Abstract: Roughening of interfaces implies the divergence of the interface width w with the system size L. For two-dimensional systems the divergence of \(w^2\) is linear in L. In the framework of a detailed capillary wave approximation and of statistical field theory we derive an expression for the asymptotic behaviour of \(w^2\) , which differs from results in the literature. It is confirmed by Monte Carlo simulations. PubDate: 2021-03-12
Abstract: We prove a mean field limit, a law of large numbers and a central limit theorem for a system of point vortices on the 2D torus at equilibrium with positive temperature. The point vortices are formal solutions of a class of equations generalising the Euler equations, and are also known in the literature as generalised inviscid SQG. The mean-field limit is a steady solution of the equations, the CLT limit is a stationary distribution of the equations. PubDate: 2021-03-12
Abstract: We consider here a new family of processes which describe particles which only can move at the speed of light c in the ordinary 3D physical space. The velocity, which randomly changes direction, can be represented as a point on the surface of a sphere of radius c and its trajectories only may connect points of this variety. A process can be constructed both by considering jumps from one point to another (velocity changes discontinuously) and by continuous velocity trajectories on the surface. We recently proposed to follow this second strategy assuming that the velocity is described by a Wiener process (which is isotropic only in the ’rest frame’) on the surface of the sphere. Using both Ito calculus and Lorentz boost rules, we succeed here in characterizing the entire Lorentz-invariant family of processes. Moreover, we highlight and describe the short-term ballistic behavior versus the long-term diffusive behavior of the particles in the 3D physical space. PubDate: 2021-03-11
Abstract: We study the point process W in \({\mathbb {R}}^d\) obtained by adding an independent Gaussian vector to each point in \({\mathbb {Z}}^d\) . Our main concern is the asymptotic size of fluctuations of the linear statistics in the large volume limit, defined as $$\begin{aligned} N(h,R) = \sum _{w\in W} h\left( \frac{w}{R}\right) , \end{aligned}$$ where \(h\in \left( L^1\cap L^2\right) ({\mathbb {R}}^d)\) is a test function and \(R\rightarrow \infty \) . We will also consider the stationary counter-part of the process W, obtained by adding to all perturbations a random vector which is uniformly distributed on \([0,1]^d\) and is independent of all the Gaussians. We focus on two main examples of interest, when the test function h is either smooth or is an indicator function of a convex set with a smooth boundary whose curvature does not vanish. PubDate: 2021-03-07
Abstract: This work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schrödinger (NLS) equation and to a classical system on a cylindrical lattice. A comparison with molecular dynamics simulations performed for the NLS model shows very good agreement. PubDate: 2021-03-06
Abstract: We extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington–Kirkpatrick spin-glass model without external magnetic field to the quantum case with a “transverse field” of strength \(\mathsf {b}\) . More precisely, if the Gaussian disorder is weak in the sense that its standard deviation \(\mathsf {v}>0\) is smaller than the temperature \(1/\beta \) , then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any \(\mathsf {b}/\mathsf {v}\ge 0\) . The macroscopic annealed free energy turns out to be non-trivial and given, for any \(\beta \mathsf {v}>0\) , by the global minimum of a certain functional of square-integrable functions on the unit square according to a Varadhan large-deviation principle. For \(\beta \mathsf {v}<1\) we determine this minimum up to the order \((\beta \mathsf {v})^{4}\) with the Taylor coefficients explicitly given as functions of \(\beta \mathsf {b}\) and with a remainder not exceeding \((\beta \mathsf {v})^{6}/16\) . As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong \(\beta \mathsf {b}\) -dependence even to lowest order. Our main tool for dealing with the non-commutativity of the spin-operator components is a probabilistic representation of the Boltzmann–Gibbs operator by a Feynman–Kac (path-integral) formula based on an independent collection of Poisson processes in the positive half-line with common rate \(\beta \mathsf {b}\) . Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman. PubDate: 2021-03-06
Abstract: One-dimensional edges of classical systems in two dimension sometimes show surprisingly rich phase transitions and critical phenomena, particularly when the bulk is at criticality. As such a model, we study the surface critical behavior of the 3-state dilute Potts model whose bulk is tuned at the tricritical point. To investigate it more precisely than in the previous works (Deng and Blöte, Phys Rev E 70:035107, 2004; Phys Rev E 71:026109, 2005), we analyze it from the viewpoint of the boundary conformal field theory (BCFT). The complete classification of the conformal boundary conditions for the minimal BCFTs discussed in Behrend et al. (Nucl Phys B 579:707, 2000) allows us to collect the twelve boundary fixed points in the tricritical 3-state Potts BCFT. Employing the tensor network renormalization method, we numerically study the surface phase diagram of the tricritical 3-state Potts model in detail, and reveal that the eleven boundary fixed points among the twelve can be realized on the lattice by controlling the external field and coupling strength at the boundary. The last unfound fixed point would be out of the physically sound region in the parameter space, similarly to the ‘new’ boundary condition in the 3-state Potts BCFT. PubDate: 2021-03-06
