We prove that the canonical partition function of a hard-core model with s molecules on a d-dimensional hypercubic lattice with interaction range R is a polynomial in the lattice lengths n_i ( i=1,… ,d ), provided that n_i>Rs for all i. The polynomial has total degree sd, and degree s in each variable. Under periodic boundary conditions, the partition function depends polynomially on the single variable N=∏ _i=1^d n_i . This polynomial characterization provides a unified framework for deriving a variety of recurrence relations satisfied by the partition function.
In this paper, we discuss the asymptotic dynamics of the distribution of the solution process to the non-autonomous McKean–Vlasov stochastic lattice plate equations with a variable time delay defined on the integer set ℤ. First, we rigorously investigate the well-posedness of solutions for these non-autonomous, distribution-dependent stochastic delay lattice plate equations. Subsequently, we establish the existence and uniqueness of pullback measure attractors for this system characterized by the solution operators. Furthermore, as an application of pullback measure attractor, we show that under appropriate conditions, the invariant measure for the system is ergodic and exponentially mixing. Finally, we characterize the upper semi-continuity of pullback measure attractors as the stochastic delay lattice plate system with distribution dependence converges to a system with distribution independence.
This review article discusses wave propagation and heat transport in one-dimensional nonlinear lattices, focusing particularly on the role of symmetry. The concept of continuous translational symmetry in nonlinear lattices has recently been introduced. Furthermore, a nonlinear lattice possessing this symmetry has been explicitly constructed. This paper describes this symmetric lattice and its various properties, focusing especially on the propagation of discrete breather (DB). Effects due to the lattice discreteness emerge in traveling DBs in ordinary nonlinear lattices. In contrast, the constructed lattice removes these discrete effects, showing that the lack of this symmetry is the origin of the discrete effect. The paper also outlines recent progress regarding the construction of nonlinear lattices without umklapp processes, which possess similar symmetry, and their heat transport. Numerical results indicate that this similar symmetry vanishes thermal resistance by eliminating the umklapp process. These findings suggest a unified perspective that symmetry determines both the mobility and the heat transport of nonlinear waves in lattices.
A measure of correlations for statistical systems is advanced, called correlation index. The measure is general, being applicable to quantum and classical, equilibrium and nonequilibrium statistical systems. After formulating the general notion of correlation indices for arbitrary trace-class operators, the calculation of the correlation indices are illustrated by several examples of density operators and reduced density matrices. The computation of the correlation indices for equilibrium systems is exemplified for bosonic systems with the Bose-Einstein condensate, superconducting systems, and for spin systems. Correlation indices for nonequilibrium systems are considered for nonlinear dynamics of spin and pseudospin systems.
Constructing a physically realizable fractional-order random walk model within finite spaces and heterogeneous potential fields has long remained a persistent challenge in the study of anomalous diffusion. In this paper, a novel composite continuous-time random walk model is investigated to characterize anomalous diffusion in a potential field over a finite domain under non-Markovian temporal conditions, in which the composite jumps that incrementally sample spatial forces are considered for the trapped particles in potential field. Concurrently, we consider both biexponential and power-law waiting time distributions to capture trapping effects in complex environments. In the continuous limit, we rigorously derive the macroscopic evolution equations corresponding to the master equation of this random walk: a spatial fractional Fokker-Planck equation for biexponential waiting times and a time-space fractional Fokker-Planck equation for power-law waiting times. Here, the spatial fractional derivative is defined via spectral fractional Fokker-Planck operator, endowing it with a clear physical interpretation in finite domains; the time fractional derivative takes the Caputo form, characterizing the non-Markovian nature of power-law waiting times. We derived the analytical solution of the propagator for the diffusion equation under reflecting and absorbing boundary conditions, and conducted a discussion on the first-passage time under double absorbing boundaries within a finite domain. By comparing the analytical solutions of the propagator and first-passage time probability density function with Monte Carlo simulation results, we validated the correctness of the theoretical derivation. This model may hold significant value for understanding complex transport processes.
