arXiv subjects

Lamberto Rondoni

Publications and source records attributed to Lamberto Rondoni.

At least 73 records · Page 4Linked to original sources

arXiv2014

On the foundations of statistical mechanics: ergodicity, many degrees of freedom and inference

The present paper is meant to give a simple introduction to the problem of the connection between microscopic dynamics and statistical laws. For sake of simplicity, we mostly refer to non-dissipative dynamics, since dissipation adds technical difficulties to the conceptual issues, although part of our discussion extends beyond this limit. In particular, the relevance of chaos and ergodicity is here confronted with that of the large number of degrees of freedom. In Section 2, we review the microscopic connection, along the lines of Boltzmann's approach, and of its further developments. In Section 3, we discuss the falsifiability of statistical mechanics and its role as statistical inference. In particular we argue that the Maximum entropy priciple is in general not a predictive tool.

cond-mat.stat-mech↗

arXiv2014

Role of Ergodicity in the Transient Fluctuation Relation and a New Relation for a Dissipative Non-Chaotic Map

Deterministic dynamical systems such as the baker maps are useful to shed light on some of the conditions verified by deterministic models in non-equilibrium statistical physics. We investigate a 2D dynamical system, enjoying a weak form of reversibility, with peculiar basins of attraction and steady states. In particular, we test the conditions required for the validity of the Transient Fluctuation Relation. Our analysis illustrates by means of concrete examples why ergodicity of the equilibrium dynamics seems to be a necessary condition for the Transient Fluctuation Relation to hold. This investigation then leads to the numerical verification of a kind of transient relation which, differently from the usual Transient Fluctuation Relation (FR), holds only asymptotically. At the same time, it is not a steady state fluctuation relation, because no fluctuations are present in the steady state.

math.DS↗

arXiv2014

Focus on some Nonequilibrium Issues

A mathematical framework for the physics of nonequilibrium phenomena is gradually being developed. This review is meant to shed light on some aspects of Response Theory, on the theory of Fluctuation Relations, on the so-called "t-mixing" condition, and on the use of Large Deviation techniques in the description of stochastic diffusion processes.

cond-mat.stat-mech↗

arXiv2013

Optimal protocols and the Jarzynski equality

The Jarzynski Equality is a well-known and widely used identity, relating the free energy difference between two states of a system to the work done over some arbitrary, nonequilibrium transformation between the two states. Despite being valid for both stochastic and deterministic systems, we show that the optimal transformation protocol for the deterministic case seems to differ from that predicated from an analysis of the stochastic dynamics. In addition, it is shown that for certain situations, more dissipative processes can sometimes lead to better numerical results for the free energy differences.

cond-mat.stat-mech↗

arXiv2013

A trivial non-chaotic map lattice asymptotically indistiguishable from a Lévy walk

In search for mathematically tractable models of anomalous diffusion, we introduce a simple dynamical system consisting of a chain of coupled maps of the interval whose Lyapunov exponents vanish everywhere. The volume preserving property and the vanishing Lyapunov exponents are intended to mimic the dynamics of polygonal billiards, which are known to give rise to anomalous diffusion, but which are too complicated to be analyzed as thoroughly as desired. Depending on the value taken by a single parameter α, our map experiences sub-diffusion, super-diffusion or normal diffusion. Therefore its transport properties can be compared with those of given Lévy walks describing transport in quenched disordered media. Fixing α so that the mean square displacement generated by our map and that generated by the corresponding Lévy walk asymptotically coincide, we prove that all moments of the corresponding asymptotic distributions coincide as well, hence all observables which are expressed in terms of the moments coincide.

math-ph↗

arXiv2013

Effects of breaking vibrational energy equipartition on measurements of temperature in macroscopic oscillators subject to heat flux

When the energy content of a resonant mode of a crystalline solid in thermodynamic equilibrium is directly measured, assuming that quantum effects can be neglected it coincides with temperature except for a proportionality factor. This is due to the principle of energy equipartition and the equilibrium hypothesis. However, most natural systems found in nature are not in thermodynamic equilibrium and thus the principle cannot be granted. We measured the extent to which the low-frequency modes of vibration of a solid can defy energy equipartition, in presence of a steady state heat flux, even close to equilibrium. We found, experimentally and numerically, that the energy separately associated with low frequency normal modes strongly depends on the heat flux, and decouples sensibly from temperature. A 4% in the relative temperature difference across the object around room temperature suffices to excite two modes of a macroscopic oscillator, as if they were at equilibrium, respectively, at temperatures about 20% and a factor 3.5 higher. We interpret the result in terms of new flux-mediated correlations between modes in the nonequilibrium state, which are absent at equilibrium.

