arXiv subjects

Hugo Touchette

Publications and source records attributed to Hugo Touchette.

At least 37 records · Page 2Linked to original sources

arXiv2018

A Generalized Multifractal Formalism for the Estimation of Nonconcave Multifractal Spectra

Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called multifractal spectrum. The practical estimation of the multifractal spectrum fundamentally relies on exploiting the scale dependence of statistical properties of appropriate multiscale quantities, such as wavelet leaders, that can be robustly computed from discrete data. Despite successes of multifractal analysis in various real-world applications, current estimation procedures remain essentially limited to providing concave upper-bound estimates, while there is a priori no reason for the multifractal spectrum to be a concave function. This work addresses this severe practical limitation and proposes a novel formalism for multifractal analysis that enables nonconcave multifractal spectra to be estimated in a stable way. The key contributions reside in the development and theoretical study of a generalized multifractal formalism to assess the multiscale statistics of wavelet leaders, and in devising a practical algorithm that permits this formalism to be applied to real-world data, allowing for the estimation of nonconcave multifractal spectra. Numerical experiments are conducted on several synthetic multifractal processes as well as on a real-world remote-sensing image and demonstrate the benefits of the proposed multifractal formalism over the state of the art.

math.FA↗

arXiv2018

Dynamical phase transition in drifted Brownian motion

We study the occupation fluctuations of drifted Brownian motion in a closed interval, and show that they undergo a dynamical phase transition in the long-time limit without an additional low-noise limit. This phase transition is similar to wetting and depinning transitions, and arises here as a switching between paths of the random motion leading to different occupations. For low occupations, the motion essentially stays in the interval for some fraction of time before escaping, while for high occupations the motion is confined in an ergodic way in the interval. This is confirmed by studying a confined version of the model, which points to a further link between the dynamical phase transition and quantum phase transitions. Other variations of the model, including the geometric Brownian motion used in finance, are considered to discuss the role of recurrent and transient motion in dynamical phase transitions.

cond-mat.stat-mech↗

arXiv2018

Spectral properties of simple classical and quantum reset processes

We study the spectral properties of classical and quantum Markovian processes that are reset at random times to a specific configuration or state with a reset rate that is independent of the current state of the system. We demonstrate that this simple reset dynamics causes a uniform shift in the eigenvalues of the Markov generator, excluding the zero mode corresponding to the stationary state, which has the effect of accelerating or even inducing relaxation to a stationary state. Based on this result, we provide expressions for the stationary state and probability current of the reset process in terms of weighted sums over dynamical modes of the reset-free process. We also discuss the effect of resets on processes that display metastability. We illustrate our results with two classical stochastic processes, the totally asymmetric random walk and the one-dimensional Brownian motion, as well as two quantum models: a particle coherently hopping on a chain and the dissipative transverse field Ising model, known to exhibit metastability.

cond-mat.stat-mech↗

arXiv2018

Asymptotic equivalence of probability measures and stochastic processes

Let $P_n$ and $Q_n$ be two probability measures representing two different probabilistic models of some system (e.g., an $n$-particle equilibrium system, a set of random graphs with $n$ vertices, or a stochastic process evolving over a time $n$) and let $M_n$ be a random variable representing a 'macrostate' or 'global observable' of that system. We provide sufficient conditions, based on the Radon-Nikodym derivative of $P_n$ and $Q_n$, for the set of typical values of $M_n$ obtained relative to $P_n$ to be the same as the set of typical values obtained relative to $Q_n$ in the limit $n\rightarrow\infty$. This extends to general probability measures and stochastic processes the well-known thermodynamic-limit equivalence of the microcanonical and canonical ensembles, related mathematically to the asymptotic equivalence of conditional and exponentially-tilted measures. In this more general sense, two probability measures that are asymptotically equivalent predict the same typical or macroscopic properties of the system they are meant to model.

cond-mat.stat-mech↗

arXiv2017

Process interpretation of current entropic bounds

We show for Markov diffusion processes that the quadratic entropic bound, recently derived for the rate functions of nonequilibrium currents, can be seen as being produced by an effective process that creates current fluctuations in a sub-optimal way by modifying only the non-reversible part of the drift or force of the process considered while keeping its reversible part constant. This provides a clear interpretation of the bound in terms of a physical process, which explains, among other things, its relation to the fluctuation relation, linear response, and reversible limits. The existence of more general quadratic bounds, and related uncertainty relations, for physical quantities other than currents is also discussed.