The mean-field limit of interacting particle systems with singular interaction potentials under reflecting boundary conditions in non-convex domains is considered. To rigorously connect the microscopic particle system with its mesoscopic counterpart, we construct the empirical measure and establish stability by means of weighted exponential functionals. The main technical difficulties arise from the combination of non-convex geometry and singular particle interactions. To overcome the latter, we employ a smoothing (mollification) procedure of the interaction kernel, and then pass to the singular limit. For the boundary contribution, we exploit the exact shape of the boundary to keep the reflection terms under control. By combining moment bounds, stopping-time arguments, and mean-square error estimates for the empirical measure, we prove propagation of chaos and convergence to the reflected McKean-Vlasov equation.
In this paper, we study strong solutions and decay estimates for an incompressible Vlasov-magnetohydrodynamic (Vlasov-MHD) model arising in magnetized plasmas. This model consists of the Vlasov equation and the incompressible MHD equations, which interact mutually through the Lorentz force. It can be readily verified that the system has two equilibria: (f̅,u̅,B̅)=(0,0,0) and ( f̃,ũ,B̃)=((2π )^- 3/2e^- |v|^2/2,0,0) . For any prescribed time interval, we establish the existence and uniqueness of strong solutions under appropriately small initial perturbations that depend on the length of the interval. In both the whole-space and periodic settings, we further derive the corresponding decay estimates for the fluid velocity and magnetic field over the interval of existence. The primary difficulty stems from the lack of drag-type or Fokker–Planck dissipation in the kinetic equation, along with the trilinear coupling generated by the Lorentz force. To address this issue, we combine the characteristic method, support propagation for the kinetic component, refined energy estimates, and Fourier analysis.
This study focuses on establishing the large deviation principle (LDP) for a complex-valued fractional Ginzburg-Landau equation (GLE) influenced by nonlinear noise over the unbounded domain ℝ^d . Initially, we prove the well-posedness of the related deterministic control system and obtain the uniform tail estimates for its solutions. Subsequently, we show that the set of controlled solutions exhibits strong convergence when the controls are equipped with the weak topology. By examining the distributional convergence of stochastic solutions as ε→ 0 , we rigorously derive the LDP using the weak convergence method. A key challenge arises from the absence of compact Sobolev embeddings in unbounded domains. To address this, uniform tail estimates are introduced, which ensure the tightness of the induced probability measures and the precompactness of control paths. Additionally, the nonlinear drift term (μ + iν )|X|^p-2X is handled under the admissible constraint |(p-2)ν | ≤ 2μ√(p-1) , guaranteeing uniqueness, continuity, and convergence of solutions. The validity of the results is preserved under the replacement of (-Δ )^α by the classical Laplace operator. Lastly, numerical experiments are conducted to demonstrate the impact of noise intensity on solution behavior and to provide computational confirmation of the LDP.
We study a finite-velocity random motion on the real line with alternating velocities -v and c, subject to stochastic resetting to the origin. At each velocity-switching epoch, a Bernoulli trial determines whether the particle is instantaneously reset to the origin with probability θ or it continues its motion with the opposite velocity with probability 1-θ . The resulting reset times form a renewal process. We derive the probability distributions of the position-velocity process when the intertimes between consecutive velocity changes have identical (i) exponential distribution, and (ii) Erlang-2 distribution. We also obtain explicit expressions for the moments of the position process and, in case (i), the mean-square distance between the process with resetting and an independent counterpart without resetting. We further characterize the stationary distribution, expressing it in terms of the asymptotic average of the local time of the position process. Finally, we investigate the limiting behavior of the position density under a Kac-type scaling. In both cases (i) and (ii), the position density, conditioned on random initial velocity, converges to that of the Brownian motion with Poisson resetting.
