55 research outputs found
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
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
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
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
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