arXiv subjects

Yuri Kozitsky

Publications and source records attributed to Yuri Kozitsky.

At least 19 recordsLinked to original sources

arXiv2026

The stochastic evolution of an infinite population with logistic-type interaction

An infinite population of point entities dwelling in the habitat $X=\mathds{R}^d$ is studied. Its members arrive in and depart from $X$ at random. The departure rate has a term corresponding to a logistic-type interaction between the entities. Thereby, the corresponding Kolmogorov operator $L$ has an additive quadratic term, which usually produces essential difficulties in its study. The population pure states are locally finite counting measures defined on $X$. The set of such states $Γ$ is equipped with the vague topology, which allows one to use probability measures defined thereon. The population evolution is described at two levels. At the first level, one deals with the Fokker-Planck equation for $(L,\mathcal{F},μ_0)$ where $\mathcal{F}$ is an appropriate set of bounded test functions $F:Γ\to \mathds{R}$ (domain of $L$) and $μ_0$ is an initial state, which is supposed to belong to the set $\mathcal{P}_{\rm exp}$ of sub-Poissonian probability measures on $Γ$. We prove that the Fokker-Planck equation has a unique solution $t\mapstoμ_t$, which belongs to $\mathcal{P}_{\rm exp}$. Some of the properties of this solution are also described. The second-level description yields a Markov process with cadlag paths such that its one-dimensional marginals coincide with the mentioned states $μ_t$. The process is obtained as the unique solution of the corresponding martingale problem. The results obtained are discussed and compared with those known for similar models with logistic-type interactions.

math.PR↗

arXiv2026

The Lee-Yang property of the Blume-Capel model

The Lee-Yang theory is based on the theorem concerning a property of the ferromagnetic Ising model partition function proved by T. D. Lee and C. N. Yang in 1952. It provides powerful tools for understanding the very nature of phase transitions and critical phenomena in various spin and similar models. Sometimes, this theory is applied to models for which the theorem's validity has not been proved. By the main result of E. H. Lieb and A. D. Sokal, Commun. Math. Phys. 80, 153 (1981), for a given ferromagnetic model, this validity is guaranteed by the same property of the single-spin partition function. Due to the anisotropy term $\sum_i ΔS_i^2$, the single-spin partition function of the Blume-Capel model fails to have the Lee-Yang property for $βΔ> \ln 2$. In this article, we show that the ferromagnetic interaction in such a model can induce the Lee-Yang property, even for these values of $βΔ$. To the best of our knowledge, it is the first result of this kind.

math-ph↗

arXiv2026

Staggering domino-like blast front motion in a one-dimensional cold gas

One-dimensional alternating particle systems are widely used to study interconnections between the hydrodynamics of blast waves in a gas-like medium and the Newtonian dynamics of its corpuscular constituents. We study the model in which point particles with masses $m,μ, m,μ,\dots, (m\geqμ)$ are distributed on the positive half-line $\mathbb{R}_{+}$. Their dynamics are initiated by giving a positive velocity to the leftmost particle; in its course, the particles undergo elastic collisions. For this model with $m/μ=2$, it has previously been established that the dynamics that start from random initial positions are consistent with predictions based on Euler's hydrodynamic equation. In particular, they have the following properties: (i) the position of the rightmost particle (shock front) evolves as $t^δ$ with $δ<1$; (ii) recoiled particles behind the front enter the negative half-axis; (iii) particles with locations $x\leq0$ move ballistically and eventually take over the total energy of the system. In this paper, we present numerical and analytical results for the dynamics of this model with nonrandom (typically equidistant) initial positions and various values of $m/μ$. For $m/μ=2$ and equidistant initial positions, our results qualitatively agree with those just mentioned. At the same time, we found an infinite family of numbers $\{\mathcal{M}_k\}$ such that, for $m/μ=\mathcal{M}_k$, the hydrodynamic behavior mentioned changes drastically to the following. At each moment, only a single triplet $m,μ, m$ is in motion, whereas all other particles are at rest. As a result, the shock front moves ballistically with an average velocity equal to the initial one. Such a `staggering domino-like' picture is obtained as an exact solution, which yields, in particular, explicit formulas for $\mathcal{M}_k$ and the particle velocities and positions.

cond-mat.stat-mech↗

arXiv2026

The Lee-Yang property of isotropic vector ferromagnets and lattice fields

The Lee-Yang property of a given spin model means that its partition function has purely imaginary zeros as a function of an external magnetic field. A similar property is also used in the theory of quantum anharmonic crystals and quantum lattice fields. A number of powerful analytic methods of the mathematical theory of such models employ this property. Its suitable generalization is used in the theory of models of isotropic $D$-dimensional spins (rotors) or $D$-component quantum lattice fields. So far, the (generalized) Lee-Yang property has been established only for two-dimensional isotropic models. In this work, we prove that isotropic spin and field models living on $\mathds{Z}$ have this property for all even $D$.

