Abstract: A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.

Abstract: We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport and becomes relevant for processes far from quasi-stationary regime. General discussion is illustrated by numerical analysis of the optimal memory erasure protocol for a model for micron-size particle manipulated by optical tweezers.

Abstract: We present a brief report on the different methods for finding exact solutions of nonlinear evolution equations. Explicit exact traveling wave solutions are the most amenable besides implicit and parametric ones. It is shown that most of methods that exist in the literature are equivalent to the “generalized mapping method” that unifies them. By using this method a class of formal exact solutions for reaction diffusion equations with finite memory transport is obtained. Attention is focused to the finite-memory-transport-Fisher and Nagumo equations.

Abstract: We present a procedure to solve the inverse Ising problem, that is, to find the interactions between a set of binary variables from the measure of their equilibrium correlations. The method consists in constructing and selecting specific clusters of spins, based on their contributions to the cross-entropy of the Ising model. Small contributions are discarded to avoid overfitting and to make the computation tractable. The properties of the cluster expansion and its performances on synthetic data are studied. To make the implementation easier we give the pseudo-code of the algorithm.

Abstract: We develop a cluster expansion for the probability of full connectivity of high density random networks in confined geometries. In contrast to percolation phenomena at lower densities, boundary effects, which have previously been largely neglected, are not only relevant but dominant. We derive general analytical formulas that show a persistence of universality in a different form to percolation theory, and provide numerical confirmation. We also demonstrate the simplicity of our approach in three simple but instructive examples and discuss the practical benefits of its application to different models.

Abstract: We proposed in Beltran and Landim (J. Stat. Phys. 140:1065–1114, 2010) a new approach to prove the metastable behavior of reversible dynamics based on potential theory and local ergodicity. In this article we extend this theory to nonreversible dynamics based on the Dirichlet principle proved in Gaudilliere and Landim (arXiv: 1111.2445 , 2011). We also include in this article the proof of the metastability of a class of birth and death chains.

Abstract: We compute the joint probability density function (jpdf) P N (M,τ M ) of the maximum M and its position τ M for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For N→∞, this jpdf is peaked around $M = \sqrt{2N}$ and τ M =1/2, while the typical fluctuations behave for large N like $M - \sqrt{2N} \propto s N^{-1/6}$ and τ M −1/2∝wN −1/3 where s and w are correlated random variables. One obtains an explicit expression of the limiting jpdf P(s,w) in terms of the Tracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory and a psi-function for the Hastings-McLeod solution to the Painleve II equation. Our result yields, up to a rescaling of the random variables s and w, an expression for the jpdf of the maximum and its position for the Airy2 process minus a parabola. This latter describes the fluctuations in many different physical systems belonging to the Kardar-Parisi-Zhang (KPZ) universality class in 1+1 dimensions. In particular, the marginal probability density function (pdf) P(w) yields, up to a model dependent length scale, the distribution of the endpoint of the directed polymer in a random medium with one free end, at zero temperature. In the large w limit one shows the asymptotic behavior logP(w)∼−w 3/12.

Abstract: We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to T 1/3. We conjecture that this is the worse case correction possible.

Abstract: We study the adiabatic response of open systems governed by Lindblad evolutions. In such systems, there is an ambiguity in the assignment of observables to fluxes (rates) such as velocities and currents. For the appropriate notion of flux, the formulas for the transport coefficients are simple and explicit and are governed by the parallel transport on the manifold of instantaneous stationary states. Among our results we show that the response coefficients of open systems, whose stationary states are projections, is given by the adiabatic curvature.

Abstract: We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the Airy2 process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.

Abstract: We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Ito stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Ito (α=0), Stratonovich (α=0.5) and anti-Ito (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations.

Abstract: Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

Abstract: We present an exact calculation of the mean first-passage time to a target on the surface of a 2D or 3D spherical domain, for a molecule alternating phases of surface diffusion on the domain boundary and phases of bulk diffusion. We generalize the results of Benichou et al. in (J. Stat. Phys. 142:657, 2011) and consider a biased diffusion in a general annulus with an arbitrary number of regularly spaced targets on a partially reflecting surface. The presented approach is based on an integral equation which can be solved analytically. Numerically validated approximation schemes, which provide more tractable expressions of the mean first-passage time are also proposed. In the framework of this minimal model of surface-mediated reactions, we show analytically that the mean reaction time can be minimized as a function of the desorption rate from the surface.

Abstract: We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β=1), Hermitian (β=2), and Hermitian self-dual (β=4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “one-band” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β>0.

