Journal of Statistical Physics最新文献

We provide an analytical framework for analyzing the quality of stochastic Verlet-type integrators for simulating the Langevin equation. Focusing only on basic objective measures, we consider the ability of an integrator to correctly simulate two characteristic configurational quantities of transport, a) diffusion on a flat surface and b) drift on a tilted planar surface, as well as c) statistical sampling of a harmonic potential. For any stochastic Verlet-type integrator expressed in its configurational form, we develop closed form expressions to directly assess these three most basic quantities as a function of the applied time step. The applicability of the analysis is exemplified through twelve representative integrators developed over the past five decades, and algorithm performance is conveniently visualized through the three characteristic measures for each integrator. The GJ set of integrators stands out as the only option for correctly simulating diffusion, drift, and Boltzmann distribution in linear systems, and we therefore suggest that this general method is the one best suited for high quality thermodynamic simulations of nonlinear and complex systems, including for relatively high time steps compared to simulations with other integrators.

In this work, we investigate the phase behavior of single-site adsorption models on one-dimensional (1D) lattice at nonzero temperatures, incorporating long-range intermolecular interactions up to the 12th neighbor. A comparative analysis of the models with different intermolecular potentials such as monotonic repulsive and attractive potentials with varying rates of decay, as well as a non-monotonic Lennard–Jones and oscillating potentials is performed. To accurately determine the thermodynamic properties of these systems, a tensor network approach to the well-known transfer matrix method is implemented. Characteristics of the models calculated in such way satisfy the conditions of thermodynamic equilibrium and belong to a formally infinite, truly thermodynamic, system. Using this approach, we have confirmed the remnants of “devil’s staircase” of phase transitions at nonzero temperatures in 1D system with repulsive interactions monotonically decreasing as ({r}^{-p}), where (p=1, 2, 3). The cutoff radius of such interactions is shown to influence both the set of possible structures and their stability range. 1D systems with monotonically attractive, Lennard–Jones, and oscillating potentials demonstrate the first-order phase transition associated with condensation of lattice gas. In these cases, the type of intermolecular potential, the decay rate of the monotonic potential, and its cutoff radius do not qualitatively impact the phase behavior of the system. These results can be useful for an interpretation of experimental data in studies of adsorption in 1D adsorbents such as nanotubes and microporous solids.

Sinaĭ initiated the study of random walks with persistently positive area processes, motivated by shock waves in solutions to the inviscid Burgers’ equation. We find the precise asymptotic probability that the area process of a random walk bridge is an excursion. A key ingredient is an analogue of Sparre Andersen’s classical formula. The asymptotics are related to von Sterneck’s subset counting formulas from additive number theory. Our results sharpen bounds by Aurzada, Dereich and Lifshits and respond to a question of Caravenna and Deuschel, which arose in their study of the wetting model. In this context, Sinaĭ excursions are a class of random polymer chains exhibiting entropic repulsion.

We consider constrained-degree percolation on the hypercubic lattice. This is a continuous-time model defined by a sequence ((U_e)_{e}) of i.i.d. uniform random variables and a positive integer k, referred to as the constraint. The model evolves as follows: each edge e attempts to open at a random time (U_e), independently of all other edges. It succeeds if, at time (U_e), both of its end-vertices have degrees strictly smaller than k. It is known [21] that this model undergoes a phase transition when (dge 3) for most nontrivial values of k. In this work, we prove that, for any fixed constraint, the number of infinite clusters at any time (tin [0,1)) is almost surely either 0 or 1. We also show that the law of the process is differentiable with respect to time for local events, extending a result of [30]. As a consequence of these two results, we prove that the percolation function is continuous in the supercritical regime (tin (t_c,1)), where (t_c) denotes the percolation critical threshold. Finally, we show that the two-point connectivity function is bounded away from zero in the supercritical regime.

