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In this note we deduce a new lower bound for the convergence radius of the Virial series of a continuous system of classical particles interacting via a stable and tempered pair potential using the estimates on the Mayer coefficients obtained in the recent paper by Procacci and Yuhjtman (Lett Math Phys 107:31-46, 2017). This corrects the wrongly optimistic lower bound for the same radius claimed (but not proved) in the above cited paper (in Remark 2 below Theorem 1). The lower bound for the convergence radius of the Virial series provided here represents a strong improvement on the classical estimate given by Lebowitz and Penrose in 1964. [ABSTRACT FROM AUTHOR]

We discuss microscopic mechanisms of complex network growth, with the special emphasis of how these mechanisms can be evaluated from the measurements on real networks. As an example we consider the network of citations to scientific papers. Contrary to common belief that its growth is determined by the linear preferential attachment, our microscopic measurements show that it is driven by the nonlinear autocatalytic growth. This invalidates the scale-free hypothesis for the citation network. The nonlinearity is responsible for a dramatic dynamical phase transition: while the citation lifetime of majority of papers is 6-10 years, the highly-cited papers have practically infinite lifetime. [ABSTRACT FROM AUTHOR]

We summarize the papers published by Einstein in the Annalen der Physik in the years 1902-1904 on the derivation of the properties of thermal equilibrium on the basis of the mechanical equations of motion and of the calculus of probabilities. We point out the line of thought that led Einstein to an especially economical foundation of the discipline, and to focus on fluctuations of the energy as a possible tool for establishing the validity of this foundation. We also sketch a comparison of Einstein's approach with that of Gibbs, suggesting that although they obtained similar results, they had different motivations and interpreted them in very different ways. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large deviation principle of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space R n . Since the standard Sobolev embeddings on R n are not compact and the spectrum of the Laplace operator on R n are not discrete, there are many issues for proving the large deviations of invariant measures in the case of unbounded domains, including the difficulties for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni large deviations on compact sets as well as the exponential tightness on compact sets. Currently, there is no result available in the literature on the large deviations of invariant measures for stochastic PDEs on unbounded domains, and this paper is the first one to deal with this issue. The non-compactness of the standard Sobolev embeddings on R n is circumvented by the idea of uniform tail-ends estimates together with the arguments of weighted spaces. [ABSTRACT FROM AUTHOR]

This paper proposes an improved cumulant collision model for the pseudo-potential lattice Boltzmann method (LBM) to increase the stability of multiphase flow simulations involving low viscosities. This model is based on the work of Kharmiani et al. in (J Stat Phys 175: 47, 2019), which can be extended regardless of the collision model. The original cumulant collision model (Geier et al. in Comput Math Appl 70:507, 2015) causes a non-physical shape of droplets in pseudo-potential LBM because only the first-order central moments are considered in the forcing scheme. The improved cumulant collision model proposed in this paper applies the central moment forcing scheme to the original cumulant model to cover the high-order central moments. Several numerical simulations were carried out to validate the proposed model. First, the problem of a stationary liquid layer was solved, where the proposed model was demonstrated to be thermodynamically consistent. Second, the problem of a stationary droplet was solved, where the result agreed well with Laplace’s law. Third, the problem of a droplet impact on a liquid film was solved, where the crown radius agreed well with the analytical and numerical results available. Fourth, the simulation results carried out with the raw moment, central moment, and the proposed improved cumulant collision models were compared, as the liquid and vapor viscosities were gradually lowered. With all else being equal, only the lattice Boltzmann method with the proposed improved cumulant collision model was able to successfully simulate a density ratio of 720 and a Reynolds number of 8.7 × 10 4 . [ABSTRACT FROM AUTHOR]

We show that a modification of the proof of our paper Coquille et al. (J. Stat. Phys. 172(5), 1210–1222 (2018)), in the spirit of Fröhlich and Pfister (Commun. Math. Phys. 81, 277–298 (1981)), shows delocalisation in the long-range Discrete Gaussian Chain, and generalisations thereof, for any decay power α > 2 and at all temperatures. The argument proceeds by contradiction: any shift-invariant and localised measure (in the L 1 sense), is a convex combination of ergodic localised measures. But the latter cannot exist: on one hand, by the ergodic theorem, the average of the field over growing boxes would be almost surely bounded ; on the other hand the measure would be absolutely continuous with respect to its height-shifted translates, as a simple relative entropy computation shows. This leads to a contradiction and answers, in a non-quantitative way, an open question stated in a recent paper of C. Garban (Invisibility of the integers for the discrete Gaussian Chain via a caffarelli-silvestre extension of the discrete fractional laplacian. Preprint , (2023)). [ABSTRACT FROM AUTHOR]

The paper presents the low temperature expansion of the 2D Ising model in the presence of the magnetic field in powers of x = exp (- J / (k T)) and z = exp (B / (k T)) with full polynomials in z up to x 88 and full polynomials in x 4 up to z - 60 , in the latter case the polynomials are explicitly given. The new result presented in the paper is an expansion not in inverse powers of z but in (z 2 + x 8) - k where the subsequent coefficients (polynomials in x 4 ) turn out to be divisible by increasing powers of (1 - x 4) . This result gives a hint about the intriguing ‘off-diagonal’ correlations in the Ising model mixing the B ≠ 0 contributions with the usual low temperature B = 0 expansion what may be useful on the road to find the full analytic expression for the partition function of the Ising model with non-vanishing magnetic field. The paper describes both the analytic expansions of the partition function and the efficient combinatorial methods to get the coefficients of the expansion. [ABSTRACT FROM AUTHOR]

