Journal of Statistical Physics最新文献

A new “Percolation with Clustering” (PWC) model is introduced, where (the probabilities of) site percolation configurations on the leaf set of a binary tree are rewarded exponentially according to a generic function, which measures the degree of clustering in the configuration. Conditions on such “clustering function” are given for the existence of a limiting free energy and a wetting transition, namely the existence of a non-trivial percolation parameter threshold above and only above which the set of “dry” (open) sites have an asymptotic density. Several examples of clustering functions are given and studied using the general theory. The results here will be used in a sequel paper to study the wetting transition for the discrete Gaussian free field on the tree subject to a hard wall constraint.

We consider the Kardar-Parisi-Zhang equation on the interval [0, L] with Neumann type boundary conditions and boundary parameters u, v. We show that the k-th order cumulant of the height behaves as (c_k(L,u,v), t) in the large time limit (t rightarrow +infty ), and we compute the coefficients (c_k(L,u,v)). We obtain an expression for the upper tail large deviation function of the height. We also consider the limit of large L, with (u=tilde{u}/sqrt{L}), (u={tilde{v}}/sqrt{L}), which should give the same quantities for the two parameter family (({tilde{u}}, {tilde{v}})) KPZ fixed point on the interval. We employ two complementary methods. On the one hand we adapt to the interval the replica Bethe ansatz method pioneered by Brunet and Derrida for the periodic case. On the other hand, we perform a scaling limit using previous results available for the open ASEP. The latter method allows to express the cumulants of the KPZ equation in terms a functional equation involving an integral operator.

In this paper, we study the homogeneous inelastic Boltzmann equation for hard spheres. We first prove that the solution f(t, v) is bounded pointwise from above by (C_{f_0}langle t rangle ^3) and establish that the cooling time is infinite (( T_c = +infty )) under the condition ( f_0 in L^1_2 cap L^{infty }_{s} ) for ( s > 2 ). Away from zero velocity, we further prove that ( f(t,v)le C_{f_0, |v|} langle t rangle ) for (v ne 0) at any time ( t > 0 ). This time-dependent pointwise upper bound is natural in the cooling process, as we expect the density near ( v = 0 ) to grow rapidly. We also establish an upper bound that depends on the coefficient of normal restitution constant, (alpha in (0,1]). This upper bound becomes constant when (alpha = 1), restoring the known upper bound for elastic collisions [8]. Consequently, through these results, we obtain Maxwellian upper bounds on the solutions at each time.

The Boltzmann equation is one of the most famous equations and has vast applications in modern science. In the current study, we take the randomness of binary collisions into consideration and generalize the classical Boltzmann equation into a stochastic framework. The corresponding Kolmogorov forward equations and Liouville equation in either discrete or continuous time and state space are derived respectively, whose characteristic line gives the Boltzmann equation as a consequence of the law of large numbers. Then the large deviations principle for these equations is established, which not only explains the probabilistic origin of the H-theorem in the Boltzmann equation, but also provides a natural way to incorporate the Boltzmann equation into a broader Hamiltonian structure. The so-called Hamilton-Boltzmann equation enjoys many significant merits, like time reversibility, the conservation laws of mass, momentum and energy, Maxwellian-Boltzmann distribution as the equilibrium solution, etc. We also present results under the diffusive limit in parallel. Finally, the macroscopic hydrodynamic models including 13 moments are derived with respect to our Hamilton-Boltzmann equation under the BGK approximation. We expect our study can inspire new insights into the classical Boltzmann equation from either the stochastic aspect or a Hamiltonian view.

We say of an isolated macroscopic quantum system in a pure state (psi ) that it is in macroscopic thermal equilibrium (MATE) if (psi ) lies in or close to a suitable subspace (mathcal {H}_textrm{eq}) of Hilbert space. It is known that every initial state (psi _0) will eventually reach and stay there most of the time (“thermalize”) if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation (H_theta ^textrm{fF}) of the Hamiltonian (H_0^textrm{fF}) of (Ngg 1) free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of (H_0^textrm{fF}). Here, we first point out that also for degenerate Hamiltonians all (psi _0) thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for (H_0^textrm{fF}). Inspired by the fact that there is one eigenbasis of (H_0^textrm{fF}) for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given (H_0) that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of (H_0) lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, (H=H_0+lambda V) with (lambda ll 1), for most perturbations V the perturbed Hamiltonian H satisfies ETH and all states thermalize.

