Journal of Statistical Physics最新文献

The dispersionless limit of the standard Eliashberg theory of superconductivity is studied, in which the effective electron-electron interactions are mediated by Einstein phonons of frequency (Omega >0), equipped with electron-phonon coupling strength (lambda ). The general results on (T_c) for phonons with non-trivial dispersion relation, obtained in a previous paper by the authors, (II), then become amenable to a detailed evaluation. The results are based on the traditional notion that the phase transition between normal and superconductivity coincides with the linear stability boundary (mathscr {S}_{!c}) of the normal state region against perturbations toward the superconducting region. The variational principle for (mathscr {S}_{!c}), obtained in (II), simplifies as follows: If ((lambda ,Omega ,T)in mathscr {S}_{!c}), then (lambda = 1/mathfrak {h}(varpi )), where (varpi :=Omega /2pi T), and where (mathfrak {h}(varpi )>0) is the top eigenvalue of a compact self-adjoint operator (mathfrak {H}(varpi )) on (ell ^2) sequences; (mathfrak {H}(varpi )) is the dispersionless limit (P(domega )rightarrow delta (omega -Omega )domega ) of the operator (mathfrak {K}(P,T)) of (II). It is shown that when (varpi le sqrt{2}), then the map (varpi mapsto mathfrak {h}(varpi )) is invertible. For sufficiently large (lambda ) ((lambda >0.77) will do) this yields the following: (i) the existence of a critical temperature (T_c(lambda ,Omega ) = Omega f(lambda )); (ii) an ordered sequence of lower bounds on (f(lambda )) that converges to (f(lambda )). Also obtained is an upper bound on (T_c(lambda ,Omega )), which is not optimal yet agrees with the asymptotic behavior (T_c(lambda ,Omega ) sim C Omega sqrt{lambda }) for large enough (lambda ), given (Omega ), though with a constant C that is a factor (approx 2.034) larger than the optimal constant (frac{1}{2pi }mathfrak {g}(2)^frac{1}{2} =0.1827262477...), with (mathfrak {g}(gamma )>0) the largest eigenvalue of the compact self-adjoint operator (mathfrak {G}(gamma )) for the (gamma ) model, determined rigorously in the first one, (I), of this series of papers on (T_c) by the authors.

The recent works [5] and [15] have studied the spectral properties of the Dyson model in the absence of an external field. This paper is a continuation of [5] and aims to bridge the gap in the literature by investigating the Dyson model in a field. In this paper, we prove that, for high temperatures or strong magnetic fields, there exists a non-negative, integrable (with respect to the unique half-line Gibbs measure) eigenfunction of the transfer operator for the Dyson model if (alpha in (frac{3}{2},2]). However, unlike in the zeromagnetic- field case, this eigenfunction is not continuous.

We develop a formulation of global thermodynamics for equilibrium systems under the influence of weak gravity. The free energy for simple fluids is extended to include a dependence on ((T, V, N, mtextit{g}L)), where L represents the vertical system length in the direction of gravity. A central idea in this formulation is to uniquely fix the reference point of the gravitational potential, ensuring a consistent thermodynamic framework. Using this framework, we derive the probability density of thermodynamic quantities, which allows us to define a variational function for determining equilibrium liquid-gas coexistence under gravity when the interface is flat. The resulting free energy landscape, derived from the variational function, reveals the local stability of liquid-gas configurations. Specifically, the liquid phase resides at the lower portion of the system due to gravity, while the inverted configuration (with liquid on top) is also locally stable in this landscape. Furthermore, we characterize the transition between these liquid-gas configurations as a first-order phase transition using the thermodynamic free energy of ((T,V,N,mtextit{g}L)). Finally, we validate the predictions of global thermodynamics through molecular dynamics simulations, demonstrating the applicability and accuracy of the proposed framework.