Abstract: We discuss the combined effects of overdamped motion in a quenched random potential and diffusion, in one dimension, in the limit where the diffusion coefficient is small. Our analysis considers the statistics of the mean first-passage time T(x) to reach position x, arising from different realisations of the random potential. Specifically, we contrast the median \({\bar{T}}(x)\) , which is an informative description of the typical course of the motion, with the expectation value \(\langle T(x)\rangle \) , which is dominated by rare events where there is an exceptionally high barrier to diffusion. We show that at relatively short times the median \({\bar{T}}(x)\) is explained by a ‘flooding’ model, where T(x) is predominantly determined by the highest barriers which are encountered before reaching position x. These highest barriers are quantified using methods of extreme value statistics. PubDate: 2021-03-05
Abstract: We present criteria for statistical stability of attracting sets for vector fields using dynamical conditions on the corresponding generated flows. These conditions are easily verified for all singular-hyperbolic attracting sets of \(C^2\) vector fields using known results, providing robust examples of statistically stable singular attracting sets (encompassing in particular the Lorenz and geometrical Lorenz attractors). These conditions are shown to hold also on the persistent but non-robust family of contracting Lorenz flows (also known as Rovella attractors), providing examples of statistical stability among members of non-open families of dynamical systems. In both instances, our conditions avoid the use of detailed information about perturbations of the one-dimensional induced dynamics on specially chosen Poincaré sections. PubDate: 2021-03-03
Abstract: Recently authors have introduced the idea of training discrete weights neural networks using a mix between classical simulated annealing and a replica ansatz known from the statistical physics literature. Among other points, they claim their method is able to find robust configurations. In this paper, we analyze this so called “replicated simulated annealing” algorithm. In particular, we give criteria to guarantee its convergence, and study when it successfully samples from configurations. We also perform experiments using synthetic and real data bases. PubDate: 2021-03-02
Abstract: We derive the existence of infinite level GREM-like K-processes by taking the limit of a sequence of finite level versions of such processes as the number of levels diverges. The main step in the derivation is obtaining the convergence of the sequence of underlying finite level clock processes. This is accomplished by perturbing these processes so as to turn them into martingales, and resorting to martingale convergence to obtain convergence for the perturbed clock processes; nontriviality of the limit requires a specific choice of parameters of the original process; we conclude the step by showing that the perturbation washes away in the limit. The perturbation is done by inserting suitable factors into the expression of the clocks, as well as rescaling the resulting expression suitably; the existence of such factors is itself established through martingale convergence. PubDate: 2021-03-02
Abstract: We consider the abelian stochastic sandpile model. In this model, a site is deemed unstable when it contains more than one particle. Each unstable site, independently, is toppled at rate 1, sending two of its particles to neighbouring sites chosen independently. We show that when the initial average density is less than 1/2, the system locally fixates almost surely. We achieve this bound by analysing the parity of the total number of times each site is visited by a large number of particles under the sandpile dynamics. PubDate: 2021-03-02
Abstract: Weber–Fechner laws are phenomenological relations describing a logarithmic relation between perception and sensory stimulus in a great variety of organisms. While firmly established, a theoretical argument for those laws in terms of relevant models or from statistical physics is largely missing. We present such a discussion in terms of response theory for nonequilibrium systems, where the induced displacement or current, which stands for the perceived stimulus, crucially depends on the change in time-symmetric reactivities. Stationary nonequilibria may indeed generate extra currents by changing the dynamical activity. The argument finishes by understanding how the extra dynamical activity logarithmically encodes the actual stimulus. PubDate: 2021-03-01
