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We summarize the papers published by Einstein in the Annalen der Physik in the years 1902-04 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., Comment: 22 pages, submitted to JSP

We prove the existence of an open and dense subset of mapsfeDiff ω ∞ (S2) which have positive topological entropy. It follows that these maps have infinitely many hyperbolic periodic points and an exponential growth rate of hyperbolic periodic points. The proof is an application of Pixton's theorem

In a companion paper (Ref. 5) Fritz studies the time evolution of a one-dimensional point system which was introduced by Spitzer as a model of traffic. In the present paper we improve some of the results of Ref. 5 by using a different approach. Our results are obtained in a very simple and straightforward way but the techniques employed require conditions somewhat stronger than those assumed in Ref. 5.

To demonstrate the crystallinity of the soft disk ground state, a modified treatment of this model is given, as Radin's proof of a crucial lemma can be shown to be incorrect.

A simple proof is pointed out for the asymptotic exponential decay of then-step survival probability of a random walk on a finite lattice with traps in the limit asn → ∞. Some bounds are mentioned, which are valid for finiten and for symmetric random walks.

In addition to the comparison of the Gibbs negentropy ηG = ∫ Π ln Π to the Shannon negentropy ηs = ∑ EiΠi ln(EiΠi), discussed in the cited paper, a comparison of the former to ηs′=ηs−∑Ei lnEi the decrease in uncertainty, is of general interest from an information-theoretic standpoint. It is shown that ηG and ηs′ are similar in the situation in question. The additivity properties of ηG are briefly discussed.

The proof of a recently enunciated theorem on the occurrence of a gas-liquid phase transition is shown to be defective in several respects. Also, it is argued that the theorem itself very probably does not hold.

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., 20 pages

The prediction of climate change and its impact on extreme weather events is one of the great societal and intellectual challenges of our time. The first part of the problem is to make the distinction between weather and climate. The second part is to understand the dynamics of the fluctuations of the physical variables. The third part is to predict how the variances of the fluctuations are affected by statistical correlations in their fluctuating dynamics. This paper investigates a framework called LA SALT which can meet all three parts of the challenge for the problem of climate change. As a tractable example of this framework, we consider the Euler–Boussinesq (EB) equations for an incompressible stratified fluid flowing under gravity in a vertical plane with no other external forcing. All three parts of the problem are solved for this case. In fact, for this problem, the framework also delivers global well-posedness of the dynamics of the physical variables and closed dynamical equations for the moments of their fluctuations. Thus, in a well-posed mathematical setting, the framework developed in this paper shows that the mean field dynamics combines with an intricate array of correlations in the fluctuation dynamics to drive the evolution of the mean statistics. The results of the framework for 2D EB model analysis define its climate, as well as climate change, weather dynamics, and change of weather statistics, all in the context of a model system of SPDEs with unique global strong solutions.

The main result of the paper is a global asymptotic stability result for solutions to the Lifschitz-Slyozov-Wagner (LSW) system of equations. This extends some local asymptotic stability results of Niethammer-Vel\'{a}zquez (2006). The method of proof is along similar lines to the one used in a previous paper of the authors. This previous paper proves global asymptotic stability for a class of infinite dimensional dynamical systems for which no Lyapounov function is (apparently) available., Comment: 60 pages