cond-mat.stat-mech↗

arXiv2017

Phase transitions in large deviations of reset processes

We study the large deviations of additive quantities, such as energy or current, in stochastic processes with intermittent reset. Via a mapping from a discrete-time reset process to the Poland-Scheraga model for DNA denaturation, we derive conditions for observing first-order or continuous dynamical phase transitions in the fluctuations of such quantities and confirm these conditions on simple random walk examples. These results apply to reset Markov processes, but also show more generally that subleading terms in generating functions can lead to non-analyticities in large deviation functions of 'compound processes' or 'random evolutions' switching stochastically between two or more subprocesses.

cond-mat.stat-mech↗

arXiv2017

A minimal model of dynamical phase transition

We calculate the large deviation functions characterizing the long-time fluctuations of the occupation of drifted Brownian motion and show that these functions have non-analytic points. This provides the first example of dynamical phase transition that appears in a simple, homogeneous Markov process without an additional low-noise, large-volume or hydrodynamic scaling limit.

cond-mat.stat-mech↗

arXiv2016

Current large deviations for driven periodic diffusions

We study the large deviations of the time-integrated current for a driven diffusion on the circle, often used as a model of nonequilibrium systems. We obtain the large deviation functions describing the current fluctuations using a Fourier-Bloch decomposition of the so-called tilted generator and also construct from this decomposition the effective (biased, auxiliary or driven) Markov process describing the diffusion as current fluctuations are observed in time. This effective process provides a clear physical explanation of the various fluctuation regimes observed. It is used here to obtain an upper bound on the current large deviation function, which we compare to a recently-derived entropic bound, and to study the low-noise limit of large deviations.

cond-mat.stat-mech↗

arXiv2016

Diffusions conditioned on occupation measures

A Markov process fluctuating away from its typical behavior can be represented in the long-time limit by another Markov process, called the effective or driven process, having the same stationary states as the original process conditioned on the fluctuation observed. We construct here this driven process for diffusions spending an atypical fraction of their evolution in some region of state space, corresponding mathematically to stochastic differential equations conditioned on occupation measures. As an illustration, we consider the Langevin equation conditioned on staying for a fraction of time in different intervals of the real line, including the positive half-line which leads to a generalization of the Brownian meander problem. Other applications related to quasi-stationary distributions, metastable states, noisy chemical reactions, queues, and random walks are discussed.

cond-mat.stat-mech↗

arXiv2015

Large deviations for Markov processes with resetting

Markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches, foraging, and population dynamics. Here we study the large deviations of time-additive functions or observables of Markov processes with resetting. By deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables, characterizing the likelihood of their fluctuations in the long-time limit. We consider as an illustration the large deviations of the area of the Ornstein-Uhlenbeck process with resetting. Other applications involving diffusions, random walks, and jump processes with resetting or catastrophes are discussed.

cond-mat.stat-mech↗

arXiv2015

Variational and optimal control representations of conditioned and driven processes

We have shown recently that a Markov process conditioned on rare events involving time-integrated random variables can be described in the long-time limit by an effective Markov process, called the driven process, which is given mathematically by a generalization of Doob's $h$-transform. We show here that this driven process can be represented in two other ways: first, as a process satisfying various variational principles involving large deviation functions and relative entropies and, second, as an optimal stochastic control process minimizing a cost function also related to large deviation functions. These interpretations of the driven process generalize and unify many previous results on maximum entropy approaches to nonequilibrium systems, spectral characterizations of positive operators, and control approaches to large deviation theory. They also lead, as briefly discussed, to new methods for analytically or numerically approximating large deviation functions.

