Journal of Statistical Physics最新文献

We systematically develop beneficial and practical velocity measures for accurate and efficient statistical simulations of the Langevin equation with direct applications to computational statistical mechanics and molecular dynamics sampling. Recognizing that the existing velocity measures for the most statistically accurate discrete-time Verlet-type algorithms are inconsistent with the simulated configurational coordinate, we seek to create and analyze new velocity companions that both improve existing methods as well as offer practical options for implementation in existing computer codes. The work is based on the set of GJ methods that, of all methods, for any time step within the stability criteria correctly reproduces the most basic statistical features of a Langevin system; namely correct Boltzmann distribution for harmonic potentials and correct transport in the form of drift and diffusion for linear potentials. Several new and improved velocities exhibiting correct drift are identified, and we expand on an earlier conclusion that, generally, only half-step velocities can exhibit correct, time-step independent Maxwell–Boltzmann distributions. Specific practical and efficient algorithms are given in familiar forms, and these are used to numerically validate the analytically derived expectations. One especially simple algorithm is highlighted, and the ability of one of the new on-site velocities to produce statistically correct averages for a particular damping value is specified.

This paper’s objective is to improve the existing proof of the derivation of the Rayleigh–Boltzmann equation from the nonideal Rayleigh gas (Bodineau et al. in Invent Math 203:493–553, 2016), yielding a far faster convergence rate. This equation is a linear version of the Boltzmann equation, describing the behavior of a small fraction of tagged particles having been perturbed from thermodynamic equilibrium. This linear equation, derived from the microscopic Newton laws as suggested by the Hilbert’s sixth problem, is much better understood than the quadratic Boltzmann equation, and even enable results on long time scales for the kinetic description of gas dynamics. The present paper improves the physically poor convergence rate that had been previously proved, into a much more satisfactory rate which is more than exponentially better.

The survival probability for a periodic non-autonomous Ornstein–Uhlenbeck process is calculated analytically using two different methods. The first uses an asymptotic approach. We treat the associated Kolmogorov Backward Equation with an absorbing boundary by dividing the domain into an interior region, centered around the origin, and a “boundary layer” near the absorbing boundary. In each region we determine the leading-order analytical solutions, and construct a uniformly valid solution over the entire domain using asymptotic matching. In the second method we examine the integral relationship between the probability density function and the mean first passage time probability density function. These allow us to determine approximate analytical forms for the exit rate. The validity of the solutions derived from both methods is assessed numerically, and we find the asymptotic method to be superior.

In this paper, we prove the existence of a crystallization transition for a family of hard-core particle models on periodic graphs in dimension (dge 2). We consider only models featuring a single species of particles, which in particular forbids the particles from rotation and reflection, and establish a criterion under which crystallization occurs at sufficiently high densities. The criterion is more general than that in Jauslin and Lebowitz (Commun Math Phys 364:655–682, 2018), as it allows models in which particles do not tile the space in the close-packing configurations, such as discrete hard-disk models. To prove crystallization, we prove that the pressure is analytic in the inverse of the fugacity for large enough complex fugacities, using Pirogov–Sinai theory. One of the main new tools used for this result is the definition of a local density, based on a discrete generalization of Voronoi cells. We illustrate the criterion by proving that it applies to three examples: staircase models and the radius 2.5 hard-disk model on (mathbb Z^{2}), and a heptacube model on (mathbb Z^{3}).

A dry frictional interface loaded in shear often displays stick–slip. The amplitude of this cycle depends on the probability that a microscopic event nucleates a rupture and on the rate at which microscopic events are triggered. The latter is determined by the distribution of soft spots, P(x), which is the density of microscopic regions that yield if the shear load is increased by some amount x. In minimal models of a frictional interface—that include disorder, inertia and long-range elasticity—we discovered an ‘armouring’ mechanism by which the interface is greatly stabilised after a large slip event: P(x) then vanishes at small argument as (P(x)sim x^theta ) (de Geus et al., Proc Natl Acad Sci USA 116(48):23977-23983, 2019. https://doi.org/10.1073/pnas.1906551116). The exponent (theta ) is non-zero only in the presence of inertia (otherwise (theta =0)). It was found to depend on the statistics of the disorder in the model, a phenomenon that was not explained. Here, we show that a single-particle toy model with inertia and disorder captures the existence of a non-trivial exponent (theta >0), which we can analytically relate to the statistics of the disorder.