We investigate the structure of non-equilibrium steady states (NESS) for a class of exactly solvable models in the setting of a chain with left and right reservoirs. Inspired by recent results on the harmonic model Large deviations and additivity principle for the open harmonic process, (2023), (JSP 191(1):10, 2024). we focus on models in which the NESS is a mixture of equilibrium product measures, and where the probability measure which describes the mixture has a spatial Markovian property. We completely characterize the structure of such mixture measures, and show that under natural scaling and translation invariance properties, the only possible mixture measures are coinciding with the Dirichlet process found in Carinci Gioia, Franceschini Chiara, Frassek Rouven, Giardinà Cristian, Redig Frank. Large deviations and additivity principle for the open harmonic process, (2023), in the context of the harmonic model.

One of the important questions in statistical mechanics is how irreversibility (time’s arrow) occurs when Newton equations of motion are time reversal invariant. One objection to irreversibility is based on Poincaré’s recursion theorem: a classical Hamiltonian confined system returns after some time, so-called Poincaré recurrence time (PRT), close to its initial configuration. Boltzmann’s reply was that for a (N sim 10^{23} ) macroscopic number of particles, PRT is very large and exceeds the age of the universe. In this paper we compute for the first time, using molecular dynamics, a typical recurrence time T(N) for a realistic case of a gas of N particles. We find that (T(N) sim N^z exp (y N) ) and determine the exponents y and z for different values of the particle density and temperature. We also compute y analytically using Boltzmann’s hypotheses. We find an excellent agreement with the numerical results. This agreement validates Boltzmann’s hypotheses, not yet mathematically proven. We establish that T(N) exceeds the age of the Universe for a relatively small number of particles, much smaller than ( 10^{23} ).

This paper reveals the intrinsic structure of Matrix Product States (MPS) by establishing their deep connection to entangled hidden Markov models (EHMMs). It is demonstrated that a significant class of MPS can be derived as the outcomes of EHMMs, showcasing their underlying quantum correlations. Additionally, a lower bound is derived for the relative entropy between the EHMM-observation process and the corresponding MPS, providing a quantitative measure of their informational divergence. Conversely, it is shown that every MPS is naturally associated with an EHMM, further highlighting the interplay between these frameworks. These results are supported by illustrative examples from quantum information, emphasizing their importance in understanding entanglement, quantum correlations, and tensor network representations.

In this paper, we introduce a variation of the elephant random walk whose steps are polynomially decaying. At each time k, the walker’s step size is (k^{-gamma }) with (gamma >0). We investigate effects of the step size exponent (gamma ) and the memory parameter (alpha in [-1,1]) on the long-time behavior of the walker. For fixed (alpha ), it admits phase transition from divergence to convergence (localization) at (gamma _{c}(alpha )=max {alpha ,1/2}). This means that large enough memory effect can shift the critical point for localization. Moreover, we obtain quantitative limit theorems which provide a detailed picture of the long-time behavior of the walker.

We prove new concentration inequalities for quantum spin systems which apply to any local observable measured on any product state or on any state with exponentially decaying correlations. Our results do not require the spins to be arranged in a regular lattice, and cover the case of observables that contain terms acting on spins at arbitrary distance. Moreover, we introduce a local (W_1) distance, which quantifies the distinguishability of two states with respect to local observables. We prove a transportation-cost inequality stating that the local (W_1) distance between a generic state and a state with exponentially decaying correlations is upper bounded by a function of their relative entropy. Finally, we apply such inequality to prove the equivalence between the canonical and microcanonical ensembles of quantum statistical mechanics and the weak eigenstate thermalization hypothesis for the Hamiltonians whose Gibbs states have exponentially decaying correlations.

We study the weakly asymmetric simple exclusion process on the integer lattice. Under suitable constraints on the strength of the weak asymmetry of the dynamics, we prove moderate deviation principles for the fluctuation fields when the process starts from stationary measures. As an application, we obtain sample path moderate deviation principles for the occupation time of the process in one dimension.

We obtain new lower bounds on the critical points for various models of oriented percolation. The method relies on establishing a stochastic domination of the percolation processes by multitype Galton-Watson trees. This approach can be applied to classical bond and site oriented percolation on (mathbb {Z}^2), as well as to other lattices, such as inhomogeneous ones, and in three dimensions.