arXiv subjects

Angelo Vulpiani

Publications and source records attributed to Angelo Vulpiani.

At least 91 records · Page 5Linked to original sources

arXiv2008

Coarse graining of master equations with fast and slow states

We propose a general method for simplifying master equations by eliminating from the description rapidly evolving states. The physical recipe we impose is the suppression of these states and a renormalization of the rates of all the surviving states. In some cases, this decimation procedure can be analytically carried out and is consistent with other analytical approaches, like in the problem of the random walk in a double-well potential. We discuss the application of our method to nontrivial examples: diffusion in a lattice with defects and a model of an enzymatic reaction outside the steady state regime.

cond-mat.soft↗

arXiv2008

Role of chaos for the validity of statistical mechanics laws: diffusion and conduction

Several years after the pioneering work by Fermi Pasta and Ulam, fundamental questions about the link between dynamical and statistical properties remain still open in modern statistical mechanics. Particularly controversial is the role of deterministic chaos for the validity and consistency of statistical approaches. This contribution reexamines such a debated issue taking inspiration from the problem of diffusion and heat conduction in deterministic systems. Is microscopic chaos a necessary ingredient to observe such macroscopic phenomena?

nlin.CD↗

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↗

arXiv2007

Transport properties in chaotic and non-chaotic many particles systems

Two deterministic models for Brownian motion are investigated by means of numerical simulations and kinetic theory arguments. The first model consists of a heavy hard disk immersed in a rarefied gas of smaller and lighter hard disks acting as a thermal bath. The second is the same except for the shape of the particles, which is now square. The basic difference of these two systems lies in the interaction: hard core elastic collisions make the dynamics of the disks chaotic whereas that of squares is not. Remarkably, this difference is not reflected in the transport properties of the two systems: simulations show that the diffusion coefficients, velocity correlations and response functions of the heavy impurity are in agreement with kinetic theory for both the chaotic and the non-chaotic model. The relaxation to equilibrium, however, is very sensitive to the kind of interaction. These observations are used to reconsider and discuss some issues connected to chaos, statistical mechanics and diffusion.

nlin.CD↗

arXiv2007

Fluctuation-Response Relation and modeling in systems with fast and slow dynamics

We show how a general formulation of the Fluctuation-Response Relation is able to describe in detail the connection between response properties to external perturbations and spontaneous fluctuations in systems with fast and slow variables. The method is tested by using the 360-variable Lorenz-96 model, where slow and fast variables are coupled to one another with reciprocal feedback, and a simplified low dimensional system. In the Fluctuation-Response context, the influence of the fast dynamics on the slow dynamics relies in a non trivial behavior of a suitable quadratic response function. This has important consequences for the modeling of the slow dynamics in terms of a Langevin equation: beyond a certain intrinsic time interval even the optimal model can give just statistical prediction.

nlin.CD↗

arXiv2007

Macroscopic equations for the adiabatic piston

A simplified version of a classical problem in thermodynamics -- the adiabatic piston -- is discussed in the framework of kinetic theory. We consider the limit of gases whose relaxation time is extremely fast so that the gases contained on the left and right chambers of the piston are always in equilibrium (that is the molecules are uniformly distributed and their velocities obey the Maxwell-Boltzmann distribution) after any collision with the piston. Then by using kinetic theory we derive the collision statistics from which we obtain a set of ordinary differential equations for the evolution of the macroscopic observables (namely the piston average velocity and position, the velocity variance and the temperatures of the two compartments). The dynamics of these equations is compared with simulations of an ideal gas and a microscopic model of gas settled to verify the assumptions used in the derivation. We show that the equations predict an evolution for the macroscopic variables which catches the basic features of the problem. The results here presented recover those derived, using a different approach, by Gruber, Pache and Lesne in J. Stat. Phys. 108, 669 (2002) and 112, 1177 (2003).

nlin.CD↗

arXiv2007

Position and velocity space diffusion of test particles in stochastic electromagnetic fields

The two--dimensional diffusive dynamics of test particles in a random electromagnetic field is studied. The synthetic electromagnetic fluctuations are generated through randomly placed magnetised ``clouds'' oscillating with a frequency $ω$. We investigate the mean square displacements of particles in both position and velocity spaces. As $ω$ increases the particles undergo standard (Brownian--like) motion, anomalous diffusion and ballistic motion in position space. Although in general the diffusion properties in velocity space are not trivially related to those in position space, we find that energization is present only when particles display anomalous diffusion in position space. The anomalous character of the diffusion is only in the non--standard values of the scaling exponents while the process is Gaussian.

cond-mat.stat-mech↗

arXiv2007

Discreteness effects in a reacting system of particles with finite interaction radius

An autocatalytic reacting system with particles interacting at a finite distance is studied. We investigate the effects of the discrete-particle character of the model on properties like reaction rate, quenching phenomenon and front propagation, focusing on differences with respect to the continuous case. We introduce a renormalized reaction rate depending both on the interaction radius and the particle density, and we relate it to macroscopic observables (e.g., front speed and front thickness) of the system.

cond-mat.stat-mech↗

arXiv2006

Relevance of initial and final conditions for the Fluctuation Relation in Markov processes

Numerical observations on a Markov chain and on the continuous Markov process performed by a granular tracer show that the ``usual'' fluctuation relation for a given observable is not verified for finite (but arbitrarily large) times. This suggests that some terms which are usually expected to be negligible, i.e. ``border terms'' dependent only on initial and final states, in fact cannot be neglected. Furthermore, the Markov chain and the granular tracer behave in a quite similar fashion.