math-ph↗

arXiv2025

Breakdown of hydrodynamics in a one-dimensional cold gas

The following model is studied analytically and numerically: point particles with masses $m,μ,m, \dots$ ($m\geqμ$) are distributed over the positive half-axis. Their dynamics is initiated by giving a positive velocity to the particle located at the origin; in its course the particles undergo elastic collisions. We show that, for certain values of $m/μ$, starting from the initial state where the particles are equidistant the system evolves in a hydrodynamic way: (i) the rightmost particle (blast front) moves as $t^δ$ with $δ< 1$; (ii) recoiled particles behind the front enter the negative half-axis; (iii) the splatter -- the particles with locations $x\leq 0$ -- moves in the ballistic way and eventually takes over the whole energy of the system. These results agree with those obtained in S. Chakraborti et al, SciPost Phys. 2022, 13, 074, for $m/μ=2$ and random initial particle positions. At the same time, we explicitly found the collection of positive numbers $\{\mathcal{M}_i, i \in \mathbf{N} \}$ such that, for $m/μ= \mathcal{M}_i$, $i\leq 700$, the following holds: (a) the splatter is absent; (b) the number of simultaneously moving particles is at most three; (c) the blast front moves in the ballistic way. However, if, similarly as in S. Chakraborti et al, the particle positions are sampled from a uniformly distributed ensemble, for $m/μ= \mathcal{M}_i$ the system evolves in a hydrodynamic way.

cond-mat.stat-mech↗

arXiv2025

On the Lee-Yang property of some ferromagnets

According to the Lieb-Sokal theorem, the partition function, $Z$, of a ferromagnetic spin model has the Lee-Yang property if the single-spin partition function has it. In this note, it is shown that for some spin models a ferromagnetic interaction can induce the Lee-Yang property of $Z$ even if the single-spin partition function fails to have it. In particular, this holds for the Blume-Capel model and for the annealed states of the $s=\pm 1$ site dilute Ising model with a neares-neighbor interaction on $\mathds{Z}^d$, as well as with interactions defined by a hierarchical structure similar to that of Dyson's hierarchical model.

math-ph↗

arXiv2023

Lusin spaces as images of locally compact Polish spaces

A Lusin space is a Hausdorff space being the image of a Polish space under a continuous bijection. Such spaces have multiple applications, in particular, as state spaces of various stochastic systems. In this work, we consider the spaces obtained as the images of a noncompact and locally compact Polish space $(X, \mathcal{T})$, which we call $c$-Lusin. The main result is the statement that a $c$-Lusin space $Y=f(X)$, can be written as $Z\cup Y_1$, where $Z$ is a locally compact Polish space whereas $Y_1$ is $c$-Lusin. At the same time, $Y_1$ is the set of the discontinuity points of $f^{-1}$ which is a closed subset of $Y$. Moreover, $Y_1$ is nowhere dense if (and only if) $Y$ is a Baire space. By the same arguments, $Y_1$ can also be decomposed as $Z_1 \cup Y_2$ with the properties as above. In the case where $f$ can be extended to a continuous map $f:X\cup \{\infty\} \to Y$, and thus $Y_1$ is a singleton, we construct a metric on $X$ such that the corresponding metric space is compact and homeomorphic to the $c$-Lusin space $(f(X), \mathcal{T}')$.

math.GN↗

arXiv2023

Uniqueness of Markov random fields with higher-order dependencies

Markov random fields on a countable set $\sf V$ are studied. They are canonically set by a specification $γ$, for which the dependence structure is defined by a pre-modification $(h_e)_{e\in {\sf E}}$ -- a consistent family of functions $h_e : S^e\to [0,+\infty)$, where $S$ is a standard Borel space and $\sf E$ is an infinite collection of finite $e\subset {\sf V}$. Different $e$ may contain distinct number of elements, which, in particular, means that the dependence graph ${\sf H}=({\sf V}, {\sf E})$ is a hypergraph. Given $e\in {\sf E}$, let $δ(e)$ be the logarithmic oscillation of $h_e$. The result of this work is the assertion that the set of all fields $\mathcal{G}(γ)$ is a singleton whenever $δ(e)$ satisfies a condition, a particular version of which can be $δ(e) \leq \varkappa g(n_{\sf L}(e))$, holding for all $e$ and some $\sf H$-specific $\varkappa\in (0,1)$. Here $g$ is an increasing function, e.g., $g(n) = a+\log n$, and $n_{\sf L}(e)$ is the degree of $e$ in the line-graph ${\sf L}({\sf H})$, which may grow ad infinitum. This uniqueness condition is essentially less restrictive than those based on classical Dobrushin's methods, according to which either of $|e|$, $n_{\sf L}(e)$ and $δ(e)$ should be globally bounded. We also prove that its fulfilment implies that the unique element of $\mathcal{G}(γ)$ is globally Markov.

