Infinite Particle Systems Research Articles

Extremal invariant distributions of infinite Brownian particle systems with rank dependent drifts

Extremal invariant distributions of infinite Brownian particle systems with rank dependent drifts

Vanishing self-diffusivity in Ginibre interacting Brownian motions in two dimensions

AbstractWe prove that the tagged particles of infinitely many Brownian particles in $$ {\mathbb {R}} ^2$$ R 2 interacting via a logarithmic (two-dimensional Coulomb) potential with inverse temperature $$ \beta = 2 $$ β = 2 are sub-diffusive. The associated unlabeled diffusion is reversible with respect to the Ginibre random point field, and the dynamics are thus referred to as the Ginibre interacting Brownian motion. If the interacting Brownian particles have interaction potential $$ \Psi $$ Ψ of Ruelle class and the total system starts in a translation invariant equilibrium state, then the tagged particles are always diffusive if the dimension $$ d$$ d of the space $$ {\mathbb {R}}^{d} $$ R d is greater than or equal to two. That is, the tagged particles are always non-degenerate under diffusive scaling. Our result is, therefore, contrary to known results. The Ginibre random point field has various levels of geometric rigidity. Our results reveal that the geometric property of infinite particle systems affects the dynamical property of the associated stochastic dynamics.

Stochastic differential equations for infinite particle systems of jump type with long range interactions

Infinite-dimensional stochastic differential equations (ISDEs) describing systems with an infinite number of particles are considered. Each particle undergoes a Lévy process, and the interaction between particles is determined by the long-range interaction potential. The potential is of Ruelle's class or logarithmic. We discuss the existence and uniqueness of strong solutions of the ISDEs.

Orthogonal intertwiners for infinite particle systems in the continuum

This article focuses on a system of sticky Brownian motions, also known as Howitt–Warren martingale problem, and correlated Brownian motions and shows that infinite-dimensional orthogonal polynomials intertwine the dynamics of infinitely many particles and their n-particle evolution. The proof is based on two assumptions about the model: information about the reversible measures for the n-particle dynamics and consistency. Additionally, explicit formulas for the polynomials are used, including a new explicit formula for infinite-dimensional Meixner polynomials, the orthogonal polynomials with respect to the Pascal process. As an application of the intertwining relations, new reversible measures for the infinite-particle dynamics are obtained.

Stochastic analysis of infinite particle systems—A new development in classical stochastic analysis and dynamical universality of random matrices

We examine the stochastic dynamics of infinite particle systems moving in R d \mathbb {R}^{d} . The classical stochastic analysis pursues the stochastic dynamics of a single particle. We explain how to extend the classical stochastic analysis when the object becomes an infinite particle system. The equilibrium states of a system of infinite number of particles resulting from random matrices have the logarithmic function as an interaction potential. Hence, we develop a theory that applies to interaction potentials having a dominating influence over long distances. As an application, we show that the universality of point processes related to random matrices holds for stochastic dynamics.

Dynamical fluctuations in the Riesz gas.

We consider an infinite system of particles on a line performing identical Brownian motions and interacting through the |x-y|^{-s} Riesz potential, causing the overdamped motion of particles. We investigate fluctuations of the integrated current and the position of a tagged particle. We show that for 0<s<1, the standard deviations of both quantities grow as t^{s/2(1+s)}. When s>1, the interactions are effectively short-ranged, and the universal subdiffusive t^{1/4} growth emerges with only amplitude depending on the exponent s. We also show that the two-time correlations of the tagged-particle position have the same form as for fractional Brownian motion.

Existence of traveling solitary waves in Fermi-Pasta-Ulam-type systems with saturable nonlinearities on 2D-lattice

The article deals with the Fermi–Pasta–Ulam type systems with saturable nonlinearities that describes an infinite systems of particles on a two dimensional lattice. The main result concerns the existence of solitary traveling waves solutions with vanishing relative displacement profiles. By means of critical point theory, we obtain sufficient conditions for the existence of such solutions.

A Markov process for a continuum infinite particle system with attraction

A Markov process for a continuum infinite particle system with attraction

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 afterward. 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.

Existence of Periodic Traveling Waves in Fermi–Pasta–Ulam Type Systems on 2D-Lattice with Saturable Nonlinearities

The Fermi–Pasta–Ulam-type systems with saturable nonlinearities, namely, infinite systems of particles on a two dimensional lattice, have been considered. The main result concerns the existence of traveling-wave solutions with periodic relative displacement profiles. By means of critical point theory, sufficient conditions for the existence of such solutions have been obtained.

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 $\chi$ -- a Markov process is constructed in an explicit way. Along with its location $x\in X$, each particle is characterized by age $\alpha\geq 0$ -- time since arriving. As the state space one takes the set of marked configurations $\widehat{\Gamma}$, 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 $\chi$ and a departure rate $m(x,\alpha)\geq 0$, and acting on bounded continuous functions $F:\widehat{\Gamma}\to \mathds{R}$. 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.

Existence of periodic traveling waves in Fermi-Pasta-Ulam type systems on 2D-lattice with saturable nonlinearities

The Fermi-Pasta-Ulam-type systems with saturable nonlinearities, namely, infinite systems of particles on a two dimensional lattice, have been considered. The main result concerns the existence of traveling-wave solutions with periodic relative displacement profiles. By means of critical point theory, sufficient conditions for the existence of such solutions have been obtained.