cond-mat.stat-mech↗

arXiv2006

Absorbing processes in Richardson diffusion: analytical results

We consider the recently addressed problem of a passive particle (a predator), being the center of a ``sphere of interception'' of radius $R$ and able to absorb other passive particles (the preys) entering into the sphere. Assuming that all the particles are advected by a turbulent flow and that, in particular, the Richardson equation properly describes the relative dispersion, we calculate an analytical expression for the flux into the sphere as a function of time, assuming an initial constant density of preys outside the sphere. In the same framework, we show that the distribution of times of first passage into the sphere has a $t^{-5/2}$ power law tail, seen in contrast to the $t^{-3/2}$ appearing in standard 3D diffusion. We also discuss the correction due to the integral length scale on the results in the stationary case.

nlin.CD↗

arXiv2005

Properties making a chaotic system a good Pseudo Random Number Generator

We discuss two properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers: high value of Kolmogorov-Sinai entropy and high-dimensionality. We propose the multi dimensional Anosov symplectic (cat) map as a Pseudo Random Number Generator. We show what chaotic features of this map are useful for generating Pseudo Random Numbers and investigate numerically which of them survive in the discrete version of the map. Testing and comparisons with other generators are performed.

nlin.CD↗

arXiv2005

Brownian motion and diffusion: from stochastic processes to chaos and beyond

One century after Einstein's work, Brownian Motion still remains both a fundamental open issue and a continous source of inspiration for many areas of natural sciences. We first present a discussion about stochastic and deterministic approaches proposed in the literature to model the Brownian Motion and more general diffusive behaviours. Then, we focus on the problems concerning the determination of the microscopic nature of diffusion by means of data analysis. Finally, we discuss the general conditions required for the onset of large scale diffusive motion.

nlin.CD↗

arXiv2005

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.

nlin.CD↗

arXiv2005

Models of fluidized granular materials: examples of non-equilibrium stationary states

We review some models of granular materials fluidized by means of external forces such as: random homogeneous forcing with damping, vibrating plates, flow in an inclined channel and flow in a double well potential. All these systems show the presence of density correlations and non-Gaussian velocity distributions. These models are useful to understand the role of a kinetically defined ``temperature'' (in this case the so-called {\em granular temperature}) in a non-equilibrium stationary state. In the homogeneously randomly driven gas the granular temperature is {\em different} from that of the driving bath. Moreover two different granular materials mixed together may stay in a stationary state with different temperatures. At the same time, granular temperature determines (as in equilibrium systems) the escape time from a well in a double well potential.

cond-mat.soft↗

arXiv2004

The production rate of the coarse grained Gibbs entropy and the Kolmogorov-Sinai entropy: a real connection ?

We discuss the connection between the Kolmogorov-Sinai entropy, $h_{KS}$, and the production rate of the coarse grained Gibbs entropy, $r_G$. Detailed numerical computations show that the (often accepted) identification of the two quantities does not hold in systems with intermittent behavior and/or very different characteristic times and in systems presenting pseudo-chaos. The basic reason of this fact is in the asymptotic (with respect to time) nature of $h_{KS}$, while $r_G$ is a quantity related to short time features of a system.

nlin.CD↗

arXiv2004

Minimal Stochastic Model for Fermi's Acceleration

We introduce a simple stochastic system able to generate anomalous diffusion both for position and velocity. The model represents a viable description of the Fermi's acceleration mechanism and it is amenable to analytical treatment through a linear Boltzmann equation. The asymptotic probability distribution functions (PDF) for velocity and position are explicitly derived. The diffusion process is highly non-Gaussian and the time growth of moments is characterized by only two exponents $ν_x$ and $ν_v$. The diffusion process is anomalous (non Gaussian) but with a defined scaling properties i.e. $P(|{\bf x}|,t) = 1/t^{ν_x}F_x(|{\bf x}|/t^{ν_x})$ and similarly for velocity.

cond-mat.stat-mech↗

arXiv2003

Evidence for a k^{-5/3} spectrum from the EOLE Lagrangian balloons in the low stratosphere

The EOLE Experiment is revisited to study turbulent processes in the lower stratosphere circulation from a Lagrangian viewpoint and resolve a discrepancy on the slope of the atmospheric energy spectrum between the work of Morel and Larcheveque (1974) and recent studies using aircraft data. Relative dispersion of balloon pairs is studied by calculating the Finite Scale Lyapunov Exponent, an exit time-based technique which is particularly efficient in cases where processes with different spatial scales are interfering. Our main result is to reconciliate the EOLE dataset with recent studies supporting a k^{-5/3} energy spectrum in the range 100-1000 km. Our results also show exponential separation at smaller scale, with characteristic time of order 1 day, and agree with the standard diffusion of about 10^7 m^2/s at large scales. A still open question is the origin of a k^{-5/3} spectrum in the mesoscale range, between 100 and 1000 km.

physics.ao-ph↗

arXiv2003

Complexity Characterization of Dynamical Systems Through Predictability

Some aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. A special attention is devoted to finite-resolution effects on predictability, which can be accounted with suitable generalization of the standard indicators. The problems involved in systems with intrinsic randomness is discussed, with emphasis on the important problems of distinguishing chaos from noise and of modeling the system.

nlin.CD↗