arXiv subjects

Ilias Kachapov

Publications and source records attributed to Ilias Kachapov.

3 recordsLinked to original sources

arXiv2022

Formalizing Elements of Probabilistic Mechanics

In this paper we create a model of particle motion on a three-dimensional lattice using discrete random walk with small steps. We rigorously construct a probability space of the particle trajectories. Unlike deterministic approach in classical mechanics, here we use probabilistic properties of particle movement to formally derive analogues of Newton's first and second laws of motion. Similar probabilistic models can potentially be applied to justify laws of thermodynamics in a consistent manner.

math.PR↗

arXiv2018

Alternative proof of existence of Gibbs measure at high temperature

Mathematical models in equilibrium statistical mechanics describe physical systems with many particles interacting with an external force and with one another. Gibbs measure is a fundamental concept in this theory. In existing literature infinite-volume models are constructed as limits of finite models and existence of Gibbs measure for them is proven through DLR formalism. The general existence proofs are quite complicated and involve topology and cluster expansion. In this paper we develop a more transparent and more constructive proof of existence of infinite Gibbs measure for a particular case of interaction model at high temperature. The proof is based on a limiting procedure and involves estimates of series of semi-invariants and graph-related estimates.

math.PR↗

arXiv2015

Application of semi-invariants to proof of the central limit theorem on a lattice

Statistical mechanics describes interaction between particles of a physical system. Particle properties of the system can be modelled with a random field on a lattice and studied at different distance scales using renormalization group transformation. Here we consider a thermodynamic limit of Ising model with weak interaction and we use semi-invariants to prove that a random field transformed by renormalization group converges in distribution to an independent field with Gaussian distribution as the distance scale infinitely increases; it is a generalization of the central limit theorem to the Ising model.

math.PR↗