One-dimensional Particle Systems Research Articles

Species segregation in one-dimensional granular-system simulations

We present one-dimensional molecular dynamics simulations of a two-species, initially uniform, freely evolving granular system. Colliding particles swap their relative position with a 50% probability allowing for the initial spatial ordering of the particles to evolve in time and frictional forces to operate. Unlike one-dimensional systems of identical particles, two-species one-dimensional systems of quasi-elastic particles are ergodic and the particles' velocity distributions tend to evolve towards Maxwell-Boltzmann distributions. Under such conditions, standard fluid equations with merely an additional sink term in the energy equation, reflecting the non-elasticity of the interparticle collisions, provide an excellent means to investigate the system's evolution. According to the predictions of fluid theory we find that the clustering instability is dominated by a non-propagating mode at a wavelength of the order 10 pi L/N epsilon , where N is the total number of particles, L the spatial extent of the system and epsilon the inelasticity coefficient. The typical fluid velocities at the time of inelastic collapse are seen to be supersonic, unless N epsilon<or= 10 pi . Species segregation, driven by the frictional force occurs as a result of the strong temperature gradients within clusters which pushes the light particles towards the clusters' edges and the heavy particles towards the center. Segregation within clusters is complete at the time of inelastic collapse.

Accumulation on the boundary for a one-dimensional stochastic particle system

We consider an infinite particle system on the positive half-line, with particles moving independently of each other. When a particle hits the boundary, it immediately disappears and the boundary moves to the right by some fixed quantity (the particle size). We study the speed of the boundary movement (growth). Possible applications are dynamics of traffic jam growth, growth of a thrombus in a vessel, and epitaxy. Nontrivial mathematics concerns the correlation between particle dynamics and boundary growth.

Phase separation in one-dimensional stochastic particle systems?

A class of interacting particle systems modelling driven diffusive systems with short range interactions has been suggested to exhibit macroscopic phase separation in d = 1 dimensions. Unlike all previously studied models exhibiting similar phenomena, there the phase separated state is fluctuating in the bulk of the macroscopic domains. We discuss a recently introduced sufficient criterion for the existence of such phase separation and point out some assumptions which require rigorous proof. We also introduce a new model for strong phase separation into essentially nonfluctuation states. We informally describe its exact invariant measure.

Collapse of periodic orbits in a driven inelastic particle system

The dynamical behavior of a one-dimensional inelastic particle system with particles of unequal mass traveling between two walls is investigated. The system is driven by adding energy at one of the walls while the other wall is stationary and does not add energy. By deriving analytic solutions for the periodic orbits of this system, we show that there are a countable infinity of critical mass ratios at which the particle dynamics become highly degenerate in the following sense. As the mass ratio passes through these critical points, large numbers of stable periodic orbits can collapse onto a single trivial orbit. We show that the widely studied equal-mass systems represent one of these critical points and are therefore such a degenerate case. We also show that in the elastic limit the number of orbits that collapse onto the single trivial orbit can become arbitrarily large.

Euler hydrodynamics of one-dimensional attractive particle systems

We consider attractive irreducible conservative particle systems on $\mathbb{Z}$, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1--30], from which we relax the assumption that the process has explicit invariant measures.

Improvement of an Estimate of the Global Existence Theorem for Solutions of the Bogoliubov Equations

For a one-dimensional system of particles interacting like elastic balls, we improve an estimate of the existence theorem for the global solution of the Cauchy problem for the Bogoliubov equations with initial data from the space of sequences of measurable functions.

Hamiltonian model of heat conductivity and Fourier law

We investigate the stationary nonequilibrium states of a quasi one-dimensional system of heavy particles whose interaction is mediated by purely elastic collisions with light particles, in contact at the boundary with two heat baths with fixed temperatures T + and T - . It is shown that Fourier law is satisfied with a thermal conductivity proportional to T ( x ) where T ( x ) is the local temperature. Entropy flux and entropy production are also investigated.

Aggregation rates in one-dimensional stochastic systems with adhesion and gravitation

We consider one-dimensional systems of self-gravitating sticky particles with random initial data and describe the process of aggregation in terms of the largest cluster size Ln at any fixed time prior to the critical time. The asymptotic behavior of Ln is also analyzed for sequences of times tending to the critical time. A phenomenon of phase transition shows up, namely, for small initial particle speeds (“cold” gas) Ln has logarithmic order of growth while higher speeds (“warm” gas) yield polynomial rates for Ln.

Influence of nonlinear interactions and boundary effects on the propagation of energy in a one-dimensional particle chain

A one-dimensional system (chain) of particles interacting with each other and with the substrate has been investigated by the method of computer modeling. Particles at the boundaries of the chain, which are considered separately, represent self-oscillating systems. The influence of the parameters of a system on the transfer of energy by the chain has been studied; the behavior of the system in the transient and steady-state regimes of motion has been investigated. It has been shown that the stationary states of the system are characterized by a nonuniform distribution of the average particle velocities (temperatures) squared over the chain’s length. A correlation between the increase in the thermal resistance of a system and the enhancement of its structurization has been established.

Periodic orbits of a one-dimensional inelastic particle system

The dynamical behavior of a one-dimensional inelastic particle system with two particles of different masses traveling between two walls is investigated. Energy is added at only one of the walls, which is oscillating, while the other wall is stationary. We show that if the particle nearer to the stationary wall is slightly lighter than the other particle and collisions between particles tend to the elastic limit, there are an infinite number of stable orbits. We also show that the widely studied situation of equal masses is an extremely special case, in which all the orbits are degenerate and collapse to a single trivial orbit in which one of the particles is trapped against the stationary wall. To cite this article: J.J. Wylie, Q. Zhang, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Spin of the ground state and the flux phase problem on the ring

As a continuation of our previous work, we derive the optimal flux phase which minimizes the ground state energy in the one-dimensional many particle systems, when the number of particles is odd in the absence of on-site interaction and external potential. Moreover, we study the relationship between the flux on the ring and the spin of the ground state through which we derive some information on the sum of the lowest eigenvalues of one-particle Hamiltonians.