We construct a unitarily invariant Hermitian matrix ensemble whose fixed-time eigenvalue law coincides with the Karlin–McGregor law for non-intersecting Brownian bridges with arbitrary finite multiplicities at both endpoints. This provides an explicit matrix-ensemble realization of the known mixed-type multiple orthogonal polynomial and Riemann–Hilbert description of the general multi-start/multi-end problem. We then derive several exact finite-n consequences of this construction. These include a path-space lift as an orbital Hermitian Brownian bridge and a reduction of the partition function to a single compact HCIZ integral with explicit t-dependence. We also compare the one-sided reduction with the Gaussian external-field ensemble, showing that, although the two ensembles are spectrally equivalent, their angular statistics are different. Finally, we derive fixed-time Schwinger–Dyson identities and associated resolvent relations for the dressed ensemble.
This study investigates the impact of seasonal weather variations on malaria transmission by developing and analyzing both deterministic and stochastic non-autonomous models. We first show that the global dynamics of the deterministic model are governed by the basic reproduction number R_0 : when R_0<1 , the disease-free periodic solution is globally asymptotically stable; when R_0 > 1 , the disease is uniformly persistent in the population. Building on the same biological assumptions, we further formulate a time nonhomogeneous stochastic model and develop an analytical solution for the probability of disease extinction by applying multi-type branching process approximation and backward Kolmogorov differential equations. Our analytical solution reveals that the probability of disease extinction varies periodically with the initial time and depends on the initial number of infected individuals. Moreover, we derive analytical expressions for the mean and variance of the time to disease extinction. Additionally, this study investigates the malaria situation in Nigeria over the period 2018-2024. Our model demonstrates a strong fit with real-world data on both monthly cases and monthly cumulative cases during this period. Numerical simulations confirm the analytical results and illustrate the impact of the treatment rate on long-term disease dynamics and the probability of disease extinction. Further, we analyze two important indicators: the total infectivity time and the total treatment time. The theoretical analytical solutions are strongly supported by numerical simulations, which show close agreement between the branching process approximation and Monte Carlo estimates, thereby validating the reliability and predictive capability of the proposed stochastic modeling approach. These findings may offer useful insights for malaria control.
We study an extension of the classical Cucker-Smale model by introducing a visual field into the interaction mechanism through the characteristic function, which captures directional and asymmetric perception among agents. For both discrete and continuous time systems, we derive sufficient conditions for flocking, showing that convergence emerges under suitable assumptions on the initial configuration and system parameters only. We further show that these conditions reduce to the classical Cucker-Smale results in the limiting case which means that the visual field of each agent turns to 2π under some initial assumptions. Numerical simulations are finally provided to illustrate our theoretical results.
For the discrete nonlinear quasi-periodic wave equation on ℤ^d , we establish the long-time stability of small-amplitude solutions. Although the perturbation lacks gauge invariance, we prove, via a partial normal form reduction, that the solutions remain localized for polynomially long times. Our result applies to the entire phase space, without restricting the number of active modes in the initial data, in particular, the initial data need not be finitely supported.
According to Born (Atomic Physics, page 103), spectra are “quantum phenomena, which from a classical standpoint are perfectly unintelligible”. However we illustrate results on classical calculations of infrared spectra of ionic crystals (actually LiF) which show that the situation is much more complex. Indeed it turns out that: 1) At room temperature and at higher ones (up to 1060 K) the classical computations reproduce the experimental data, even better than the presently available quantum ones do; 2) At lower temperatures (even at 7.5 K), the classical computations reproduce pretty well the data, if one accepts the idea advanced in 1916 by Nernst (the inventor of the third principle) that zero-point energy has room in classical physics too. It is eventually pointed out that the mentioned results might be regarded as a first step towards an implementation of the Einstein Classical Program, which aims at deducing quantum physics (admittedly the correct theory) from a realistic theory. In fact, we are considering the Einstein classical program in the extreme version in which the realistic theory is just (essentially, see below) classical electrodynamics of matter in bulk, involving phase space orbits, solutions of Newton equations. An Appendix is devoted to illustrate the Nernst approach, which concerns also the relation between equipartition and Planck's law.