Abstract: We consider the one dimensional symmetric simple exclusion process with a slow bond. In this model, particles cross each bond at rate \(N^2\) , except one particular bond, the slow bond, where the rate is N. Above, N is the scaling parameter. This model has been considered in the context of hydrodynamic limits, fluctuations and large deviations. We investigate moderate deviations from hydrodynamics and obtain a moderate deviation principle. PubDate: 2021-02-26
Abstract: Dynamical systems with \(\varepsilon \) small random perturbations appear in both continuous mechanical motions and discrete stochastic chemical kinetics. The present work provides a detailed analysis of the central limit theorem (CLT), with a time-inhomogeneous Gaussian process, near a deterministic limit cycle in \(\mathbb {R}^n\) . Based on respectively the theory of random perturbations of dynamical systems and the WKB approximation that codes the large deviations principle (LDP), results are developed in parallel from both standpoints of stochastic trajectories and transition probability density and their relations are elucidated. We show rigorously the correspondence between the local Gaussian fluctuations and the curvature of the large deviation rate function near its infimum, connecting the CLT and the LDP of diffusion processes. We study uniform asymptotic behavior of stochastic limit cycles through the interchange of limits of time \(t\rightarrow \infty \) and \(\varepsilon \rightarrow 0\) . Three further characterizations of stochastic limit cycle oscillators are obtained: (i) An approximation of the probability flux near the cycle; (ii) Two special features of the vector field for the cyclic motion; (iii) A local entropy balance equation along the cycle with clear physical meanings. Lastly and different from the standard treatment, the origin of the \(\varepsilon \) in the theory is justified by a novel scaling hypothesis via constructing a sequence of stochastic differential equations. PubDate: 2021-02-25
Abstract: We generalize an idea in the works of Landauer and Bennett on computations, and Hill’s in chemical kinetics, to emphasize the importance of kinetic cycles in mesoscopic nonequilibrium thermodynamics (NET). For continuous stochastic systems, a NET in phase space is formulated in terms of cycle affinity \(\nabla \wedge \big (\mathbf{D}^{-1}\mathbf{b}\big )\) and vorticity potential \(\mathbf{A}(\mathbf{x})\) of the stationary flux \(\mathbf{J}^{*}=\nabla \times \mathbf{A}\) . Each bivectorial cycle couples two transport processes represented by vectors and gives rise to Onsager’s notion of reciprocality; the scalar product of the two bivectors \(\mathbf{A}\cdot \nabla \wedge \big (\mathbf{D}^{-1}\mathbf{b}\big )\) is the rate of local entropy production in the nonequilibrium steady state. An Onsager operator that relates vorticity to cycle affinity is introduced. PubDate: 2021-02-24
Abstract: We consider fast oscillating random perturbations of dynamical systems in regions where one can introduce action-angle-type coordinates. In an appropriate time scale, the evolution of first integrals, under the assumption that the set of resonance tori is small enough, is approximated by a diffusion process. If action-angle coordinates can be introduced only piece-wise, the limiting diffusion process should be considered on an open-book space. Such a process can be described by differential operators, one in each page, supplemented by some gluing conditions at the binding of the open book. PubDate: 2021-02-23
Abstract: We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation–dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products and number theory. PubDate: 2021-02-22
Abstract: Many integrable statistical mechanical models possess a fractional-spin conserved current. Such currents have been constructed by utilising quantum-group algebras and ideas from “discrete holomorphicity”. I find them naturally and much more generally using a braided tensor category, a topological structure arising in knot invariants, anyons and conformal field theory. I derive a simple constraint on the Boltzmann weights admitting a conserved current, generalising one found using quantum-group algebras. The resulting trigonometric weights are typically those of a critical integrable lattice model, so the method here gives a linear way of “Baxterising”, i.e. building a solution of the Yang-Baxter equation out of topological data. It also illuminates why many models do not admit a solution. I discuss many examples in geometric and local models, including (perhaps) a new solution. PubDate: 2021-02-18
Abstract: We study the behaviour of an interacting particle system, related to the Bak–Sneppen model and Jante’s law process defined in Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018). Let \(N\ge 3\) vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called fitness (we use this term, as it is quite standard for Bak–Sneppen models). Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution \(\zeta \) . We show that in case where \(\zeta \) is a finitely supported or continuous uniform distribution, all the fitnesses except one converge to the same value. PubDate: 2021-02-15
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