arXiv subjects

Carlos Mejía-Monasterio

Publications and source records attributed to Carlos Mejía-Monasterio.

At least 19 recordsLinked to original sources

arXiv2025

From anomalous diffusion in polygons to a transport locking relation

We study particle transport in a class of open channels of finite length, made of identical cells of connected open polygonal billiards with parallel boundaries. In these systems the Mean Square Displacement (MSD) grows in time faster than linearly. We show that irrespective of the geometry of these channels, the distribution of the first return times decays algebraically with two different exponents, separated by a crossover region that is determined by the MSD. We find that the distribution of first return times satisfies a simple scaling form. In turn, the transmission coefficient, defined as the fraction of trajectories that starting at the cell at the origin escape the channel through the other boundary, decays algebraically with the size of the system, and, as a signature of non-recurrent transport, sometimes slower. From these two processes we derive a locking relation among the scaling exponents for the asymptotic behavior of the MSD, the times of first return to the origin and the way transmission decays with the system size, showing that these three processes are interdependent. The locking relation holds for diffusive processes, as well as for fractional Brownian motion with arbitrary Hurst exponent. We argue that the locking relation may be valid for many other transport processes, Markovian or not, with finite MSD.

cond-mat.stat-mech↗

arXiv2024

Universal hyper-scaling relations, power-law tails, and data analysis for strong anomalous diffusion

Strong anomalous diffusion is {often} characterized by a piecewise-linear spectrum of the moments of displacement. The spectrum is characterized by slopes $ξ$ and $ζ$ for small and large moments, respectively, and by the critical moment $α$ of the crossover. The exponents $ξ$ and $ζ$ characterize the asymptotic scaling of the bulk and the tails of the probability distribution function of displacements, respectively. Here, we adopt asymptotic theory to match the behaviors at intermediate scales. The resulting constraint explains how distributions with algebraic tails imply strong anomalous diffusion, and it relates $α$ to the corresponding power law. Our theory provides novel relations between exponents characterizing strong anomalous diffusion, and it yields explicit expressions for the leading-order corrections to the asymptotic power-law behavior of the moments of displacement. They provide the time scale that must be surpassed to clearly discriminate the leading-order power law from its sub-leading corrections. This insight allows us to point out sources of systematic errors in their numerical estimates. Rather than separately fitting an exponent for each moment we devise a robust scheme to determine $ξ$, $ζ$ and $α$. The findings are supported by numerical and analytical results on five different models exhibiting strong anomalous diffusion.

math-ph↗

arXiv2024

Thermal transport in long-range interacting harmonic chains perturbed by long-range conservative noise

We study non-equilibrium properties of a chain of $N$ oscillators with both long-ranged harmonic interactions and long-range conservative noise that exchange momenta of particle pairs. We derive exact expressions for the (deterministic) energy-current auto-correlation at equilibrium, based on the kinetic approximation of the normal mode dynamics. In all cases the decay is algebraic in the thermodynamic limit. We distinguish four distinct regimes of correlation decay depending on the exponents controlling the range of deterministic and stochastic interactions. Surprisingly, we find that long-range noise breaks down the long-range correlations characteristic of low dimensional models, suggesting a normal regime in which heat transport becomes diffusive. For finite systems, we do also derive exact expressions for the finite-size corrections to the algebraic decay of the correlation. In certain regimes, these corrections are considerably large, rendering hard the estimation of transport properties from numerical data for the finite chains. Our results are tested against numerical simulations, performed with an efficient algorithm.