cond-mat.stat-mech↗

arXiv2013

Statistical distribution of bonding distances in a unidimensional solid

We study a Fermi-Pasta-Ulam-like chain with realistic potentials, which models a unidimensional solid in contact with heat baths at some temperature. We formulate an explicit analytical expression for the probability density of bonding distances between neighbor particles, which depends on temperature similarly to the distribution of velocities. Its validity is verified with a striking accuracy through simulations.

cond-mat.stat-mech↗

arXiv2013

Brownian Motion and General Relativity

We construct a model of Brownian Motion on a pseudo-Riemannian manifold associated with general relativity. There are two aspects of the problem: The first is to define a sequence of stopping times associated with the Brownian "kicks" or impulses. The second is to define the dynamics of the particle along geodesics in between the Brownian kicks. When these two aspects are taken together, we can associate various distributions with the motion. We will find that the statistics of space-time events will obey a temperature dependent four dimensional Gaussian distribution defined over the quaternions which locally can be identified with Minkowski space. Analogously, the statistics of the 4-velocities will obey a kind of Maxwell-Juttner distribution. In contrast to previous work, our processes are characterized by two independent proper time variables defined with respect to the laboratory frame: a discrete one corresponding to the stopping times when the impulses take place and a continuous one corresponding to the geodesic motion in-between impulses. The subsequent distributions are then solutions of partial differential equations which contain derivatives with respect to both time variables.

physics.gen-ph↗

arXiv2013

Current in a quantum driven thermostatted system with off-diagonal disorder

We analyze a one-dimensional quantum model with off-diagonal disorder, consisting of a sequence of potential energy barriers whose width is a random variable either uniformly or "half-normally" distributed, subjected to an external electric field. We shed light on how the microscopic disorder affects the value of the transmission coefficient, and on the structure of the fluctuations around the solutions corresponding to the regular lattice configuration. We also characterize the asymptotic limit obtained by letting the number of barriers diverge, while their total width is kept constant. Thus, we explain the novelty of our method with respect to the standard thermodynamic limit discussed in the literature, and also evidence the onset of a large deviations principle for the transmission coefficient.

cond-mat.stat-mech↗

arXiv2012

Fluctuations in quantum one-dimensional thermostatted systems with off-diagonal disorder

We analyze a one dimensional quantum model with off-diagonal disorder, consisting of a sequence of potential energy barriers whose width is a random variable either uniformly or normally distributed. We investigate how the disorder and the energy distribution (due to a thermostat at room temperature) affect the resulting value of the transmission coefficient, and discuss the structure of the fluctuations of such coefficient at different length scales and the onset of different size-dependent regimes. Our analysis suggests an efficient way to detect tunneling resonances in a regime of off-diagonal disorder.

cond-mat.stat-mech↗

arXiv2012

Fluctuation-dissipation relation for chaotic non-Hamiltonian systems

In dissipative dynamical systems phase space volumes contract, on average. Therefore, the invariant measure on the attractor is singular with respect to the Lebesgue measure. As noted by Ruelle, a generic perturbation pushes the state out of the attractor, hence the statistical features of the perturbation and, in particular, of the relaxation, cannot be understood solely in terms of the unperturbed dynamics on the attractor. This remark seems to seriously limit the applicability of the standard fluctuation dissipation procedure in the statistical mechanics of nonequilibrium (dissipative) systems. In this paper we show that the singular character of the steady state does not constitute a serious limitation in the case of systems with many degrees of freedom. The reason is that one typically deals with projected dynamics, and these are associated with regular probability distributions in the corresponding lower dimensional spaces.

cond-mat.stat-mech↗

arXiv2011

Equilibrium, fluctuation relations and transport for irreversible deterministic dynamics

