Large Deviations in Renewal Models of Statistical Mechanics

In Ref. [1] the author has recently established sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a separable Banach space. The renewal model has been there identified with constrained and non-constrained pinning models of polymers, which amount to Gibbs changes of measure of a classical renewal process. In this paper we show that the constrained pinning model is the common mathematical structure to the Poland-Scheraga model of DNA denaturation and to some relevant one-dimensional lattice models of Statistical Mechanics, such as the Fisher-Felderhof model of fluids, the Wako-Sait\^o-Mu\~noz-Eaton model of protein folding, and the Tokar-Dreyss\'e model of strained epitaxy. Then, in the framework of the constrained pinning model, we develop an analytical characterization of the large deviation principles for cumulative rewards corresponding to multivariate deterministic rewards that are uniquely determined by, and at most of the order of magnitude of, the time elapsed between consecutive renewals. In particular, we outline the explicit calculation of the rate functions and successively we identify the conditions that prevent them from being analytic and that underlie affine stretches in their graphs. Finally, we apply the general theory to the number of renewals. From the point of view of Equilibrium Statistical Physics and Statistical Mechanics, cumulative rewards of the above type are the extensive observables that enter the thermodynamic description of the system. The number of renewals, which turns out to be the commonly adopted order parameter for the Poland-Scheraga model and for also the renewal models of Statistical Mechanics, is one of these observables

Apparent multifractality of self-similar L\'evy processes

Scaling properties of time series are usually studied in terms of the scaling laws of empirical moments, which are the time average estimates of moments of the dynamic variable. Nonlinearities in the scaling function of empirical moments are generally regarded as a sign of multifractality in the data. We show that, except for the Brownian motion, this method fails to disclose the correct monofractal nature of self-similar L\'evy processes. We prove that for this class of processes it produces apparent multifractality characterised by a piecewise-linear scaling function with two different regimes, which match at the stability index of the considered process. This result is motivated by previous numerical evidence. It is obtained by introducing an appropriate stochastic normalisation which is able to cure empirical moments, without hiding their dependence on time, when moments they aim at estimating do not exist

An exactly solvable model for a beta-hairpin with random interactions

I investigate a disordered version of a simplified model of protein folding, with binary degrees of freedom, applied to an ideal beta-hairpin structure. Disorder is introduced by assuming that the contact energies are independent and identically distributed random variables. The equilibrium free-energy of the model is studied, performing the exact calculation of its quenched value and proving the self-averaging feature.Comment: 9 page

On the Mean Residence Time in Stochastic Lattice-Gas Models

A heuristic law widely used in fluid dynamics for steady flows states that the amount of a fluid in a control volume is the product of the fluid influx and the mean time that the particles of the fluid spend in the volume, or mean residence time. We rigorously prove that if the mean residence time is introduced in terms of sample-path averages, then stochastic lattice-gas models with general injection, diffusion, and extraction dynamics verify this law. Only mild assumptions are needed in order to make the particles distinguishable so that their residence time can be unambiguously defined. We use our general result to obtain explicit expressions of the mean residence time for the Ising model on a ring with Glauber + Kawasaki dynamics and for the totally asymmetric simple exclusion process with open boundaries

Large Deviations in Discrete-Time Renewal Theory

We establish sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. The framework we consider is the pinning model of polymers, which amounts to a Gibbs change of measure of a classical renewal process and includes it as a special case. We first tackle the problem in a constrained pinning model, where one of the renewals occurs at a given time, by an argument based on convexity and super-additivity. We then transfer the results to the original pinning model by resorting to conditioning

Dynamical transition in the TASEP with Langmuir kinetics: mean-field theory

We develop a mean-field theory for the totally asymmetric simple exclusion process (TASEP) with open boundaries, in order to investigate the so-called dynamical transition. The latter phenomenon appears as a singularity in the relaxation rate of the system toward its non-equilibrium steady state. In the high-density (low-density) phase, the relaxation rate becomes independent of the injection (extraction) rate, at a certain critical value of the parameter itself, and this transition is not accompanied by any qualitative change in the steady-state behavior. We characterize the relaxation rate by providing rigorous bounds, which become tight in the thermodynamic limit. These results are generalized to the TASEP with Langmuir kinetics, where particles can also bind to empty sites or unbind from occupied ones, in the symmetric case of equal binding/unbinding rates. The theory predicts a dynamical transition to occur in this case as well.Comment: 37 pages (including 16 appendix pages), 6 figures. Submitted to Journal of Physics

Large deviation principles for renewal-reward processes

We establish a sharp large deviation principle for renewal-reward processes, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. In fact, we demonstrate a weak large deviation principle without assuming any exponential moment condition on the law of waiting times and rewards by resorting to a sharp version of Cram\'er's theorem. We also exhibit sufficient conditions for exponential tightness of renewal-reward processes, which leads to a full large deviation principle

A simplified exactly solvable model for beta-amyloid aggregation

We propose an exactly solvable simplified statistical mechanical model for the thermodynamics of beta-amyloid aggregation, generalizing a well-studied model for protein folding. The monomer concentration is explicitly taken into account as well as a non trivial dependence on the microscopic degrees of freedom of the single peptide chain, both in the alpha-helix folded isolated state and in the fibrillar one. The phase diagram of the model is studied and compared to the outcome of fibril formation experiments which is qualitatively reproduced.Comment: 4 pages, 2 figure

Large Fluctuations and Transport Properties of the L\'evy-Lorentz gas

The L\'evy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d.\ random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this paper we study the large fluctuations of the continuous-time process and the resulting transport properties of the model, both annealed and quenched, confirming and extending previous work by physicists that pertain to the annealed framework. Specifically, focusing on the particle displacement, and under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity, we prove a uniform large deviation principle for the annealed fluctuations and present the asymptotics of annealed moments, demonstrating annealed superdiffusion. Then, we provide an upper large deviation bound for the quenched fluctuations and the asymptotics of quenched moments, showing that, somehow unexpectedly, the asymptotically stable diffusive regime conditional on a typical arrangement of the scatterers is normal diffusion, and not superdiffusion. Although the L\'evy-Lorentz gas seems to be accepted as a model for anomalous diffusion, our findings lead to the conclusion that superdiffusion is a metastable behavior, which develops into normal diffusion on long timescales, and raise a new question about how the transition from the quenched normal diffusion to the annealed superdiffusion occurs

Statistical fluctuations under resetting: rigorous results

In this paper we investigate the normal and the large fluctuations of additive functionals of the Brownian motion under a general non-Poissonian resetting mechanism. Cumulative functionals of regenerative processes are very close to renewal-reward processes and inherit most of the properties of the latter. Here we review and use the classical law of large numbers and central limit theorem for renewal-reward processes to obtain same theorems for additive functionals of the reset Brownian motion. Then, we establish large deviation principles for these functionals by illustrating and applying a large deviation theory for renewal-reward processes that has been recently developed by the author. We discuss applications of the general results to the positive occupation time, the area, and the absolute area. While introducing advanced tools from renewal theory, we demonstrate that a rich phenomenology accounting for dynamical phase transitions emerges when one goes beyond Poissonian resetting.Comment: Submitted to the special issue of Journal of Physics A: Mathematical and Theoretical on "Stochastic Resetting: Theory and Applications