Identifying key nodes in complex networks is crucial for understanding their robustness and vulnerability. Nevertheless, this task remains an open and challenging problem. Traditional centrality measures often rely on single-scale topological information, which limits their ability to fully characterize node importance. To address this issue, this paper proposes a multiscale structural contraction method based on discrete curvature for key node identification. From a geometric perspective, discrete curvature is employed to quantify the tightness of connections between nodes guiding the iterative contraction of local structures and progressively constructing a multiscale representation of the network. By integrating structural information preserved across different scales and mapping the aggregated multiscale evaluation back to the original graph, a comprehensive ranking of node importance is generated. Experimental results demonstrate that the proposed method outperforms existing baseline approaches on both real-world and synthetic networks, enabling more accurate and robust identification of structurally critical nodes.
We investigate a class of stochastic evolution equations with monotone coefficients and finite delay, motivated by non-equilibrium physical systems in which both memory effects and stochastic perturbations play essential roles. The inherent non-Markovian nature induced by delay challenges the application of classical ergodic theory. To overcome this challenge, we introduce the concept of recurrent entrance measures, which extends the classical notion of invariant measures to non-autonomous settings. By generalizing Shcherbakov’s comparability principle to a stochastic variational framework, we prove that these measures inherit the full spectrum of recurrence properties (e.g., stationary, periodic, quasi-periodic, almost periodic, Levitan almost periodic, Bohr almost automorphic, pseudo-periodic, Birkhoff recurrent, pseudo-recurrent and Poisson stable) from the system’s coefficients, marking a shift from pathwise to statistical descriptions of recurrence. For autonomous systems, we prove uniform exponential mixing in the Wasserstein metric and further establish strong laws of large numbers and central limit theorems for solution maps. Our results provide a unified statistical framework for stochastic evolution systems with delay, with applications to physical models exhibiting memory and stochastic forcing.
Cluster synchronization, where subsets of nodes evolve coherently while others remain desynchronized, underlies a wide range of coordinated phenomena in complex systems. In networks with multiple types of interactions—so-called hypernetworks—each coupling layer can promote distinct synchronization patterns, making it challenging to predict how coherent groups form and remain stable. Existing symmetry-based methods often require explicit group-theoretic identification, which becomes infeasible for large or structurally heterogeneous systems. Here, we introduce a spectral framework that predicts the hierarchical formation and stability of synchronization clusters in hypernetworks composed of commuting interaction layers. By exploiting the shared eigenspace of layer-specific Laplacian matrices, the framework analytically determines the sequence of cluster formation and decouples their stability via a generalized Master Stability Function. Numerical experiments with Chen and Lorenz oscillators confirm the theoretical predictions, revealing a one-to-one correspondence between spectral organization and the hierarchical emergence of synchronization clusters. This approach provides a unified, computationally efficient method for analyzing and controlling clustered coordination in systems governed by multiple, structurally compatible interaction mechanisms.
We analyze a sufficient framework to predetermine the success or failure of stochastic asymptotic tracking in pursuer-evader dynamics. Specifically, we formulate a stochastic extension of the deterministic model studied in literature, which shares a similar interaction structure but lacks positive invariance. Within this setting, we establish conditions under which stochastic asymptotic tracking succeeds (pursuer winning) or fails (evader winning), providing both qualitative and quantitative characterizations of these regimes.
From N+1 random points on a line we wish to select M+1 points so as to maximize the minimal spacing between them. We consider an initial configuration with independent and identically distributed spacings. Equivalently, the points are arrival times of a generic renewal process. For general spacing distributions, and for all M≤ N , we derive exact distributional identities for the maximal minimal spacing and obtain its asymptotic behavior. The problem admits a reformulation in terms of a threshold-resetting random walk. The walk advances by successive random increments and is reset to the origin upon exceeding a fixed threshold. The probability that the optimal spacing exceeds a given value coincides with the probability that the walk completes at least M reset cycles within N steps. This yields an exact representation in terms of first-passage functionals of the walk. The same mapping suggests a numerical scheme for the max-min spacing problem in the regime of large N and M, whose accuracy is tested against the exact results obtained here.