math.PR↗

arXiv2023

On the Statistical Mechanics of Large Populations

There exists a wide variety of works on the dynamics of large populations ranging from simple heuristic modeling to those based on advanced computer supported methods. Their interconnections, however, remain mostly vague, which significantly limits the effectiveness of using computer methods in this domain. The aim of the present publication is to propose a concept based on the experience elaborated in the nonequilibrium statistical mechanics of interacting physical particles. Its key aspect is to explicitly describe micro-states of populations of interacting entities as probability measures and then to link this description to its macroscopic counterpart based on kinetic-like equations, suitable for solving numerically. The pivotal notion introduced here is a sub-Poissonian state where the large n asymptotic of the probability of finding n particles in a given vessel is similar to that for noninteracting entities, for which macro- and microscopic descriptions are equivalent. To illustrate the concept, an individual based model of an infinite population of interacting entities is proposed and analyzed. For this population, its evolution preserves sub-Poissonian states, that allows one to describe it through the correlation functions of such states for which a chain of evolution equations is obtained. The corresponding kinetic equation is derived and numerically solved and analyzed.

math.DS↗

arXiv2022

A Markov process for a continuum infinite particle system with attraction

An infinite system of point particles placed in $\mathds{R}^d$ is studied. The particles are of two types; they perform random walks in the course of which those of distinct types repel each other. The interaction of this kind induces an effective multi-body attraction of the same type particles, which leads to the multiplicity of states of thermal equilibrium in such systems. The pure states of the system are locally finite counting measures on $\mathds{R}^d$. The set of such states $Γ^2$ is equipped with the vague topology and the corresponding Borel $σ$-field. For a special class $\mathcal{P}_{\rm exp}$ of probability measures defined on $Γ^2$, we prove the existence of a family $\{P_{t,μ}: t\geq 0, \ μ\in \mathcal{P}_{\rm exp}\}$ of probability measures defined on the space of c{à}dl{à}g paths with values in $Γ^2$, which is a unique solution of the restricted martingale problem for the mentioned stochastic dynamics. Thereby, the corresponding Markov process is specified.

math.PR↗

arXiv2021

A Markov process for an infinite age-structured population

For an infinite system of particles arriving in and departing from a habitat $X$ -- a locally compact Polish space with a positive Radon measure $χ$ -- a Markov process is constructed in an explicit way. Along with its location $x\in X$, each particle is characterized by age $α\geq 0$ -- time since arriving. As the state space one takes the set of marked configurations $\widehatΓ$, equipped with a metric that makes it a complete and separable metric space. The stochastic evolution of the system is described by a Kolmogorov operator $L$, expressed through the measure $χ$ and a departure rate $m(x,α)\geq 0$, and acting on bounded continuous functions $F:\widehatΓ\to \mathds{R}$. For this operator, we pose the martingale problem and show that it has a unique solution, explicitly constructed in the paper. We also prove that the corresponding process has a unique stationary state and is temporarily egrodic if the rate of departure is separated away from zero.

math.PR↗

arXiv2021

Evolution of states of an infinite particle system with nonlocal branching

We study the evolution of states of an infinite system of point particles dwelling in a locally compact Polish space $X$. Each particle produces at random a finite `cloud' of offsprings distributed over $X$ according to some law, and disappears afterwards. The system's states are probability measures on an appropriate space of locally finite counting measures on $X$. Their evolution is obtained by solving the corresponding Fokker-Planck equation. We prove that this equation has a unique solution and discuss some of its properties. Our pivotal idea of dealing with infinite systems consists in passing to tempered counting measures by imposing appropriate restrictions on the branching. In this approach, we first solve a nonlinear evolution equation in the space of bounded continuous functions on $X$ -- so called log-Laplace equation. Next we solve the Kolmogorov equation which is then used to solve the Fokker-Planck equation and thus describe the evolution in question.