We investigate the impact of weak collisions on Landau damping in the Vlasov-Poisson-Fokker-Planck system on a torus, specifically focusing on its proximity to a Maxwellian distribution. In the case where the Gevrey index satisfies (frac{1}{s}le 3), we establish the global stability and enhanced dissipation of small initial data, which remain unaffected by the small diffusion coefficient (nu ). For Gevrey index (frac{1}{s}>3), we prove the global stability and enhanced dissipation of initial data, whose size is on the order of (O(nu ^frac{1-3s}{3-3s})). Our analysis provides insights into the effects of enhanced dissipation and plasma echoes.

We study the emergent dynamics of the spatially extended continuum Winfree model on the whole domain and derive the uniform-in-time continuum limit from the infinite lattice model. In previous literature, the asymptotic convergence of the Winfree model has been studied in (ell ^1)-type topology such as the order parameter. Since we are dealing with the whole domain, where each point of the space represents an individual with one oscillator, it is more natural to employ (L^infty ) topology to analyze diverse emergent patterns such as an oscillator death, a phase-locking state and a quasi-steady state. Our key analysis lies in the uniform-in-time stability with respect to the initial data. From this, under a general network structure, a sufficiently large coupling strength leads to the exponential convergence of Winfree oscillators to an equilibrium. Moreover, uniform-in-time continuum limit from the infinite lattice model can be proved using the contraction property of extremal phases and stability estimates with respect to initial data and system parameters.

The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by Gács, whose counterexample is a cellular automaton that is non-ergodic under uniform random noise with sufficiently small rate. For all known counterexamples, non-ergodicity has been proved under small enough rates. Conversely, all cellular automata are ergodic with sufficiently high-rate noise. No other types of phase transitions of ergodicity are known, and the behavior of known counterexamples under intermediate noise rates is unknown. We present an example of a cellular automaton with two phase transitions. Using Gács’s result as a black box, we construct a cellular automaton that is ergodic under small noise rates, non-ergodic for slightly higher rates, and again ergodic for rates close to 1.

Statistical mechanics explains the properties of macroscopic phenomena based on the movements of microscopic particles such as atoms and molecules. Movements of microscopic particles can be represented by large-scale interacting systems. In this article, we systematically study combinatorial objects which we call interactions, given as symmetric directed graphs representing the possible transitions of states on adjacent sites of large-scale interacting systems. Such interactions underlie various standard stochastic processes such as the exclusion processes, generalized exclusion processes, multi-species exclusion processes, lattice gas with energy processes, and the multi-lane exclusion processes. We introduce the notion of equivalences of interactions using their space of conserved quantities. This allows for the classification of interactions reflecting the expected macroscopic properties. In particular, we prove that when the set of local states consists of two, three or four elements, then the number of equivalence classes of separable interactions are respectively one, two and five. We also define the wedge sums and box products of interactions, which give systematic methods for constructing new interactions from existing ones. Furthermore, we prove that the irreducibly quantified condition for interactions, which implicitly plays an important role in the theory of hydrodynamic limits, is preserved by wedge sums and box products. Our results provide a systematic method to construct and classify interactions, offering abundant examples suitable for considering hydrodynamic limits.

There is a dimensionless parameter which enters into the equation for the evolution of supersaturation in Ostwald ripening processes. This parameter is typically a large number. Here it is argued that the consequent stiffness of the equation results in the evolution of the supersaturation being unstable. The instability is evident in numerical simulations of Ostwald ripening.

We study the hydrodynamic limit for three gradient spin models: generalized Kipnis-Marchioro-Presutti (KMP), its discrete version and a family of harmonic models, under symmetric and nearest-neighbor interactions. These three models share some universal properties: occupation variables are unbounded, all these processes are of gradient type, their invariant measures are product with spatially homogeneous weights, and, notably, they are all attractive, meaning that the process preserves the partial order of measures along the dynamics. In view of hydrodynamics of large-scale interacting systems, dealing with processes taking values in unbounded configuration spaces is known to be a technically intricate problem. In the present paper, we show the hydrodynamic limit for all three models listed above in a comprehensive way, and show as a main result, that, under the diffusive time scaling, the hydrodynamic equation is given by the heat equation with model-dependent diffusion coefficient. Our novelty is showing the attractiveness for each model, which is crucial for the proof of hydrodynamics.