intuitive, notably by simulating a deterministic model system, a billiard game. In this system an initial 'bang' correlates all events, just as in the superdeterministic universe. We introduce the…
Building upon previous works by Young, Chernov-Zhang and Bruin-Melbourne-Terhesiu, we present a general scheme to improve bounds on the statistical properties (in particular, decay of correlations, and rates in the almost sure invariant…
We discuss the propagation of kinetic energy through billiard balls fixed in place along a one-dimensional segment. The number of billiard balls is assumed to be large but finite and we assume kinetic energy propagates following the usual…