One of the goals of climate science is to characterize the statistics of extreme and potentially dangerous events in the present and future climate. Extreme events like heat waves, droughts, or floods due to persisting rains are characterized by large anomalies of the time average of an observable over a long time. The framework of Donsker–Varadhan large deviation theory could therefore be useful for their analysis. In this paper we discuss how concepts and numerical algorithms developed in the context of with large deviation theory can be applied to study extreme, rare fluctuations of time averages of surface temperatures at regional scale with comprehensive numerical climate models. When performing this type of analysis, unless a rigorous study of the convergence to the large deviation limit is performed, it can be easy to be misled in thinking to have reached the asymptotic regime. In this paper we provide a systematic protocol to study the convergence of large deviation functions tailored for applications to climate problems. Referring to the existing literature on the subject, we provide explicit formulas to compute large deviation functions directly from time series of a deterministic dynamical system that can be applied to climate records, and we describe how to study the convergence. We show how using a rare event algorithm applied to a numerical model can improve the efficiency of the computation of the large deviation functions. As a case study we consider the time averaged European surface temperature obtained with the numerical climate model Plasim. We show how a precise analysis of the convergence leads to the conclusion that the large deviation limit is nor properly reached for time scales shorter than a few years, and is therefore of no practical interest to study midlatitude heat waves. Finally we show how, even in a case like this, rare event algorithms developed to study large deviation functions can be used to improve the statistics of events on time scales shorter than the one needed to reach the large deviation asymptotic regime.

We consider a spin system with pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by \citet{Baiklee16} and \citet{Baikleewu18} which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to \citet{Baiklee16} in the "paramagnetic regime" when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the corresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by \citet{Ban16}. The proof of the approximation by the linear spectral statistics part is taken from \citet{Banerjee2017}., 37 pages

In this paper, we continue our analysis of spatial versions of agent-based models for the dynamics of money that have been introduced in the statistical physics literature, focusing on two models with debts. Both models consist of systems of economical agents located on a finite connected graph representing a social network. Each agent is characterized by the number of coins she has, which can be negative in case she is in debt, and each monetary transaction consists in one coin moving from one agent to one of her neighbors. In the first model, that we name the model with individual debt limit, the agents are allowed to individually borrow up to a fixed number of coins. In the second model, that we name the model with collective debt limit, agents can borrow coins from a central bank as long as the bank is not empty, with reimbursements occurring each time an agent in debt receives a coin. Based on numerical simulations of the models on complete graphs, it was conjectured that, in the large population/temperature limits, the distribution of money converges to a shifted exponential distribution for the model with individual debt limit, and to an asymmetric Laplace distribution for the model with collective debt limit. In this paper, we prove exact formulas for the distribution of money that are valid for all possible social networks. Taking the large population/temperature limits in the formula found for the model with individual debt limit, we prove convergence to the shifted exponential distribution, thus establishing the first conjecture. Simplifying the formula found for the model with collective debt limit is more complicated, but using a computer to plot this formula shows an almost perfect fit with the Laplace distribution, which strongly supports the second conjecture., 19 pages, 2 figures

In this paper, we consider a broadcasting process in which information is propagated from a given root node on a noisy tree network, and answer the question that whether the symbols at the nth level of the tree contain non-vanishing information of the root as n goes to infinity. Although the reconstruction problem on the tree has been studied in numerous contexts including information theory, mathematical genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In the remarkable work of Borgs, Chayes, Mossel and Roch (The Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels), the exact threshold for the reconstruction problem for a binary asymmetric channel on the d-ary tree is establish, provided that the asymmetry is sufficiently small, which is the first exact reconstruction threshold obtained in roughly a decade. In this paper, by means of refined analyses of moment recursion on a weighted version of the magnetization, and concentration investigations, we rigorously give a complete answer to the question of how small it needs to be to establish the tightness of the reconstruction threshold and further determine its asymptotics of large degrees., Complete version of the conference paper "Reconstruction for the Asymmetric Ising Model on Regular Trees"

In this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter $$\varepsilon $$ ε such that, for every fixed value of the slow variable, the fast dynamics are sufficiently chaotic with ergodic invariant measure. Convergence of the slow process to the solution of a homogenized stochastic differential equation (SDE) in the limit $$\varepsilon $$ ε to zero, with explicit formulas for drift and diffusion coefficients, has so far only been obtained for the case that the fast dynamics evolve independently. In this paper we give sufficient conditions for the convergence of the first moments of the slow variable in the coupled case. Our proof is based upon a new method of stochastic regularization and functional-analytical techniques combined via a double limit procedure involving a zero-noise limit as well as considering $$\varepsilon $$ ε to zero. We also give exact formulas for the drift and diffusion coefficients for the limiting SDE. As a main application of our theory, we study weakly-coupled systems, where the coupling only occurs in lower time scales.