In a recent paper [M. Colangeli \textit{et al.}, J.\ Stat.\ Mech.\ P04021, (2011)] it was argued that the Fluctuation Relation for the phase space contraction rate $Λ$ could suitably be extended to non-reversible dissipative systems. We strengthen here those arguments, providing analytical and numerical evidence based on the properties of a simple irreversible nonequilibrium baker model. We also consider the problem of response, showing that the transport coefficients are not affected by the irreversibility of the microscopic dynamics. In addition, we prove that a form of \textit{detailed balance}, hence of equilibrium, holds in the space of relevant variables, despite the irreversibility of the phase space dynamics. This corroborates the idea that the same stochastic description, which arises from a projection onto a subspace of relevant coordinates, is compatible with quite different underlying deterministic dynamics. In other words, the details of the microscopic dynamics are largely irrelevant, for what concerns properties such as those concerning the Fluctuation Relations, the equilibrium behaviour and the response to perturbations.

math.DS↗

arXiv2011

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 Nosé-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.

cond-mat.stat-mech↗

arXiv2011

Steady state fluctuation relation and time-reversibility for non-smooth chaotic maps

Steady state fluctuation relations for dynamical systems are commonly derived under the assumption of some form of time-reversibility and of chaos. There are, however, cases in which they are observed to hold even if the usual notion of time reversal invariance is violated, e.g. for local fluctuations of Navier-Stokes systems. Here we construct and study analytically a simple non-smooth map in which the standard steady state fluctuation relation is valid, although the model violates the Anosov property of chaotic dynamical systems. Particularly, the time reversal operation is performed by a discontinuous involution, and the invariant measure is also discontinuous along the unstable manifolds. This further indicates that the validity of fluctuation relations for dynamical systems does not rely on particularly elaborate conditions, usually violated by systems of interest in physics. Indeed, even an irreversible map is proved to verify the steady state fluctuation relation.

math-ph↗

arXiv2009

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.

cond-mat.stat-mech↗

arXiv2008

The nonequilibrium Ehrenfest gas: a chaotic model with flat obstacles?

It is known that the non-equilibrium version of the Lorentz gas (a billiard with dispersing obstacles, electric field and Gaussian thermostat) is hyperbolic if the field is small. Differently the hyperbolicity of the non-equilibrium Ehrenfest gas constitutes an open problem, since its obstacles are rhombi and the techniques so far developed rely on the dispersing nature of the obstacles. We have developed analytical and numerical investigations which support the idea that this model of transport of matter has both chaotic (positive Lyapunov exponent) and non-chaotic steady states with a quite peculiar sensitive dependence on the field and on the geometry, not observed before. The associated transport behaviour is correspondingly highly irregular, with features whose understanding is of both theoretical and technological interest.

math-ph↗

arXiv2008

On the Fluctuation Relation for Nose-Hoover Boundary Thermostated Systems

We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nosé-Hoover thermostats applied at the boundaries of the system. The transient fluctuation relation, which holds only for a precise choice of the initial ensemble, is verified at all times, as expected. Times longer than the mesoscopic scale, needed for local equilibrium to be settled, are required if a different initial ensemble is considered. This shows how the transient fluctuation relation asymptotically leads to the steady state relation when, as explicitly checked in our systems, the condition found in [D.J. Searles, {\em et al.}, J. Stat. Phys. 128, 1337 (2007)], for the validity of the steady state fluctuation relation, is verified. For the steady state fluctuations of the phase space contraction rate $\zL$ and of the dissipation function $\zW$, a similar relaxation regime at shorter averaging times is found. The quantity $\zW$ satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale; the quantity $\zL$ appears to begin a monotonic convergence after such times. This is consistent with the fact that $\zW$ and $\zL$ differ by a total time derivative, and that the tails of the probability distribution function of $\zL$ are Gaussian.

cond-mat.stat-mech↗

arXiv2008

Fluctuation-Dissipation: Response Theory in Statistical Physics

General aspects of the Fluctuation-Dissipation Relation (FDR), and Response Theory are considered. After analyzing the conceptual and historical relevance of fluctuations in statistical mechanics, we illustrate the relation between the relaxation of spontaneous fluctuations, and the response to an external perturbation. These studies date back to Einstein's work on Brownian Motion, were continued by Nyquist and Onsager and culminated in Kubo's linear response theory. The FDR has been originally developed in the framework of statistical mechanics of Hamiltonian systems, nevertheless a generalized FDR holds under rather general hypotheses, regardless of the Hamiltonian, or equilibrium nature of the system. In the last decade, this subject was revived by the works on Fluctuation Relations (FR) concerning far from equilibrium systems. The connection of these works with large deviation theory is analyzed. Some examples, beyond the standard applications of statistical mechanics, where fluctuations play a major role are discussed: fluids, granular media, nano-systems and biological systems.

cond-mat.stat-mech↗