Carlos Mejía-Monasterio

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 realizations 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 behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, 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 behavior even in bounded domains, in which all moments of the first passage distribution exist.

First passages for a search by a swarm of independent random searchers

In this paper we study some aspects of search for an immobile target by a swarm of N non-communicating, randomly moving searchers (numbered by the index k, k = 1, 2,..., N), which all start their random motion simultaneously at the same point in space. For each realization of the search process, we record the unordered set of time moments {?k}, where ?k is the time of the first passage of the kth searcher to the location of the target. Clearly, ?ks are independent, identically distributed random variables with the same distribution function ?(?). We evaluate then the distribution P(?) of the random variable , where is the ensemble-averaged realization-dependent first passage time. We show that P(?) exhibits quite a non-trivial and sometimes a counterintuitive behavior. We demonstrate that in some well-studied cases (e.g. Brownian motion in finite d-dimensional domains) the mean first passage time is not a robust measure of the search efficiency, despite the fact that ?(?) has moments of arbitrary order. This implies, in particular, that even in this simplest case (not to mention complex systems and/or anomalous diffusion) first passage data extracted from a single-particle tracking should be regarded with appropriate caution because of the significant sample-to-sample fluctuations.

Refined Second Law of Thermodynamics for fast random processes

We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport and becomes relevant for processes far from quasi-stationary regime. General discussion is illustrated by numerical analysis of the optimal memory erasure protocol for a model for micron-size particle manipulated by optical tweezers.

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 an 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 nonglass-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.

Increasing thermoelectric efficiency: a dynamical systems approach.

We study the problem of thermoelectricity and propose a simp le microscopic mechanism for the increase of thermoelectric efficiency. We consider the cross transpo rt of particles and energy in open classical ergodic billiards. We show that, in the linear response regime, wher e we find exact expressions for all transport coefficients, the thermoelectric efficiency of ideal ergodic g ases can approach Carnot efficiency for sufficiently complex charge carrier molecules. Our results are clearly d emonstrated with a simple numerical simulation of a Lorentz gas of particles with internal rotational degrees of freedom.

Magnetically induced thermal rectification.

We consider far from equilibrium heat transport in chaotic billiard chains with noninteracting charged particles in the presence of nonuniform transverse magnetic field. If half of the chain is placed in a strong magnetic field, or if the strength of the magnetic field has a large gradient along the chain, heat current is shown to be asymmetric with respect to exchange of the temperatures of the heat baths. Thermal rectification factor can be arbitrarily large for sufficiently small temperature of one of the baths.

Fourier's law in a quantum spin chain and the onset of quantum chaos

We study heat transport in a nonequilibrium steady state of a quantum interacting spin chain. We provide clear numerical evidence of the validity of Fourier law. The regime of normal conductivity is shown to set in at the transition to quantum chaos.

Thermal Rectification in Billiardlike Systems

We study the thermal rectification phenomenon in billiard systems with interacting particles. This interaction induces a local dynamical response of the billiard to an external thermodynamic gradient. To explain this dynamical effect we study the steady state of an asymmetric billiard in terms of the particle and energy reflection coefficients. This allows us to obtain expressions for the region in parameter space where large thermal rectifications are expected. Our results are confirmed by extensive numerical simulations.

Optimal Protocols and Optimal Transport in Stochastic Thermodynamics

Thermodynamics of small systems has become an important field of statistical physics. Such systems are driven out of equilibrium by a control, and the question is naturally posed how such a control can be optimized. We show that optimization problems in small system thermodynamics are solved by (deterministic) optimal transport, for which very efficient numerical methods have been developed, and of which there are applications in cosmology, fluid mechanics, logistics, and many other fields. We show, in particular, that minimizing expected heat released or work done during a nonequilibrium transition in finite time is solved by the Burgers equation and mass transport by the Burgers velocity field. Our contribution hence considerably extends the range of solvable optimization problems in small system thermodynamics.

A stochastic model of anomalous heat transport: analytical solution of the steady state

We consider a one-dimensional harmonic crystal with conservative noise, in contact with two stochastic Langevin heat baths at different temperatures. The noise term consists of collisions between neighbouring oscillators that exchange their momenta, with a rate γ. The stationary equations for the covariance matrix are exactly solved in the thermodynamic limit (N → ∞). In particular, we derive an analytical expression for the temperature profile, which turns out to be independent of γ. Moreover, we obtain an exact expression for the leading term of the energy current, which scales as . Our theoretical results are finally found to be consistent with the numerical solutions of the covariance matrix for finite N.