Using mathematical statistical mechanics methods, this paper shows that decisions to heavily promote entrepreneurship beyond a certain threshold in a society would lead to an increase in the society’s Gini coefficient, and thus to more economic inequalities. More specifically, we show that, in a heterogeneous society made up of both entrepreneurs (Es) and ordinary agents (OAs), economic inequalities reach a minimum at an optimal ratio of ‘Es to OAs’. When dealing with a purely homogeneous society made up of entrepreneurs only or ordinary agents only, economic inequalities would decrease as trading activities intensify among the society’s agents. In particular, ideologies that consist in flattening the wealth/income of citizens (as it was recommended, for example, by communist regimes in the last century) to reduce economic inequalities through strict government interventions, may not lead to positive outcomes. We also show that introducing a little heterogeneity into a purely homogeneous society will help reduce economic inequalities in this society. Thus, a society that is composed of ordinary agents only will see its economic inequalities decrease if a number of entrepreneurs join this society. Vice-versa, a society of entrepreneurs only will have its inequalities reduced if some ordinary agents join this society and engage the pre-existing entrepreneurs in trading activities; one could think of southern San Francisco Bay Area, Silicon Valley, as a concrete example of such a situation. To encourage ordinary agents become entrepreneurs, governments could design and implement tax-incentive policies for their respective societies. While such tax-incentive policies would help increase the number of entrepreneurs in the targeted society, they would also have unintended consequences: the society’s middle class gets depleted at equilibrium, and the inequalities that result from implementing the aforementioned policies will be uniformly bigger than the ones that result from an equal tax redistribution policy. The paper concludes with a discussion section that raises a number of questions about socio-economic phenomena and explains how these phenomena can be accounted for using physics laws and principles.

In the present paper we continue the investigation of solutions to higher-order heat-type equations with random initial conditions, which play the important role in many applied areas. We consider the random initial conditions given by harmonizable $$\varphi $$ -sub-Gaussian processes. The main results are the bounds for the distributions of the suprema over bounded and unbounded domains for solutions of such equations. The results obtained in the paper hold, in particular, for the case of Gaussian initial condition.

This paper is concerned with general spatially explicit versions of three stochastic models for the dynamics of money that have been introduced and studied numerically by statistical physicists: the uniform reshuffling model, the immediate exchange model and the model with saving propensity. All three models consist of systems of economical agents that consecutively engage in pairwise monetary transactions. Computer simulations performed in the physics literature suggest that, when the number of agents and the average amount of money per agent are large, the distribution of money at equilibrium approaches the exponential distribution for the first model, the gamma distribution with shape parameter two for the second model and a gamma distribution whose shape parameter depends on the saving propensity for the third model. The main objective of this paper is to give rigorous proofs of and extend these conjectures to generalizations of the first two models and a variant of the third model that include local rather than global interactions, i.e., instead of choosing the two interacting agents uniformly at random from the system, the agents are located on the vertex set of a general connected graph and can only interact with their neighbors., 17 pages, 2 figures

International audience; Some recent papers relate the criticality of complex systems to their maximal capacity of information processing. In the present paper, we consider high dimensional point processes, known as age-dependent Hawkes processes, which have been used to model spiking neural networks. Using mean-field approximation, the response of the network to a stimulus is computed and we provide a notion of stimulus sensitivity. It appears that the maximal sensitivity is achieved in the sub-critical regime, yet almost critical for a range of biologically relevant parameters.