Stochastic geometry and dynamics of infinitely many particle systems—random matrices and interacting Brownian motions in infinite dimensions

We explain the general theories involved in solving an infinite-dimensional stochastic differential equation (ISDE) for interacting Brownian motions in infinite dimensions related to random matrices. Typical examples are the stochastic dynamics of infinite particle systems with logarithmic interaction potentials such as the sine, Airy, Bessel, and also for the Ginibre interacting Brownian motions. The first three are infinite-dimensional stochastic dynamics in one-dimensional space related to random matrices called Gaussian ensembles. They are the stationary distributions of interacting Brownian motions and given by the limit point processes of the distributions of eigenvalues of these random matrices. The sine, Airy, and Bessel point processes and interacting Brownian motions are thought to be geometrically and dynamically universal as the limits of bulk, soft edge, and hard edge scaling. The Ginibre point process is a rotation- and translation-invariant point process on R 2 \mathbb {R}^2 , and an equilibrium state of the Ginibre interacting Brownian motions. It is the bulk limit of the distributions of eigenvalues of non-Hermitian Gaussian random matrices. When the interacting Brownian motions constitute a one-dimensional system interacting with each other through the logarithmic potential with inverse temperature β = 2 \beta = 2 , an algebraic construction is known in which the stochastic dynamics are defined by the space-time correlation function. The approach based on the stochastic analysis (called the analytic approach) can be applied to an extremely wide class. If we apply the analytic approach to this system, we see that these two constructions give the same stochastic dynamics. From the algebraic construction, despite being an infinite interacting particle system, it is possible to represent and calculate various quantities such as moments by the correlation functions. We can thus obtain quantitative information. From the analytic construction, it is possible to represent the dynamics as a solution of an ISDE. We can obtain qualitative information such as semi-martingale properties, continuity, and non-collision properties of each particle, and the strong Markov property of the infinite particle system as a whole. Ginibre interacting Brownian motions constitute a two-dimensional infinite particle system related to non-Hermitian Gaussian random matrices. It has a logarithmic interaction potential with β = 2 \beta = 2 , but no algebraic configurations are known.The present result is the only construction.

Blast in the one-dimensional cold gas: Comparison of microscopic simulations with hydrodynamic predictions

We study the response of an infinite system of point particles on the line initially at rest to the instantaneous release of energy in a localized region. The blast generates shock waves, and we make a detailed comparison of the density, velocity, and temperature in the growing region between the shock waves predicted by Euler equations for the ideal nondissipative compressible gas and the results of direct microscopic simulations. At long times, the hydrodynamic variables acquire self-similar forms with scaling functions predicted by the Taylor–von Neumann–Sedov (TvNS) blast-wave solution. The scaling functions obtained from the microscopic dynamics show a remarkable agreement with the TvNS predictions, except at the blast core, where the TvNS solution predicts a diverging temperature, which is not observed in simulations. We show that the effect of heat conduction becomes important and present results from a numerical solution of the Navier–Stokes–Fourier equations. A different scaling form is observed in the blast core. Our microscopic model is the one-dimensional hard-point gas with binary mass distribution and alternating masses. This infinitely dilute gas has the ideal gas equation of state and is nonintegrable and known to display fast equilibration.

Infinite system of random walkers: winners and losers

We study an infinite system of particles initially occupying a half-line y ⩽ 0 and undergoing random walks on the entire line. The right-most particle is called a leader. Surprisingly, every particle except the original leader may never achieve the leadership throughout the evolution. For the equidistant initial configuration, the kth particle attains the leadership with probability e−2 k −1(ln k)−1/2 when k ≫ 1. This decay law provides a quantitative measure of the correlation between earlier misfortune proportional to the label k and eternal failure. We also show that the winner defined as the first walker overtaking the initial leader has label k ≫ 1 with probability decaying as .

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 $\Gamma$ is equipped with the vague topology and the corresponding Borel $\sigma$-field. For a special class $\mathcal{P}_{\rm exp}$ of (sub-Poissonian) probability measures on $\Gamma$, we prove the existence of a unique family $\{P_{t,\mu}: t\geq 0, \ \mu \in \mathcal{P}_{\rm exp}\}$ of probability measures on the space of cadlag paths with values in $\Gamma$ 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.

Existence of traveling waves in Fermi-Pasta-Ulam type systems on 2D-lattice

The article deals with the Fermi-Pasta-Ulam-type systems that describe infinite systems of particles on a 2D lattice. The main result concerns the existence of the solutions corresponding to traveling waves with periodic and vanishing profiles. By means of the critical point theory, the sufficient conditions for the existence of such solutions are obtained.

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 particles 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 numerical algorithm which we use to solve it is based on a space-time discretization, boundary conditions, composite Simpson and trapezoidal rules, Runge-Kutta methods, and adjustable system-size schemes. We show that, for special choices of the model parameters, the solutions manifest unusual time behavior. A numerical error analysis of the obtained results is also carried out.

Correlation of clusters: Partially truncated correlation functions and their decay

In this article, we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive Kirkwood–Salsburg-type equations for the PTCF and write the solutions of these equations as a sum of contributions labeled by certain forest graphs, the connected components of which are tree graphs. We generalize the method developed by Minlos and Poghosyan [Theor. Math. Phys. 31, 408 (1977)] in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF, which were obtained earlier for lattice spin systems.

Infinite particle systems with collective behaviour and related mesoscopic equations

We study analogs of Kuramoto and Cucker-Smale models for systems with infinite interacting particles. For such systems, we describe the way to derive the corresponding kinetic (mesoscopic) equation reflecting the statistical characteristics of the whole microscopic system in the sense of a proper scaling. This derivation has an algorithmic realization in terms of related hierarchical chains of correlation function equations. We compare the obtained mesoscopic equations for considered infinite interacting particle systems with kinetic equations derived from ordinary differential equation models of finite particle systems using the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy technique.