Blockage hydrodynamics of one-dimensional driven conservative systems

We consider an arbitrary one-dimensional conservative particle system with finite-range interactions and finite site capacity, governed on the hydrodynamic scale by a scalar conservation law with Lipschitz-continuous flux h. A finite-size perturbation restricts the local current to some maximum value ϕ. We show that the perturbed hydrodynamic behavior is entirely determined by ϕ if inf(h;ϕ) is first nondecreasing and then nonincreasing, which we believe is a necessary condition.

A heap-based algorithm for the study of one-dimensional particle systems

A heap-based algorithm for the study of one-dimensional particle systems

Particle systems with variable length

We present a new class of one-dimensional particle systems, in which the number of components may change in the process of interaction. We suggest that some of these systems display spontaneous symmetry breaking.

Velocity correlations, diffusion, and stochasticity in a one-dimensional system

We consider the motion of a test particle in a one-dimensional system of equal-mass point particles. The test particle plays the role of a microscopic "piston" that separates two hard-point gases with different concentrations and arbitrary initial velocity distributions. In the homogeneous case when the gases on either side of the piston are in the same macroscopic state, we compute and analyze the stationary velocity autocorrelation function C(t). Explicit expressions are obtained for certain typical velocity distributions, serving to elucidate in particular the asymptotic behavior of C(t). It is shown that the occurrence of a nonvanishing probability mass at zero velocity is necessary for the occurrence of a long-time tail in C(t). The conditions under which this is a t(-3) tail are determined. Turning to the inhomogeneous system with different macroscopic states on either side of the piston, we determine its effective diffusion coefficient from the asymptotic behavior of the variance of its position, as well as the leading behavior of the other moments about the mean. Finally, we present an interpretation of the effective noise arising from the dynamics of the two gases, and thence that of the stochastic process to which the position of any particle in the system reduces in the thermodynamic limit.

Mixing Properties for Mechanical Motion¶of a Charged Particle in a Random Medium

We study a one-dimensional semi-infinite system of particles driven by a constant positive force F which acts only on the leftmost particle of mass M, called the heavy particle (the h.p.), and all other particles are mechanically identical and have the same mass m < M. Particles interact through elastic collisions. At initial time all neutral particles are at rest, and the initial measure is such that the interparticle distances ξi are i.i.d. r.v. Under conditions on the distribution of ξ which imply that the minimal velocity obtained by each neutral particle after the first interaction with the h.p. is bigger than the drift of an associated Markovian dynamics (in which each neutral particle is annihilated after the first collision) we prove that the dynamics has a strong cluster property, and as a consequence, we prove existence of the discrete time limit distribution for the system as seen from the first particle, a ψ-mixing property, a drift velocity, as well as the central limit theorem for the tracer particle.

Burgers turbulence initialized by a regenerative impulse

In this article we look at a one-dimensional infinitesimal particle system governed by the completely inelastic collision rule. Considering uniformly spread mass, we feed the system with initial velocities, so that when time evolves the corresponding velocity field fulfils the inviscid Burgers equation. More precisely, we suppose here that the initial velocities are zero, except for particles located on a stationary regenerative set for which the velocity is some given constant number. We give results of a large deviation type. First, we estimate the probability that a typical particle is located at time 1 at distance at least D from its initial position, when D tends to infinity. Its behaviour is related to the left tail of the gap measure of the regenerative set. We also show the same asymptotics for the tail of the shock interval length distribution. Second, we analyse the event that a given particle stands still at time T as T tends to infinity. The data to which we relate its behaviour are the right tail of the gap measure of the regenerative set. We conclude with some results on the shock structure.

Heat Conduction in a One-Dimensional Gas of Elastically Colliding Particles of Unequal Masses

We study the nonequilibrium state of heat conduction in a one-dimensional system of hard point particles of unequal masses interacting through elastic collisions. A BBGKY-type formulation is presented and some exact results are obtained from it. Extensive numerical simulations for the two-mass problem indicate that, even for arbitrarily small mass differences, a nontrivial steady state is obtained. This state exhibits local thermal equilibrium and has a temperature profile as predicted by kinetic theory. The temperature jumps typically seen in such studies are shown to be finite-size effects. The thermal conductivity appears to have a very slow divergence with system size, different from that seen in most other systems.

Bounded Fluctuations and Translation Symmetry Breaking in One-Dimensional Particle Systems

We present general results for one-dimensional systems of point charges (signed point measures) on the line with a translation invariant distribution μ for which the variance of the total charge in an interval is uniformly bounded (instead of increasing with the interval length). When the charges are restricted to multiples of a common unit, and their average charge density does not vanish, then the boundedness of the variance implies translation-symmetry breaking—in the sense that there exists a function of the charge configuration that is nontrivially periodic under translations—and hence that μ is not “mixing.” Analogous results are formulated also for one dimensional lattice systems under some constraints on the values of the charges at the lattice sites and their averages. The general results apply to one-dimensional Coulomb systems, and to certain spin chains, putting on common grounds different instances of symmetry breaking encountered there.

The Hydrodynamic Limit of a Deterministic Particle System with Conservation of Mass and Momentum

We study the hydrodynamic limit of a deterministic one-dimensional particle system with nearest neighbour interaction and an additional regularizing force. Under its evolution mass and momentum are conserved. In the limit with Euler scaling their macroscopic distributions are shown to be governed by the compressible Navier–Stokes equations with a density dependent viscosity.