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., To appear in JSP

The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing interactions on a Cayley tree of order two. By the phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. We also study some algebraic property of the disordered phase of the model, which is a new phenomena even in a classical setting., Comment: 24 pages. arXiv admin note: text overlap with arXiv:1011.2256

In the past few years, the miscible displacements in porous media were usually simulated by some semiempirical models based on the volume averaging at the representative elementary volume scale. To better understand the microscopic mechanism of the viscous fingering phenomenon in porous media for miscible fluids, in this paper the miscible displacements processes in porous media are studied using the lattice Boltzmann method (LBM) at the pore scale. First, the code of LBM is tested by simulating the displacement process of two miscible fluids with the same viscosity between two parallel plates which is the well-known Taylor–Aris dispersion problem, and comparing the results with the theoretical predictions. Then, the effects of the Peclet number Pe, the viscosity ratio M and the structure of the porous media on the displacement phenomenon are investigated, and the location and velocity of the finger tip, the displacement efficiency are also studied. In this paper, the displacement efficiency is calculated by $$1-m$$ , here the quantity m is defined as $$m=V_M/V_T$$ , where $$V_M$$ is the volume of more viscous fluids (the displaced fluid) left behind the finger tip, $$V_T$$ is the total pore volume behind the finger tip. It can be found that the “interface” of two fluids will become clearer with the increasing of the Peclet number. As Pe and M are large enough, the viscous fingering phenomenon will occur, and in the front of the finger, “mushroom-like” pattern can be observed. Besides, with the increasing of Pe or M the quantity m will be increased too, i.e., the displacement efficiency will be decreased. While Pe (or M) is greater than a certain value, the growth rate of the quantity m will slow down. The same trend was observed for the miscible displacement in capillary tubes or Hele–Shaw cells. Besides, changing the structure of the porous media makes the finger pattern different. The present simulation results provide a good understanding of the microscopic mechanism of the miscible displacement process in porous media, and also show that the LBM can be a useful tool for investigation miscible fluids behavior in porous media.

We prove a conditional Wegner estimate for Schr\"odinger operators with random potentials of breather type. More precisely, we reduce the proof of the Wegner estimate to a scale free unique continuation principle. The relevance of such unique continuation principles has been emphasized in previous papers, in particular in recent years. We consider the standard breather model, meaning that the single site potential is the characteristic function of a ball or a cube. While our methods work for a substantially larger class of random breather potentials, we discuss in this particular paper only the standard model in order to make the arguments and ideas easily accessible.

In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker–Planck–Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543–1608, 2004) for Boltzmann equation, but the Fokker–Planck term in this paper creates some technical difficulties.

This paper is the second in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. The method is set within a normed algebra $\mathcal{N}$ of functionals of the fields. In this paper, we develop a general method---localisation---to approximate an element of $\mathcal{N}$ by a local polynomial in the fields. From the point of view of the renormalisation group, the construction of the local polynomial corresponding to $F$ in $\mathcal{N}$ amounts to the extraction of the relevant and marginal parts of $F$. We prove estimates relating $F$ and its corresponding local polynomial, in terms of the $T_{\phi}$ semi-norm introduced in part I of the series., Comment: 30 pages

Spurred by theoretical predictions from Spohn and coworkers [Phys. Rev. E {\bf 69}, 035102(R) (2004)], we rederived and extended their result heuristically as well as investigated the scaling properties of the associated Langevin equation in curved geometry with an asymmetric potential. With experimental colleagues we used STM line scans to corroborate their prediction that the fluctuations of the step bounding a facet exhibit scaling properties distinct from those of isolated steps or steps on vicinal surfaces. The correlation functions was shown to go as $t^{0.15(3)}$ decidedly different from the $t^{0.26(2)}$ behavior for fluctuations of isolated steps. From the exponents, we were able to categorize the universality, confirming the prediction that the non-linear term of the KPZ equation, long known to play a central role in non-equilibrium phenomena, can also arise from the curvature or potential-asymmetry contribution to the step free energy. We also considered, with modest Monte Carlo simulations, a toy model to show that confinement of a step by another nearby step can modify as predicted the scaling exponents of the step's fluctuations. This paper is an expansion of a celebratory talk at the 95$^{\rm th}$ Rutgers Statistical Mechanics Conference, May 2006., Comment: 13 pages, 11 figures. This paper is an expansion of a celebratory talk at the 95$^{\rm th}$ Rutgers Statistical Mechanics Conference, May 2006

