Transport in quantum multi-barrier systems as random walks on a lattice

A quantum finite multi-barrier system, with a periodic potential, is considered and exact expressions for its plane wave amplitudes are obtained using the Transfer Matrix method [10]. This quantum model is then associated with a stochastic process of independent random walks on a lattice, by properly relating the wave amplitudes with the hopping probabilities of the particles moving on the lattice and with the injection rates from external particle reservoirs. Analytical and numerical results prove that the stationary density profile of the particle system overlaps with the quantum mass density profile of the stationary Schrodinger equation, when the parameters of the two models are suitably matched. The equivalence between the quantum model and a stochastic particle system would mainly be fruitful in a disordered setup. Indeed, we also show, here, that this connection, analytically proven to hold for periodic barriers, holds even when the width of the barriers and the distance between barriers are randomly chosen

Harmonic damped oscillators with feedback. A Langevin study

We consider a system in direct contact with a thermal reservoir and which, if left unperturbed, is well described by a memory-less equilibrium Langevin equation of the second order in the time coordinate. In such conditions, the strength of the noise fluctuations is set by the damping factor, in accordance with the Fluctuation and Dissipation theorem. We study the system when it is subject to a feedback mechanism, by modifying the Langevin equation accordingly. Memory terms now arise in the time evolution, which we study in a non-equilibrium steady state. Two types of feedback schemes are considered, one focusing on time shifts and one on phase shifts, and for both cases we evaluate the power spectrum of the system's fluctuations. Our analysis finds application in feedback cooled oscillators, such as the Gravitational Wave detector AURIGA.Comment: 17 page

Anomalies, absence of local equilibrium and universality in 1-d particles systems

One dimensional systems are under intense investigation, both from theoretical and experimental points of view, since they have rather peculiar characteristics which are of both conceptual and technological interest. We analyze the dependence of the behaviour of one dimensional, time reversal invariant, nonequilibrium systems on the parameters defining their microscopic dynamics. In particular, we consider chains of identical oscillators interacting via hard core elastic collisions and harmonic potentials, driven by boundary Nos\'e-Hoover thermostats. Their behaviour mirrors qualitatively that of stochastically driven systems, showing that anomalous properties are typical of physics in one dimension. Chaos, by itslef, does not lead to standard behaviour, since it does not guarantee local thermodynamic equilibrium. A linear relation is found between density fluctuations and temperature profiles. This link and the temporal asymmetry of fluctuations of the main observables are robust against modifications of thermostat parameters and against perturbations of the dynamics.Comment: 26 pages, 16 figures, revised text, two appendices adde

Comment on `Universal relation between the Kolmogorov-Sinai entropy and the thermodynamic entropy in simple liquids'

The intriguing relations between Kolmogorov-Sinai entropy and self diffusion coefficients and the excess (thermodynamic) entropy found by Dzugutov and collaborators do not appear to hold for hard sphere and hard disks systems.Comment: 1 page revte

Deterministic model of battery, uphill currents, and nonequilibrium phase transitions

We consider point particles in a table made of two circular cavities connected by two rectangular channels, forming a closed loop under periodic boundary conditions. In the first channel, a bounce-back mechanism acts when the number of particles flowing in one direction exceeds a given threshold T. In that case, the particles invert their horizontal velocity, as if colliding with vertical walls. The second channel is divided in two halves parallel to the first but located in the opposite sides of the cavities. In the second channel, motion is free. We show that, suitably tuning the sizes of cavities of the channels and of T, nonequilibrium phase transitions take place in the N→∞ limit. This induces a stationary current in the circuit, thus modeling a kind of battery, although our model is deterministic, conservative, and time reversal invariant

Transport and nonequilibrium phase transitions in polygonal urn models

We study the deterministic dynamics of N point particles moving at a constant speed in a 2D table made of two polygonal urns connected by an active rectangular channel, which applies a feedback control on the particles, inverting the horizontal component of their velocities when their number in the channel exceeds a fixed threshold. Such a bounce-back mechanism is non-dissipative: it preserves volumes in phase space. An additional passive channel closes the billiard table forming a circuit in which a stationary current may flow. Under specific constraints on the geometry and on the initial conditions, the large N limit allows nonequilibrium phase transitions between homogeneous and inhomogeneous phases. The role of ergodicity in making a probabilistic theory applicable is discussed for both rational and irrational urns. The theoretical predictions are compared with the numerical simulation results. Connections with the dynamics of feedback-controlled biological systems are highlighted

Boltzmann entropy and chaos in a large assembly of weakly interacting systems

We introduce a high dimensional symplectic map, modeling a large system consisting of weakly interacting chaotic subsystems, as a toy model to analyze the interplay between single-particle chaotic dynamics and particles interactions in thermodynamic systems. We study the growth with time of the Boltzmann entropy, S_B, in this system as a function of the coarse graining resolution. We show that a characteristic scale emerges, and that the behavior of S_B vs t, at variance with the Gibbs entropy, does not depend on the coarse graining resolution, as far as it is finer than this scale. The interaction among particles is crucial to achieve this result, while the rate of entropy growth depends essentially on the single-particle chaotic dynamics (for t not too small). It is possible to interpret the basic features of the dynamics in terms of a suitable Markov approximation.Comment: 21 pages, 11 figures, submitted to Journal of Statistical Physic

Deterministic reversible model of non-equilibrium phase transitions and stochastic counterpart

N point particles move within a billiard table made of two circular cavities connected by a straight channel. The usual billiard dynamics is modified so that it remains deterministic, phase space volumes preserving and time reversal invariant. Particles move in straight lines and are elastically reflected at the boundary of the table, as usual, but those in a channel that are moving away from a cavity invert their motion (rebound), if their number exceeds a given threshold T. When the geometrical parameters of the billiard table are fixed, this mechanism gives rise to non--equilibrium phase transitions in the large N limit: letting T/N decrease, the homogeneous particle distribution abruptly turns into a stationary inhomogeneous one. The equivalence with a modified Ehrenfest two urn model, motivated by the ergodicity of the billiard with no rebound, allows us to obtain analytical results that accurately describe the numerical billiard simulation results. Thus, a stochastic exactly solvable model that exhibits non-equilibrium phase transitions is also introduced

Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms

The fluctuations in nonequilibrium systems are under intense theoretical and experimental investigation. Topical ``fluctuation relations'' describe symmetries of the statistical properties of certain observables, in a variety of models and phenomena. They have been derived in deterministic and, later, in stochastic frameworks. Other results first obtained for stochastic processes, and later considered in deterministic dynamics, describe the temporal evolution of fluctuations. The field has grown beyond expectation: research works and different perspectives are proposed at an ever faster pace. Indeed, understanding fluctuations is important for the emerging theory of nonequilibrium phenomena, as well as for applications, such as those of nanotechnological and biophysical interest. However, the links among the different approaches and the limitations of these approaches are not fully understood. We focus on these issues, providing: a) analysis of the theoretical models; b) discussion of the rigorous mathematical results; c) identification of the physical mechanisms underlying the validity of the theoretical predictions, for a wide range of phenomena.Comment: 44 pages, 2 figures. To appear in Nonlinearity (2007