We consider a Bose gas on the unit torus at zero temperature in the Gross-Pitaevskii regime, known to perform Bose-Einstein condensation: a macroscopic fraction of the bosons occupy the same quantum state, called condensate. We study the Bose gas’ quantum depletion, that is the number of bosons outside the condensate, and derive an explicit asymptotic formula of its generating function. Moreover, we prove an upper bound for the tails of the quantum depletion.

Entropy, its production, and its change in a dynamical system can be understood from either a fully stochastic dynamic description or a deterministic dynamics exhibiting chaotic behaviors. By taking the former approach based on the general diffusion process with diffusion (alpha ^{-1}varvec{D}(textbf{x})) and drift (textbf{b}(textbf{x})), where (alpha ) represents the “size parameter” of a system, we show that there are two distinctly different entropy balance equations. One reads (textrm{d}S^{(alpha )}/textrm{d}t = e^{(alpha )}_p + Q^{(alpha )}_{ex}) for all (alpha ). Our key result addresses the asymptotic of the entropy production rate (e^{(alpha )}_p) and heat exchange rate (Q^{(alpha )}_{ex}) up to (O(tfrac{1}{alpha }))-corrections as system’s size (alpha rightarrow infty ). It yields in particular that the “extensive”, leading (alpha )-order terms of (e^{(alpha )}_p) and (Q^{(alpha )}_{ex}) are exactly canceled out. Therefore in the asymptotic limit of (alpha rightarrow infty ), there is a second, local entropy balance equation (textrm{d}S/textrm{d}t=nabla cdot textbf{b}(textbf{x}(t))+left( varvec{D}:varvec{varSigma }^{-1}right) (textbf{x}(t))) on the order of O(1), where (alpha ^{-1}varvec{D}(textbf{x}(t))) represents the randomness generated in the dynamics usually represented by metric entropy, (alpha ^{-1}varvec{varSigma }(textbf{x}(t))) is the covariance matrix of the local Gaussian description at (textbf{x}(t)) that is a solution to the ordinary differential equation (dot{textbf{x}}=textbf{b}(textbf{x})) at time t, and (varvec{D}:varvec{varSigma }^{-1}) is the Frobenius product of (varvec{D}) and (varvec{varSigma }^{-1}). This latter equation is akin to the notions of volume-preserving conservative dynamics and entropy production in the deterministic dynamic approach to irreversible thermodynamics à la D. Ruelle [55]. Our study follows the rigorous approach and formalism of [28]; the mathematical details with sufficient care are given in the appendices.

We consider a class of random billiards in a tube, where reflection angles at collisions with the boundary of the tube are random variables rather than deterministic (and elastic) quantities. We obtain a (non-standard) Central Limit Theorem for the displacement of a particle, which marginally fails to have a second moment w.r.t. the invariant measure of the random billiard.

The distribution of the first positive position reached by a random walker starting from the origin is fundamental for understanding the statistics of extremes and records in one-dimensional random walks. We present a comprehensive study of this distribution, focusing particularly on its moments and asymptotic tail behaviour, in the case where the step distribution is continuous and symmetric, encompassing both diffusive random walks and Lévy flights.

Vladimirov defined an operator on balls in (mathbb {Q}_{p}), the p-adic numbers, analogous to the Laplace operator in the real setting. Kochubei later gave a probabilistic interpretation of this operator. The Vladimirov–Kochubei operator generates a real-time diffusion process in the ring of p-adic integers, a Brownian motion in (mathbb {Z}_{p}). The current work proves that this process is a limit of discrete-time random walks. It motivates the construction of the Vladimirov–Kochubei operator, provides further intuition about ultrametric diffusion, and gives an example of the weak convergence of stochastic processes in a profinite group.