cond-mat.stat-mech↗

arXiv2015

Convergence of large deviation estimators

We study the convergence of statistical estimators used in the estimation of large deviation functions describing the fluctuations of equilibrium, nonequilibrium, and manmade stochastic systems. We give conditions for the convergence of these estimators with sample size, based on the boundedness or unboundedness of the quantity sampled, and discuss how statistical errors should be defined in different parts of the convergence region. Our results shed light on previous reports of 'phase transitions' in the statistics of free energy estimators and establish a general framework for reliably estimating large deviation functions from simulation and experimental data and identifying parameter regions where this estimation converges.

cond-mat.stat-mech↗

arXiv2015

Equivalence and nonequivalence of ensembles: Thermodynamic, macrostate, and measure levels

We present general and rigorous results showing that the microcanonical and canonical ensembles are equivalent at all three levels of description considered in statistical mechanics - namely, thermodynamics, equilibrium macrostates, and microstate measures - whenever the microcanonical entropy is concave as a function of the energy density in the thermodynamic limit. This is proved for any classical many-particle systems for which thermodynamic functions and equilibrium macrostates exist and are defined via large deviation principles, generalizing many previous results obtained for specific classes of systems and observables. Similar results hold for other dual ensembles, such as the canonical and grand-canonical ensembles, in addition to path ensembles used for describing nonequilibrium systems driven in steady states.

cond-mat.stat-mech↗

arXiv2014

Nonequilibrium Markov processes conditioned on large deviations

We consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob's $h$-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, which is used with importance sampling to simulate rare events, and which gives rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned.

cond-mat.stat-mech↗

arXiv2014

Large deviations for correlated random variables described by a matrix product ansatz

We study the large deviations of sums of correlated random variables described by a matrix product ansatz, which generalizes the product structure of independent random variables to matrices whose non-commutativity is the source of correlations. We show with specific examples that different large deviation behaviors can be found with this ansatz. In particular, it is possible to construct sums of correlated random variables that violate the Law of Large Numbers, the Central Limit Theorem, as well as sums that have nonconvex rate functions or rate functions with linear parts or plateaux.

cond-mat.stat-mech↗

arXiv2014

Fluctuation relations for anisotropic systems

Currents of particles or energy in driven nonequilibrium steady states are known to satisfy certain symmetries, referred to as fluctuation relations, determining the ratio of the probabilities of positive fluctuations to negative ones. A generalization of these fluctuation relations has been proposed recently for extended nonequilibrium systems of dimension greater than one, assuming, crucially, that they are isotropic [P. I.\ Hurtado, C.\ Pérez-Espigares, J. J.\ del Pozo, and P. L.\ Garrido, Proc.\ Nat.\ Acad.\ Sci.\ (USA) \textbf{108}, 7704 (2011)]. Here we relax this assumption and derive a fluctuation relation for $d$-dimensional systems having anisotropic bulk driving rates. We illustrate this anisotropic fluctuation relation for the particle current fluctuations in the 2-$d$ anisotropic zero-range process, using both exact and fluctuating hydrodynamic approaches.

cond-mat.stat-mech↗

arXiv2013

Weak-noise limit of a piecewise-smooth stochastic differential equation

We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a simple model of Brownian motion with solid friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided that the singularity of the path integral associated with the non-smooth SDE is treated with some heuristics. We also show that, as in the case of smooth SDEs, the deterministic paths of the noiseless system correctly describe the behaviour of the non-smooth SDE in the low-noise limit. Finally, we consider a smooth regularisation of the piecewise-constant SDE and study to which extent this regularisation can rectify some of the problems encountered when dealing with discontinuous drifts and singularities in SDEs.

cond-mat.stat-mech↗

arXiv2013

Nonequilibrium microcanonical and canonical ensembles and their equivalence

Generalizations of the microcanonical and canonical ensembles for paths of Markov processes have been proposed recently to describe the statistical properties of nonequilibrium systems driven in steady states. Here we propose a theory of these ensembles that unifies and generalizes earlier results, and show how it is fundamentally related to the large deviation properties of nonequilibrium systems. Using this theory, we provide conditions for the equivalence of nonequilibrium ensembles, generalizing those found for equilibrium systems, construct driven physical processes that generate these ensembles, and re-derive in a simple way known and new product rules for their transition rates. A nonequilibrium diffusion model is used to illustrate these results.

cond-mat.stat-mech↗