Abstract: In this paper we study the Blackwell and Furstenberg measures, which play an important role in information theory and the study of Lyapunov exponents. For the Blackwell measure we determine parameter domains of singularity and give upper bounds for the Hausdorff dimension. For the Furstenberg measure, we establish absolute continuity for some parameter values. Our method is to analyze linear fractional iterated function schemes which are contracting on average, have no separation properties (that is, we do not assume that the open set condition holds, see Hutchinson in Indiana Univ. Math. J. 30:713–747, 1981) and, in the case of the Blackwell measure, have place dependent probabilities. In such a general setting, even an effective upper bound on the dimension of the measure is difficult to achieve.

Abstract: The stationary states of the kinetic mixed spin-2 and spin-5/2 Ising ferrimagnetic system with repulsive biquadratic coupling are examined within a mean-field approach under the presence of a time varying (sinusoidal) magnetic field. We employ Glauber-type stochastic dynamics to construct a set of coupled mean-field dynamic equations, and we solve these equations to find the phases in the system. We also investigate the thermal behavior of the dynamic order parameters to characterize the nature (continuous or discontinuous) of the dynamic phase transitions (DPTs) and obtain the DPT points. The phase diagrams are presented in the reduced magnetic field amplitude and reduced temperature plane, and a comparison is made with the results of other kinetic mixed spin systems. We also study the behavior of the dynamic magnetic hysteresis and investigate the effects of the frequency on hysteresis properties, and we find the results are in good agreement with some previous theoretical and experimental works.

Abstract: Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only—at most— local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deter- ministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of the individuals. Stochastic equations for the population do account for these fluctuation-generated effects by means of demographic noise terms but, owing to their complexity, they can be difficult (or, at times, impossible) to deal with. Even when they can be written in a simple form, they are still diffi- cult to numerically integrate due to the presence of the "square-root" intrinsic noise. In this paper, we discuss a simple way to add the effect of demographic stochasticity to three clas- sic, deterministic ecological examples where aggregation plays an important role. We study the resulting equations using a recently-introduced integration scheme especially devised to integrate numerically stochastic equations with demographic noise. Aimed at scrutinizing the ability of these stochastic examples to show aggregation, we find that the three systems not only show patchy configurations, but also undergo a phase transition belonging to the directed percolation universality class.

Abstract: The work by Ott et al. (Math. Res. Lett. 16:463–475, 2009) established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case). In fact, the present paper is the first one where non-random examples are also found, which are not small perturbations of a given map. Our approach uses martingale approximation technique in the form of Sethuraman and Varadhan (Electron. J. Probab. 10:121–1235, 2005).

Abstract: Motivated by the general problem of studying sample-to-sample fluctuations in disorder-generated multifractal patterns we attempt to investigate analytically as well as numerically the statistics of high values of the simplest model—the ideal periodic 1/f Gaussian noise. Our main object of interest is the number of points $\mathcal{N}_{M}(x)$ above a level $\frac{x}{2}V_{m}$ , with V m =2lnM standing for the leading-order typical value of the absolute maximum for the sample of M points. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of $\mathcal{N}_{M}(x)$ for 00. For the 1/f noise case we further study asymptotic values of the prefactors in scaling laws for the moments of the counting function. Our numerics shows however that one needs prohibitively large sample sizes to reach such asymptotics even with a moderate precision. This motivates us to derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism.

Abstract: We consider a macroscopic system in contact with boundary reservoirs and/or under the action of an external field. We discuss the case in which the external forcing depends explicitly on time and drives the system from a nonequilibrium state to another one. In this case the amount of energy dissipated along the transformation becomes infinite when an unbounded time window is considered. Following the general proposal by Oono and Paniconi and using results of the macroscopic fluctuation theory, we give a natural definition of a renormalized work. We then discuss its thermodynamic relevance by showing that it satisfies a Clausius inequality and that quasi static transformations minimize the renormalized work. In addition, we connect the renormalized work to the quasi potential describing the fluctuations in the stationary nonequilibrium ensemble. The latter result provides a characterization of the quasi potential that does not involve rare fluctuations.

Abstract: In this paper we construct several models with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order k≥2. We prove that each of the constructed model has at least two translational-invariant Gibbs measures.

Abstract: We study the Ising model on a Cayley tree. A wide class of new extreme Gibbs states is exhibited.