cond-mat.stat-mech↗

arXiv2024

Particle transport in open polygonal billiards: a scattering map

Polygonal billiards exhibit a rich and complex dynamical behavior. In recent years polygonal billiards have attracted great attention due to their application in the understanding of anomalous transport, but also at the fundamental level, due to its connections with diverse fields in mathematics. We explore this complexity and its consequences on the properties of particle transport in infinitely long channels made of the repetitions of an elementary open polygonal cell. Borrowing ideas from the Zemlyakov-Katok construction, we construct an interval exchange transformation classified by the singular directions of the discontinuities of the billiard flow over the translation surface associated to the elementary cell. From this, we derive an exact expression of a scattering map of the cell connecting the outgoing flow of trajectories with the unconstrained incoming flow. The scattering map is defined over a partition of the coordinate space, characterized by different families of trajectories. Furthermore, we obtain an analytical expression for the average speed of propagation of ballistic modes, describing with high accuracy the speed of propagation of ballistic fronts appearing in the tails of the distribution of the particle displacement. The symbolic hierarchy of the trajectories forming these ballistic fronts is also discussed.

nlin.CD↗

arXiv2021

Displacement Autocorrelation Functions for Strong Anomalous Diffusion: A Scaling Form, Universal Behavior, and Corrections to Scaling

Strong anomalous diffusion is characterized by asymptotic power-law growth of the moments of displacement, with exponents that do not depend linearly on the order of the moment. The exponents concerning small-order moments are dominated by random motion, while higher-order exponents grow by faster trajectories, such as ballistic excursions or "light fronts". Often such a situation is characterized by two linear dependencies of the exponents on their order. Here, we introduce a simple exactly solvable model, the Fly-and-Die (FnD) model, that sheds light on this behavior and on the consequences of light fronts on displacement autocorrelation functions in transport processes. We present analytical expressions for the moments and derive a scaling form that expresses the long-time asymptotics of the autocorrelation function $\langle x(t_1)\,x(t_2)\rangle$ in terms of the dimensionless time difference $(t_2-t_1)/t_1$. The scaling form provides a faithful collapse of numerical data for vastly different systems. This is demonstrated here for the Lorentz gas with infinite horizon, polygonal billiards with finite and infinite horizon, the Lévy-Lorentz gas, the Slicer Map, and Lévy walks. Our analysis also captures the system-specific corrections to scaling.

cond-mat.stat-mech↗

arXiv2019

Tracer Diffusion on a Crowded Random Manhattan Lattice

We study by extensive numerical simulations the dynamics of a hard-core tracer particle (TP) in presence of two competing types of disorder - frozen convection flows on a square random Manhattan lattice and a crowded dynamical environment formed by a lattice gas of mobile hard-core particles. The latter perform lattice random walks, constrained by a single-occupancy condition of each lattice site, and are either insensitive to random flows (model A) or choose the jump directions as dictated by the local directionality of bonds of the random Manhattan lattice (model B). We focus on the TP disorder-averaged mean-squared displacement, (which shows a super-diffusive behaviour $\sim t^{4/3}$, $t$ being time, in all the cases studied here), on higher moments of the TP displacement, and on the probability distribution of the TP position $X$ along the $x$-axis. Our analysis evidences that in absence of the lattice gas particles the latter has a Gaussian central part $\sim \exp(- u^2)$, where $u = X/t^{2/3}$, and exhibits slower-than-Gaussian tails $\sim \exp(-|u|^{4/3})$ for sufficiently large $t$ and $u$. Numerical data convincingly demonstrate that in presence of a crowded environment the central Gaussian part and non-Gaussian tails of the distribution persist for both models.

cond-mat.dis-nn↗

arXiv2019

Heat flux in one-dimensional systems

Understanding heat transport in one-dimensional systems remains a major challenge in theoretical physics, both from the quantum as well as from the classical point of view. In fact, steady states of one-dimensional systems are commonly characterized by macroscopic inhomogeneities, and by long range correlations, as well as large fluctuations that are typically absent in standard three-dimensional thermodynamic systems. These effects violate locality --material properties in the bulk may be strongly affected by the boundaries, leading to anomalous energy transport-- and they make more problematic the interpretation of mechanical microscopic quantities in terms of thermodynamic observables. Here, we revisit the problem of heat conduction in chains of classical nonlinear oscillators, following a Lagrangian and an Eulerian approach. The Eulerian definition of the flux is composed of a convective and a conductive component. The former component tends to prevail at large temperatures where the system behavior is increasingly gas-like. Finally, we find that the convective component tends to be negative in the presence of a negative pressure.