This paper is concerned with the stability and asymptotic stability at large time of solutions to a system of equations, which includes the Lifschitz–Slyozov–Wagner (LSW) system in the case when the initial data has compact support. The main result of the paper is a proof of weak global asymptotic stability for LSW like systems. Previously strong local asymptotic stability results were obtained by Niethammer and Velazquez for the LSW system with initial data of compact support. Comparison to a quadratic model plays an important part in the proof of the main theorem when the initial data is critical. The quadratic model extends the linear model of Carr and Penrose, and has a time invariant solution which decays exponentially at the edge of its support in the same way as the infinitely differentiable self-similar solution of the LSW model.

The purpose of this paper is to correct errors presented recently in the paper [Lv et al. J Stat Phys 149:619–628 (2012)], where the authors analyzed Fractional Fokker–Planck equation (FFPE) with space–time dependent drift \(F(x,t)=F(x)f(t)\) and diffusion \(D(x,t)=D(x)\tilde{d}(t)\) coefficients in the factorized form. We show an important drawback in the derivation of the stochastic representation of FFPE presented in the aforementioned paper, which makes the whole proof wrong. Moreover, we present a correct proof of their result in even more general case, when both drift and diffusion can have any, not necessarily factorized, form.

Financial and economic history is strewn with bubbles and crashes, booms and busts, crises and upheavals of all sorts. Understanding the origin of these events is arguably one of the most important problems in economic theory. In this paper, we review recent efforts to include heterogeneities and interactions in models of decision. We argue that the Random Field Ising model (RFIM) indeed provides a unifying framework to account for many collective socio-economic phenomena that lead to sudden ruptures and crises. We discuss different models that can capture potentially destabilising self-referential feedback loops, induced either by herding, i.e. reference to peers, or trending, i.e. reference to the past, and account for some of the phenomenology missing in the standard models. We discuss some empirically testable predictions of these models, for example robust signatures of RFIM-like herding effects, or the logarithmic decay of spatial correlations of voting patterns. One of the most striking result, inspired by statistical physics methods, is that Adam Smith's invisible hand can badly fail at solving simple coordination problems. We also insist on the issue of time-scales, that can be extremely long in some cases, and prevent socially optimal equilibria to be reached. As a theoretical challenge, the study of so-called "detailed-balance" violating decision rules is needed to decide whether conclusions based on current models (that all assume detailed-balance) are indeed robust and generic., Comment: Review paper accepted for a special issue of J Stat Phys; several minor improvements along reviewers' comments

The drastic effect of local alliances in three-party competition is investigated in democratic hierarchical bottom-up voting. The results are obtained analytically using a model which extends a sociophysics frame introduced in 1986 (Galam in J. Math. Phys. 30:426, 1986) and 1990 (Galam in J. Stat. Phys. 61:943, 1990) to study two-party systems and the spontaneous formation of democratic dictatorship. It is worth stressing that the 1990 paper was published in the Journal of Statistical Physics, the first paper of its kind in this journal. It was shown how a minority in power can preserve its leadership using bottom-up democratic elections. However such a bias holds only down to some critical value of minimum support. The results were used latter to explain the sudden collapse of European communist parties in the nineties. The extension to three-party competition reveals the mechanisms by which a very small minority party can get a substantial representation at higher levels of the hierarchy when the other two competing parties are big. Additional surprising results are obtained, which enlighten the complexity of three-party democratic bottom-up voting. In particular, the unexpected outcomes of local voting alliances are singled out. Unbalanced democratic situations are exhibited with strong asymmetries between the actual bottom support of a party and its associated share of power at the top leadership. Subtle strategies are identified for a party to maximize its hold on the top leadership. The results are also valid to describe opinion dynamics with three competing opinions.