arXiv subjects

Vicenç Méndez

Publications and source records attributed to Vicenç Méndez.

At least 19 recordsLinked to original sources

arXiv2026

Absolute area under a scaled Brownian motion with resetting

We investigate the statistical properties and ergodic behavior of the absolute area below the trajectory of a particle subject to stochastic resetting and moving according to a scaled Brownian motion with exponent $β$. We compare the contributions to the absolute area from the waiting times between resets and the backward recurrence time. Using the renewal formalism we derive the lower-order moments of the absolute area under both Poissonian and heavy-tailed power-law resetting waiting time distributions with exponent $α$. For Poissonian resetting, stochastic interruption suppresses fluctuations, driving the area to be ergodic, regardless of the value of $β$. In contrast, power-law resetting yields a rich spectrum of ergodic regimes. For $0 < α< 1$, the system displays weak ergodicity breaking and the maximum contribution dominates the area accumulated during the waiting times. For $1+β< α< 2$, we unveil an ergodic transition exclusive to the subdiffusive regime at $α=1+β$, where ergodicity is restored. These results clarify the subtle interplay between anomalous diffusion and power-law waiting times between resettings. Our results are tested by numerical simulations exhibiting an excellent agreement.

cond-mat.stat-mech↗

arXiv2026

Infinite ergodic theory and functional statistics of non-confined Feller process

We study additive observables of a non-confined Feller process characterized by a non- normalizable stationary probability density and return times with infinite mean. The process is equivalent, after a deterministic rescaling, to a squared Bessel process, but we focus here on a question: the spatial structure of its local-time field. We show that the local time admits a fac- torization in the long time limit. Its spatial profile is governed by the non-normalizable stationary density, whereas its temporal fluctuations are controlled by a single Mittag-Leffler random ampli- tude. As a consequence, normalized spatial correlations of the local time converge to a universal constant independent of the two observation levels. Occupation times of finite intervals follow as corollaries and display Darling-Kac fluctuations. In contrast, non-integrable power observables exhibit self-similar squared-Bessel-type limits rather than Mittag-Leffler statistics. This spatial- field perspective provides a unifying framework for infinite ergodic theory under state-dependent multiplicative noise.

cond-mat.stat-mech↗

arXiv2026

Ergodic properties of functionals of Gaussian processes

We derive the first two moments of generic positive stochastic functionals in terms of the one- and two-time probability density functions of the underlying random walk, and we prove ergodicity of observables in stationary random walks. These general results are applied to the half-occupation time and the occupation time in an interval of a Gaussian random walk, for which we obtain exact analytic expressions for the first two moments. We then extend the analysis to scaled Brownian motion and fractional Brownian motion, computing the ergodicity breaking parameter and establishing a simple scaling form for the probability densities of occupation times. Within the framework of infinite ergodic theory, we further identify universal properties of positive observables. All analytical predictions are fully confirmed by numerical simulations.

cond-mat.stat-mech↗

arXiv2026

Area under subdiffusive random walks

We study the statistical properties of the area and the absolute area under the trajectories of subdiffusive random walks. Using different frameworks to describe subdiffusion (as the scaled Brownian motion, fractional Brownian motion, the continuous-time random walk or the Brownian motion in heterogeneous media), we compute the first two moments, the ergodicity breaking parameter for the absolute area and infer a general scaling for the probability density functions of these functionals. We discuss the differences between the statistical properties of the area and the absolute area for the different subdiffusion models and illustrate the experimental interest of our results. Our theoretical findings are supported by Monte Carlo simulations showing an excellent agreement.

cond-mat.stat-mech↗

arXiv2025

Ergodic properties of occupation times in heterogeneous media

We investigate the ergodic properties of Brownian motion in heterogeneous media through the statistics of occupation times. Using the Feynman-Kac formalism, we derive analytical expressions for the distributions, moments, and ergodicity breaking parameters of occupation times in two models with spatially varying diffusion coefficient: a piecewise-constant profile and a power-law profile. In the piecewise model, the half occupation time and the occupation time within an interval follow asymmetric arcsine and half-Gaussian distributions, respectively, indicating non-ergodic behavior. For the power-law case, the corresponding distributions are the Lamperti and Mittag-Leffler. In both models, we identify a transition from non-ergodic to ergodic dynamics as the exponent vary. Numerical simulations fully corroborate the analytical results, demonstrating the effectiveness of the Feynman-Kac approach for quantifying ergodicity in heterogeneous diffusion processes.

