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.Comment: 5 pages, 5 figure

Superdiffusive Heat Transport in a Class of Deterministic One-dimensional Many-Particle Lorentz Gases

We study heat transport in a one-dimensional chain of a finite number N of identical cells, coupled at its boundaries to stochastic particle reservoirs. At the center of each cell, tracer particles collide with fixed scatterers, exchanging momentum. In a recent paper (Collet and Eckmann in Commun. Math. Phys. 287:1015, 2009), a spatially continuous version of this model was derived in a scaling regime where the scattering probability of the tracers is γ∼1/N, corresponding to the Grad limit. A Boltzmann-like equation describing the transport of heat was obtained. In this paper, we show numerically that the Boltzmann description obtained in Collet and Eckmann (Commun. Math. Phys. 287:1015, 2009) is indeed a bona fide limit of the particle model. Furthermore, we study the heat transport of the model when the scattering probability is 1, corresponding to deterministic dynamics. Thought as a lattice model in which particles jump between different scatterers the motion is persistent, with a persistence probability determined by the mass ratio among particles and scatterers, and a waiting time probability distribution with algebraic tails. We find that the heat and particle currents scale slower than 1/N, implying that this model exhibits anomalous heat and particle transpor

Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems

We consider nonequilibrium transport in a simple chain of identical mechanical cells in which particles move around. In each cell, there is a rotating disc, with which these particles interact, and this is the only interaction in the model. It was shown in Ref. 1 that when the cells are weakly coupled, to a good approximation, the jump rates of particles and the energy-exchange rates from cell to cell follow linear profiles. Here, we refine that study by analyzing higher-order effects which are induced by the presence of external gradients for situations in which memory effects, typical of Hamiltonian dynamics, cannot be neglected. For the steady state we propose a set of balance equations for the particle number and energy in terms of the reflection probabilities of the cell and solve it phenomenologically. Using this approximate theory we explain how these asymmetries affect various aspects of heat and particle transport in systems of the general type described above and obtain in the infinite volume limit the deviation from the theory in Ref. 1 to first-order. We verify our assumptions with extensive numerical simulation

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.Comment: 24 pages, 6 figure

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.Comment: 7 pages, 8 figures, typed in revte

Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems

We consider nonequilibrium transport in a simple chain of identical mechanical cells in which particles move around. In each cell, there is a rotating disc, with which these particles interact, and this is the only interaction in the model. It was shown in \cite{eckmann-young} that when the cells are weakly coupled, to a good approximation, the jump rates of particles and the energy-exchange rates from cell to cell follow linear profiles. Here, we refine that study by analyzing higher-order effects which are induced by the presence of external gradients for situations in which memory effects, typical of Hamiltonian dynamics, cannot be neglected. For the steady state we propose a set of balance equations for the particle number and energy in terms of the reflection probabilities of the cell and solve it phenomenologically. Using this approximate theory we explain how these asymmetries affect various aspects of heat and particle transport in systems of the general type described above and obtain in the infinite volume limit the deviation from the theory in \cite{eckmann-young} to first-order. We verify our assumptions with extensive numerical simulations.Comment: Several change

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. General discussion is illustrated by numerical analysis of a model for micron-size particle manipulated by optical tweezers.Comment: 17 page

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.Comment: 8 pages, 5 figure