Finding a ground state of a given Hamiltonian of an Ising model on a graph G = (V , E) is an important but hard problem. The standard approach for this kind of problem is the application of algorithms that rely on single-spin-flip Markov chain Monte Carlo methods, such as the simulated annealing based on Glauber or Metropolis dynamics. In this paper, we investigate a particular kind of stochastic cellular automata, in which all spins are updated independently and simultaneously. We prove that (i) if the temperature is fixed sufficiently high, then the mixing time is at most of order log | V | , and that (ii) if the temperature drops in time n as 1 / log n , then the limiting measure is uniformly distributed over the ground states. We also provide some simulations of the algorithms studied in this paper implemented on a GPU and show their superior performance compared to the conventional simulated annealing. [ABSTRACT FROM AUTHOR]

We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of Glauber-Kawasaki dynamics with speed change. The Kawasaki part describes the movement of particles through particle interactions. It is speeded up in a diffusive space-time scaling. The Glauber part governs the creation and annihilation of particles. The Glauber part is set to favor two levels of particle density. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, a mean-curvature interface flow emerges, with a homogenized 'surface tension-mobility' parameter reflecting microscopic rates. The interface separates the two levels of particle density. Similar hydrodynamic limits have been derived in two recent papers; one where the Kawasaki part describes simple nearest neighbor interactions, and one where the Kawasaki part is replaced by a zero-range process. We extend the main results of these two papers beyond nearest-neighbor interactions. The main novelty of our proof is the derivation of a 'Boltzmann-Gibbs' principle which covers a class of local particle interactions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a Boltzmann type model that describes a collective behavior of a large group of individuals. The model considers a mechanism where as two individuals collide, they adopt after the collision the same post-collisional velocity according to a distribution centered at the mid pre-collisional velocity. We show in this paper that the solutions of a spatially homogeneous model on R d converge exponentially towards the equilibrium state for the Wasserstein metric. The convergence of the solutions in the strong-norm L 1 is also proved for initial conditions satisfying a stronger regularity property. In a last part, these results are illustrated numerically. [ABSTRACT FROM AUTHOR]

Understanding the glassy nature of neural networks is pivotal both for theoretical and computational advances in Machine Learning and Theoretical Artificial Intelligence. Keeping the focus on dense associative Hebbian neural networks (i.e. Hopfield networks with polynomial interactions of even degree P > 2 ), the purpose of this paper is twofold: at first we develop rigorous mathematical approaches to address properly a statistical mechanical picture of the phenomenon of replica symmetry breaking (RSB) in these networks, then—deepening results stemmed via these routes—we aim to inspect the glassiness that they hide. In particular, regarding the methodology, we provide two techniques: the former (closer to mathematical physics in spirit) is an adaptation of the transport PDE to this case, while the latter (more probabilistic in its nature) is an extension of Guerra’s interpolation breakthrough. Beyond coherence among the results, either in replica symmetric and in the one-step replica symmetry breaking level of description, we prove the Gardner’s picture (heuristically achieved through the replica trick) and we identify the maximal storage capacity by a ground-state analysis in the Baldi-Venkatesh high-storage regime. In the second part of the paper we investigate the glassy structure of these networks: at difference with the replica symmetric scenario (RS), RSB actually stabilizes the spin-glass phase. We report huge differences w.r.t. the standard pairwise Hopfield limit: in particular, it is known that it is possible to express the free energy of the Hopfield neural network (and, in a cascade fashion, all its properties) as a linear combination of the free energies of a hard spin glass (i.e. the Sherrington–Kirkpatrick model) and a soft spin glass (the Gaussian or ”spherical” model). While this continues to hold also in the first step of RSB for the Hopfield model, this is no longer true when interactions are more than pairwise (whatever the level of description, RS or RSB). For dense networks solely the free energy of the hard spin glass survives. As the Sherrington–Kirkpatrick spin glass is full-RSB (i.e. Parisi theory holds for that model), while the Gaussian spin-glass is replica symmetric, these different representation theorems prove a huge diversity in the underlying glassiness of associative neural networks. [ABSTRACT FROM AUTHOR]

In this paper, we prove the long-time Anderson localization for the nonlinear random Schrödinger equation on Z d by using the Birkhoff normal form technique. [ABSTRACT FROM AUTHOR]

The generalized mean-field orthoplicial model is a mean-field model on a space of continuous spins on R n that are constrained to a scaled (n - 1) -dimensional ℓ 1 -sphere, equivalently a scaled (n - 1) -dimensional orthoplex, and interact through a general interaction function. The finite-volume Gibbs states of this model correspond to singular probability measures. In this paper, we use probabilistic methods to rigorously classify the infinite-volume Gibbs states of this model, and we show that they are convex combinations of product states. The predominant methods utilize the theory of large deviations, relative entropy, and equivalence of ensembles, and the key technical tools utilize exact integral representations of certain partition functions and locally uniform estimates of expectations of certain local observables. [ABSTRACT FROM AUTHOR]