We consider a one-dimensional gradient symmetric exclusion process in mild contact with boundary reservoirs. The hydrodynamic limit of the empirical measure is given by a non-linear second-order parabolic equation with non-linear Robin boundary conditions. We prove the dynamical large deviations principle.

The Riemann surface associated with counting the current between two states of an underlying Markov process is hyperelliptic. We explore the consequences of this property for the time-dependent probability of that current for Markov processes with generic transition rates. When the system is prepared in its stationary state, the relevant meromorphic differential is in particular fully characterized by the precise identification of all its poles and zeroes.

The statistical behavior of scalars passively advected by random flows exhibits intermittency in the form of anomalous multiscaling, in many ways similar to the patterns commonly observed in incompressible high-Reynolds fluids. This similarity suggests a generic dynamical mechanism underlying intermittency, though its specific nature remains unclear. Scalar turbulence is framed in a linear setting that points towards a zero-mode scenario connecting anomalous scaling to the presence of statistical conservation laws; the duality is fully substantiated within Kraichnan theory of random flows. However, extending the zero-mode scenario to nonlinear settings faces formidable technical challenges. Here, we revisit the scalar problem in the light of a hidden symmetry scenario introduced in recent deterministic turbulence studies addressing the Sabra shell model and the Navier–Stokes equations. Hidden symmetry uses a rescaling strategy based entirely on symmetry considerations, transforming the original dynamics into a rescaled (hidden) system; It yields the universality of Kolmogorov multipliers and ultimately identifies the scaling exponents as the eigenvalues of Perron-Frobenius operators. Considering a minimal shell model of scalar advection of the Kraichnan type that was previously studied by Biferale & Wirth, the present work extends the hidden symmetry approach to a stochastic setting, in order to explicitly contrast it with the zero-mode scenario. Our study indicates that the zero-mode and the multiplicative scenarios are intrinsically related. While the zero-mode approach solves the eigenvalue problem for (p {{text {th}}}) order correlation functions, Perron-Frobenius (multiplicative) scenario defines a similar eigenvalue problem in terms of (p{text {th}}) order measures. For systems of the Kraichnan type, the first approach provides a quantitative chararacterization of intermittency, while the second approach highlights the universal connection between the scalar case and a larger class of hydrodynamic models.

We study a rumour propagation model along the lines of Lebensztayn and Rodriguez (Stat Probab Lett 78(14):2130–2136, 2008) as a long-range percolation model on (mathbb {Z}). We begin by showing a sharp phase transition-type behaviour in the sense of exponential decay of the survival time of the rumour cluster in the sub-critical phase. In the super-critical phase, under the assumption that radius of influence r.v. has (2+epsilon ) moment finite (for some (epsilon >0)), we show that the rightmost vertex in the rumour cluster has a deterministic speed in the sense that after appropriate scaling, the location of the rightmost vertex converges a.s. to a deterministic positive constant. Under the assumption that radius of influence r.v. has (4+epsilon ) moment finite, we obtain a central limit theorem for appropriately scaled and centered rightmost vertex. Later, we introduce a rumour propagation model with reactivation. For this section, we work with a family of exponentially decaying i.i.d. radius of influence r.v.’s, and we obtain the speed result for the scaled rightmost position of the rumour cluster. Each of these results is novel, in the sense that such properties have never been established before in the context of the rumour propagation model on (mathbb {Z}), to the best of our knowledge.

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the problem valid at any spatial dimension. The aim of this paper is to extend this general analysis to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solution of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provide a clear picture of the role played by the time-fractional derivatives in this kind of random motions. They display an anomalous behavior and are useful to describe several complex systems arising in statistical physics and biology. In particular, we focus on the one-dimensional random flight, called telegraph process, studying the time-fractional version of the classical telegraph equation and providing a suitable interpretation of its stochastic solutions.