cond-mat.stat-mech↗

arXiv2016

Dynamical mechanisms leading to equilibration in two-component gases

Demonstrating how microscopic dynamics cause large systems to approach thermal equilibrium remains an elusive, longstanding, and actively-pursued goal of statistical mechanics. We identify here a dynamical mechanism for thermalization in a general class of two-component dynamical Lorentz gases, and prove that each component, even when maintained in a non-equilibrium state itself, can drive the other to a thermal state with a well-defined effective temperature.

cond-mat.stat-mech↗

arXiv2014

Limits on the performance of Infotaxis under inaccurate modelling of the environment

We study the performance of infotaxis search strategy measured by the rate of success and mean search time, under changes in the environment parameters such as diffusivity, rate of emission or wind velocity. We also investigate the drop of performance caused by an innacurate modelling of the environment. Our findings show that infotaxis remains robust as long as the estimated parameters fall within a certain range around their true values, but the success rate quickly drops making infotaxis no longer feasible if the searcher agent severely underestimates or overestimates the real environment parameters. This study places some limits on the performance of infotaxis, and thus it has practical consequences for the design of infotaxis based machines to track and detect an emitting source of chemicals or volatile substances.

cs.ET↗

arXiv2014

Geometry-induced fluctuations of olfactory searches in bounded domains

In olfactory search an immobile target emits chemical molecules at constant rate. The molecules are transported by the medium which is assumed to be turbulent. Considering a searcher able to detect such chemical signals and whose motion follows the infotaxis strategy, we study the statistics of the first-passage time to the target when the searcher moves on a finite two-dimensional lattice of different geometries. Far from the target, where the concentration of chemicals is low the direction of the searcher's first movement is determined by the geometry of the domain and the topology of the lattice, inducing strong fluctuations on the average search time with respect to the initial position of the searcher. The domain is partitioned in well defined regions characterized by the direction of the first movement. If the search starts over the interface between two different regions, large fluctuations in the search time are observed.

cond-mat.stat-mech↗

arXiv2013

Geometry-induced superdiffusion in driven crowded systems

Recent Molecular Dynamics simulations of glass-forming liquids revealed superdiffusive fluctuations associated with the position of a tracer particle (TP) driven by an external force. Such anomalous response, whose mechanism remains elusive, has been observed up to now only in systems close to their glass transition, suggesting that this could be one of its hallmarks. Here, we show that the presence of superdiffusion is in actual fact much more general, provided that the system is crowded and geometrically confined. We present and solve analytically a minimal model consisting of a driven TP in a dense, crowded medium in which the motion of particles is mediated by the diffusion of packing defects, called vacancies. For such non glass-forming systems, our analysis predicts a long-lived superdiffusion which ultimately crosses over to giant diffusive behavior. We find that this trait is present in confined geometries, for example long capillaries and stripes, and emerges as a universal response of crowded environments to an external force. These findings are confirmed by numerical simulations of systems as varied as lattice-gases, dense liquids and granular fluids.

cond-mat.stat-mech↗

arXiv2013

Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory

Modern developments in microscopy and image processing are revolutionising areas of physics, chemistry, and biology as nanoscale objects can be tracked with unprecedented accuracy. However, the price paid for having a direct visualisation of a single particle trajectory with high temporal and spatial resolution is a consequent lack of statistics. This naturally calls for reliable analytical tools which will allow one to extract the properties specific to a statistical ensemble from just a single trajectory. In this article we briefly survey different analytical methods currently used to determine the ensemble average diffusion coefficient from single particle data and then focus specifically on weighted least-squares estimators, seeking the weight functions for which such estimators are ergodic. Finally, we address the question of the effects of disorder on such estimators.