cond-mat.stat-mech↗

arXiv2024

Occupation time statistics for non-Markovian random walks

We study the occupation time statistics for non-Markovian random walkers based on the formalism of the generalized master equation for the Continuous-Time Random Walk. We also explore the case when the random walker additionally undergoes a stochastic resetting dynamics. We derive and solve the backward Feynman-Kac equation to find the characteristic function for the occupation time in an interval and for the half occupation time in the semi-infinite domain. We analyze the behaviour of the PDFs, the moments, the limiting distributions and the ergodic properties for both occupation times when the underlying random walk is normal or anomalous. For the half occupation time, we revisit the famous arcsine law and examine its validity pertaining to various regimes of the rest period of the walker. Our results have been verified with numerical simulations exhibiting an excellent agreement.

cond-mat.stat-mech↗

arXiv2024

Intermittent random walks under stochastic resetting

We analyze a one-dimensional intermittent random walk on an unbounded domain in the presence of stochastic resetting. In this process, the walker alternates between local intensive search, diffusion, and rapid ballistic relocations in which it does not react to the target. We demonstrate that Poissonian resetting leads to the existence of a non-equilibrium steady state. We calculate the distribution of the first arrival time to a target along with its mean and show the existence of an optimal reset rate. In particular, we prove that the initial condition of the walker, i.e., either starting diffusely or relocating, can significantly affect the long-time properties of the search process. Moreover, we demonstrate the presence of distinct parameter regimes for the global optimization of the mean first arrival time when ballistic and diffusive movements are in direct competition.

cond-mat.stat-mech↗

arXiv2023

Dynamic redundancy as a mechanism to optimize collective random searches

We explore the case of a group of random walkers looking for a target randomly located in space, such that the number of walkers is not constant but new ones can join the search, or those that are active can abandon it, with constant rates $r_b$ and $r_d$, respectively. Exact analytical solutions are provided both for the fastest-first-passage time and for the collective search time required to reach the target, in the seminal case of Brownian walkers with $r_d=0$. We prove that even for such a simplified situation there exists an optimal rate $r_b$ at which walkers should join the search to minimize the collective effort required to reach the target. We discuss how these results open a new line to understand the optimal regulation of cooperative random searches, e.g. for the case of biological foraging in social species.

cond-mat.stat-mech↗

arXiv2023

Ergodic properties of Brownian motion under stochastic resetting

We study ergodic properties of one-dimensional Brownian motion with resetting. Using generic classes of statistics of times between resets, we find respectively for thin/fat tailed distributions, the normalized/non-normalised invariant density of this process. The former case corresponds to known results in the resetting literature and the latter to infinite ergodic theory. Two types of ergodic transitions are found in this system. The first is when the mean waiting time between resets diverges, when standard ergodic theory switches to infinite ergodic theory. The second is when the mean of the square root of time between resets diverges and the properties of the invariant density are drastically modified. We then find a fractional integral equation describing the density of particles. This finite time tool is particularly useful close to the ergodic transition where convergence to asymptotic limits is logarithmically slow. Our study implies rich ergodic behaviors for this non-equilibrium process which should hold far beyond the case of Brownian motion analyzed here.

cond-mat.stat-mech↗

arXiv2022

Conditioned backward and forward times of diffusion with stochastic resetting: a renewal theory approach

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward and forward times for stochastic processes with resetting to be the times since the last and until the next reset, given that the current state of the system is known. We focus on studying diffusion under Markovian and non-Markovian resetting. For these cases, we find the conditioned backward and forward time PDFs, comparing them with numerical simulations of the process. In particular, we find that for power-law reset time PDFs with asymptotic form $φ(t)\sim t^{-1-α}$, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of $α$. This is due to the composition between the long-time scaling of diffusion $P(x,t)\sim 1/\sqrt{t}$ and the reset time PDF.

cond-mat.stat-mech↗

arXiv2022

Non-standard diffusion under Markovian resetting in bounded domains

We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results restricted to Markovian random walks, we here find that, depending on the value of the relative standard deviation of the waiting time probability, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset to the boundary, or (iii) always beneficial.

cond-mat.stat-mech↗

arXiv2021

Phase transition in non-Markovian animal exploration model with preferential returns

We study a non-Markovian and nonstationary model of animal mobility incorporating both exploration and memory in the form of preferential returns. We derive exact results for the probability of visiting a given number of sites and develop a practical WKB approximation to treat the nonstationary problem. We further show that this model adequately describes empirical movement data of Egyptian fruit bats (Rousettus aegyptiacus) when accounting for inter-individual variation in the population. Finally, we study the probability of visiting any site a given number of times and derive the corresponding mean-field equation. Here, we find a remarkable phase transition occurring at preferential returns which scale linearly with past visits. Following empirical evidence, we suggest that this phase transition reflects a trade-off between extensive and intensive foraging modes.

cond-mat.stat-mech↗

arXiv2021

Informational entropy thresholds as a physical mechanism to explain power-law time distributions in sequential decision-making