math.PR↗

arXiv2021

A Markov process for an infinite interacting particle system in the continuum

An infinite system of point particles placed in $\mathds{R}^d$ is studied. Its constituents perform random jumps with mutual repulsion described by a translation-invariant jump kernel and interaction potential, respectively. The pure states of the system are locally finite subsets of $\mathds{R}^d$, which can also be interpreted as locally finite Radon measures. The set of all such measures $Γ$ is equipped with the vague topology and the corresponding Borel $σ$-field. For a special class $\mathcal{P}_{\rm exp}$ of (sub-Poissonian) probability measures on $Γ$, we prove the existence of a unique family $\{P_{t,μ}: t\geq 0, \ μ\in \mathcal{P}_{\rm exp}\}$ of probability measures on the space of cadlag paths with values in $Γ$ that solves a restricted initial-value martingale problem for the mentioned system. Thereby, a Markov process with cadlag paths is specified which describes the stochastic dynamics of this particle system.

math.PR↗

arXiv2020

Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumps

An algorithm is proposed for finding numerical solutions of a kinetic equation that describes an infinite system of point articles placed in $\mathbb{R}^d (d \geq 1)$. The particles perform random jumps with pair wise repulsion, in the course of which they can also merge. The kinetic equation is an essentially nonlinear and nonlocal integro-differential equation, which can hardly be solved analytically. The derivation of the algorithm is based on the use of space-time discretization, boundary conditions, composite Simpson and trapezoidal rules, Runge-Kutta methods, adjustable system-size schemes, etc. The algorithm is then applied to the one-dimensional version of the equation with various initial conditions. It is shown that for special choices of the model parameters, the solutions may have unexpectable time behaviour. A numerical error analysis of the obtained results is also carried out.

math.DS↗

arXiv2020

Uniqueness of Gibbs fields with unbounded random interactions on unbounded degree graphs

Gibbs fields with continuous spins are studied, the underlying graphs of which can be of unbounded vertex degree and the spin-spin pair interaction potentials are random and unbounded. A high-temperature uniqueness of such fields is proved to hold under the following conditions: (a) the vertex degree is of tempered growth, i.e., controlled in a certain way; (b) the interaction potentials $W_{xy}$ are such that $\|W_{xy}\|=\sup_{σ,σ'} |W_{xy}(σ, σ')|$ are independent (for different edges $\langle x, y \rangle$), identically distributed and exponentially integrable random variables.

math-ph↗

arXiv2020

Infinite populations of migrants as complex systems: self-regulation

A model is proposed and studied describing an infinite population of point migrants arriving in and departing from $X\subseteq \mathbf{R}^d$, $d\geq 1$. Both these acts occur at random with state-dependent rates. That is, depending on their geometry the existing migrants repel and attract the newcomers, which makes the population a complex system. Its states are probability measures on an appropriate configuration space, and their evolution $μ_0 \to μ_t$ is obtained by solving the corresponding Fokker-Planck equation. The main result is the conclusion that this evolution of states preserves their sub-Poissonicity, and hence a local self-regulation (suppression of clustering) takes place due to the inter-particle repulsion -- no matter of how small range. Further possibilities to study the proposed model with the help of this result are also discussed.

math.DS↗

arXiv2020

Modeling tumor growth: a simple individual-based model and its analysis

Initiation and development of a malignant tumor is a complex phenomenon that has critical stages determining its long time behavior. This phenomenon is mathematically described by means of various models: from simple heuristic models to those employing stochastic processes. In this chapter, we discuss some aspects of such modeling by analyzing a simple individual-based model, in which tumor cells are presented as point particles drifting in $\mathbf{R}_{+}:=[0,+\infty)$ towards the origin with unit speed. At the origin, each of them splits into two new particles that instantly appear in $\mathbf{R}_{+}$ at random positions. During their drift the particles are subject to a random death before splitting. In this model, trait $x\in \mathbf{R}_{+}$ of a given cell corresponds to time to its division and the death is caused by therapeutic factors. On its base we demonstrate how to derive a condition -- involving the therapy related death rate and cell cycle distribution parameters -- under which the tumor size remains bounded in time, which practically means combating the disease.

q-bio.PE↗

arXiv2020

Dynamics of an infinite age-structured particle system

The Markov evolution is studied of an infinite age-structured population of migrants arriving in and departing from a continuous habitat $X \subseteq\mathds{R}^d$ -- at random and independently of each other. Each population member is characterized by its age $a\geq 0$ (time of presence in the population) and location $x\in X$. The population states are probability measures on the space of the corresponding marked configurations. The result of the paper is constructing the evolution $μ_0 \to μ_t$ of such states by solving a standard Fokker-Planck equation for this models. We also found a stationary state $μ$ existing if the emigration rate is separated away from zero. It is then shown that $μ_t$ weakly converges to $μ$ as $t\to +\infty$.

math.DS↗