Abstract: Stochastic differential equations (SDE) are widely used in modeling stochastic dynamics in literature. However, SDE alone is not enough to determine a unique process. A specified interpretation for stochastic integration is needed. Different interpretations specify different dynamics. Recently, a new interpretation of SDE is put forward by one of us. This interpretation has a built-in Boltzmann-Gibbs distribution and shows the existence of potential function for general processes, which reveals both local and global dynamics. Despite its powerful property, its relation with classical ones in arbitrary dimension remains obscure. In this paper, we will clarify such connection and derive the concise relation between the new interpretation and Ito process. We point out that the derived relation is experimentally testable.

Abstract: A time-evolution of a slightly rarefied monoatomic gas, namely a gas for small Knudsen numbers, which is perturbed slowly and slightly from a reference uniform equilibrium state at rest is investigated on the basis of the linearized Boltzmann equation. By a systematic asymptotic analysis, a set of fluid-dynamic-type equations and its boundary conditions that describe the gas behavior up to the second order of the Knudsen number are derived. The developed theory covers a general intermolecular potential and a gas-surface interaction. It is shown that (i) the compressibility of the gas manifests itself from the leading order in the energy equation and from the first order in the continuity equation; (ii) although the momentum equation is the Stokes equation, it contains a double Laplacian of the leading order flow velocity as a source term at the second order; (iii) a double Laplacian source term also appears in the energy equation at the second order; (iv) the slip and jump conditions are the same as those in the time-independent case up to the first order, and the difference occurs at the second order in the jump conditions as the terms of the divergence of flow velocity and of the Laplacian of temperature. Numerical values of all the slip and jump coefficients are obtained for a hard-sphere gas by the use of a symmetric relation developed recently.

Abstract: We calculate the 'one-point function', meaning the marginal probability density function for any single eigenvalue, of real and complex Wishart correlation matrices. No ex- plicit expression had been obtained for the real case so far. We succeed in doing so by using supersymmetry techniques to express the one-point function of real Wishart correlation ma- trices as a twofold integral. The result can be viewed as a resummation of a series of Jack polynomials in a non-trivial case. We illustrate our formula by numerical simulations. We also rederive a known expression for the one-point function of complex Wishart correlation matrices.

Abstract: We show how kinetic theory, the statistics of classical particles obeying Newtonian dynamics, can be formulated as a field theory. The field theory can be organized to produce a self-consistent perturbation theory expansion in an effective interaction potential. The need for a self-consistent approach is suggested by our interest in investigating ergodic-nonergodic transitions in dense fluids. The formal structure we develop has been implemented in detail for the simpler case of Smoluchowski dynamics. One aspect of the approach is the identification of a core problem spanned by the variables ρ the number density and B a response density. In this paper we set up the perturbation theory expansion with explicit development at zeroth and first order. We also determine all of the cumulants in the noninteracting limit among the core variables ρ and B.

Abstract: We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q≥3 states and show that it undergoes a critical slowdown at an inverse-temperature β s (q) strictly lower than the critical β c (q) for uniqueness of the thermodynamic limit. The dynamical critical β s (q) is the spinodal point marking the onset of metastability. We prove that when β<β s (q) the mixing time is asymptotically C(β,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At β=β s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n 4/3. For β>β s (q) the mixing time is exponentially large in n. Furthermore, as β↑β s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n −2/3) around β s . These results form the first complete analysis of mixing around the critical dynamical temperature—including the critical power law—for a model with a first order phase transition.

Abstract: The compacting of a column of grains has been studied using a one-dimensional Ising model with long range directed interactions in which down and up spins represent orientations of the grain having or not having an associated void. When the column is not shaken (zero “temperature”) the motion becomes highly constrained and under most circumstances we find that the generator of the stochastic dynamics assumes an unusual form: many eigenvalues become degenerate, but the associated multi-dimensional invariant spaces have but a single eigenvector. There is no spectral expansion and a Jordan form must be used. Many properties of the dynamics are established here analytically; some are not. General issues associated with the Jordan form are also taken up.

Abstract: Here we study thermodynamic properties of an important class of single-chain magnets (SCMs), where alternate units are isotropic and anisotropic with anisotropy axes being non-collinear. This class of SCMs shows slow relaxation at low temperatures which results from the interplay of two different relaxation mechanisms, namely dynamical and thermal. Here anisotropy is assumed to be large and negative, as a result, anisotropic units behave like canted spins at low temperatures; but even then simple Ising-type model does not capture the essential physics of the system due to quantum mechanical nature of the isotropic units. We here show how statistical behavior of this class of SCMs can be studied using a transfer matrix (TM) method. We also, for the first time, discuss in detail how weak inter-chain interactions can be treated by a TM method. The finite size effect is also discussed which becomes important for low temperature dynamics. At the end of this paper, we apply this technique to study a real helical chain magnet.