While frameworks based on physical grounds (like the Drift-Diffusion Model) have been exhaustively used in psychology and neuroscience to describe perceptual decision-making in humans, analogous approaches for more complex situations like sequential (tree-like) decision making are still absent. For such scenarios, which involve a reflective prospection of future options to reach a decision, we offer a plausible mechanism based on the internal computation of the Shannon's entropy for the different options available to the subjects. When a threshold in the entropy is reached this will trigger the decision, which means that the amount of information that has been gathered through sensory evidence is enough to assess the options accurately. Experimental evidence in favour of this mechanism is provided by exploring human performances during navigation through a maze on the computer screen monitored with the help of eye-trackers. In particular, our analysis allows us to prove that: (i) prospection is effectively being used by humans during such navigation tasks, and a quantification of the level of prospection used is attainable, (ii) the distribution of decision times during the task exhibits power-law tails, a feature that our entropy-based mechanism is able to explain, in contrast to classical decision-making frameworks.

physics.soc-ph↗

arXiv2020

Continuous time random walks under Markovian resetting

We investigate the effects of markovian resseting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power law probability density functions. We prove the existence of a non-equilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target and the characteristic transport exponents.

cond-mat.stat-mech↗

arXiv2020

Stochastic resetting on comb-like structures

We study a diffusion process on a three-dimensional comb under stochastic resetting. We consider three different types of resetting: global resetting from any point in the comb to the initial position, resetting from a finger to the corresponding backbone and resetting from secondary fingers to the main fingers. The transient dynamics along the backbone in all three cases is different due to the different resetting mechanisms, finding a wide range of dynamics for the mean squared displacement. For the particular geometry studied herein, we compute the stationary solution and the mean square displacement and find that the global resetting breaks the transport in the three directions. Regarding the resetting to the backbone, the transport is broken in two directions but it is enhanced in the main axis. Finally, the resetting to the fingers enhances the transport in the backbone and the main fingers but reaches a steady value for the mean squared displacement in the secondary fingers.

cond-mat.stat-mech↗

arXiv2019

Optimal management of impaired self-avoiding random walks for minimizing spatial coverage

Self-avoidance is a common mechanism to improve the efficiency of a random walker for covering a spatial domain. However, how this efficiency decreases when self-avoidance is impaired or limited by other processes has remained largely unexplored. Here we use simulations to study the case when the self-avoiding signal left by a walker both (i) saturates after successive revisits to a site, and (ii) evaporates, or dissappears, after some characteristic time. We surprisingly reveal that the mean cover time becomes minimum for intermediate values of the evaporation time, leading to the existence of a nontrivial optimum management of the self-avoiding signal. We argue that this is a consequence of complex blocking effects caused by the interplay with the signal saturation and, remarkably, we show that the optimum becomes more and more significant as the domain size increases.

cond-mat.stat-mech↗

arXiv2019

Universal scaling in bidirectional flows of self-avoiding agents

The analysis of the radial distribution function of a system provides a possible procedure for uncovering interaction rules between individuals out of collective movement patterns. This approach from classical statistical mechanics has revealed recently the existence of a universal scaling in systems of pedestrians, provided the potential of interaction $V(τ)$ is conveniently defined in the space of the times-to-collision $τ$ [Phys. Rev. Lett. \textbf{113}, 238701 (2014)]. Here we significantly extend this result by comparing numerically the performance of completely different rules of self-avoidance in bidirectional systems and proving that all of them collapse to a common scaling both in the disordered phase ($V(τ) \sim τ^{-2}$) and in the lane-formation regime ($V(τ) \sim τ^{-1}$), so suggesting that these scalings represent actually a universal feature of any self-avoiding bidirectional flow.

cond-mat.soft↗

arXiv2019

Transport properties of random walks under stochastic non-instantaneous resetting

Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce non-instantaneous resetting as a two-state model being a combination of an exploring state where the walker moves randomly according to a propagator and a returning state where the walker performs a ballistic motion with constant velocity towards the origin. We study the emerging transport properties for two types of reset time probability density functions (PDFs): exponential and Pareto. In the first case, we find the stationary distribution and a general expression for the stationary mean square displacement (MSD) in terms of the propagator. We find that the stationary MSD may increase, decrease or remain constant with the returning velocity. This depends on the moments of the propagator. Regarding the Pareto resetting PDF we also study the stationary distribution and the asymptotic scaling of the MSD for diffusive motion. In this case, we see that the resetting modifies the transport regime, making the overall transport sub-diffusive and even reaching a stationary MSD., i.e., a stochastic localization. This phenomena is also observed in diffusion under instantaneous Pareto resetting. We check the main results with stochastic simulations of the process.

cond-mat.stat-mech↗