cond-mat.stat-mech↗

arXiv2012

Heat release by controlled continuous-time Markov jump processes

We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Bellman-Jacobi-type. As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. Thus, our results may be interpreted as a refined version of the second law of thermodynamics.

cond-mat.stat-mech↗

arXiv2012

Optimal fits of diffusion constants from single time data points of Brownian trajectories

Experimental methods based on single particle tracking (SPT) are being increasingly employed in the physical and biological sciences, where nanoscale objects are visualized with high temporal and spatial resolution. SPT can probe interactions between a particle and its environment but the price to be paid is the absence of ensemble averaging and a consequent lack of statistics. Here we address the benchmark question of how to accurately extract the diffusion constant of one single Brownian trajectory. We analyze a class of estimators based on weighted functionals of the square displacement. For a certain choice of the weight function these functionals provide the true ensemble averaged diffusion coefficient, with a precision that increases with the trajectory resolution.

cond-mat.stat-mech↗

arXiv2012

Anomalous field-induced growth of fluctuations in dynamics of a biased intruder moving in a quiescent medium

We present exact results on the dynamics of a biased, by an external force ${\bf F}$, intruder (BI) in a two-dimensional lattice gas of unbiased, randomly moving hard-core particles. Going beyond the usual analysis of the force-velocity relation, we study the probability distribution $P({\bf R}_n)$ of the BI displacement ${\bf R}_n$ at time {\it n}. We show that despite the fact that the BI drives the gas to a non-equilibrium steady-state, $P({\bf R}_n)$ converges to a Gaussian distribution as $n \to \infty$. We find that the variance $σ_x^2$ of $P({\bf R}_n)$ along ${\bf F}$ exhibits a weakly superdiffusive growth $σ_x^2 \sim ν_1 \, n \, \ln(n)$, and a usual diffusive growth, $σ_y^2 \sim ν_2 \, n$, in the perpendicular direction. We determine $ν_1$ and $ν_2$ exactly for arbitrary bias, in the lowest order in the density of vacancies, and show that $ν_1 \sim |{\bf F}|^2$ for small bias, which signifies that superdiffusive behaviour emerges beyond the linear-response approximation. Monte Carlo simulations confirm our analytical results, and reveal a striking field-induced superdiffusive behavior $σ_x^2 \sim n^{3/2}$ for infinitely long 2D stripes and 3D capillaries.

cond-mat.stat-mech↗

arXiv2012

First passages in bounded domains: When is the mean first passage time meaningful?

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(ω) distribution of the random variable ω=τ_1/(τ_1+τ_2), which is a measure for how similar the first passage times τ_1 and τ_2 are of two independent realisations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(ω) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behaviour. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behaviour, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(ω), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behaviour even in bounded domains, in which all moments of the first passage distribution exist.

cond-mat.stat-mech↗

arXiv2012

A non-perturbative renormalization group study of the stochastic Navier--Stokes equation

We study the renormalization group flow of the average action of the stochastic Navier--Stokes equation with power-law forcing. Using Galilean invariance we introduce a non-perturbative approximation adapted to the zero frequency sector of the theory in the parametric range of the Hölder exponent $4-2\,\varepsilon$ of the forcing where real-space local interactions are relevant. In any spatial dimension $d$, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for $\varepsilon=2$ as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a \emph{saturation} in the $\varepsilon$-dependence of the scaling dimension of the eddy diffusivity at $\varepsilon=3/2$ when, according to perturbative renormalization, the velocity field becomes infra-red relevant.

cond-mat.stat-mech↗

arXiv2012

Optimal estimates of the diffusion coefficient of a single Brownian trajectory

Modern developments in microscopy and image processing are revolutionizing areas of physics, chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy. The goal of single particle tracking is to determine the interaction between the particle and its environment. The price paid for having a direct visualization of a single particle is a consequent lack of statistics. Here we address the optimal way of extracting diffusion constants from single trajectories for pure Brownian motion. It is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate. Furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true (average) value.

cond-mat.stat-mech↗