arXiv subjects

Krzysztof Pilorz

Publications and source records attributed to Krzysztof Pilorz.

7 recordsLinked to original sources

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↗

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↗

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↗

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

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↗

arXiv2018

Random jumps and coalescence in the continuum: evolution of states of an infinite system

The dynamics of an infinite continuum system of randomly jumping and coalescing point particles is studied. The states of the system are probability measures on the corresponding configuration space $Γ$ the evolution of which is constructed in the following way. The evolution of observables $F_0\to F_t$ is obtained from a Kolmogorov-type evolution equation. Then the evolution of states $μ_0\to μ_t$ is defined by the relation $μ_0(F_t) =μ_t(F_0)$ for $F_0$ belonging to a measure-defining class of functions. The main result of the paper is the proof of the existence of the evolution of this type for a bounded time horizon.

math.DS↗

arXiv2015

A kinetic equation for repulsive coalescing random jumps in continuum

A continuum individual-based model of hopping and coalescing particles is introduced and studied. Its microscopic dynamics are described by a hierarchy of evolution equations obtained in the paper. Then the passage from the micro- to mesoscopic dynamics is performed by means of a Vlasov-type scaling. The existence and uniqueness of the solutions of the corresponding kinetic equation are proved.

math.DS↗