Phase transitions induced by thermal fluctuations

The present study generalizes the model of extended stochastic systems with a field-dependent kinetic coefficient [M. Ibanes, J. Garcia-Ojalvo, R. Toral, J.M. Sancho, Phys. Rev. Lett. 87, 020601 (2001)] to systems with symmetric and asymmetric bistable potentials. It is found that in systems with a relaxational flow and a symmetric local potential, reentrant phase transitions can be observed. In the case of an asymmetric local potential, a hysteresis-like behaviour in the order parameter appears. It is shown that such phase transitions can be controlled by the constant that governs relaxation flow, noise intensity and spatial coupling intensity.     

Heat and fluctuations from order to chaos

The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e. existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as 10²³ degrees of freedom systems, i.e. for simple as well as very complex systems, and reflecting the Hamiltonian nature of the microscopic motion. In Nonequilibrium Thermodynamics theorems of comparable generality do not seem to be available. Yet it is possible to find general, model independent, properties valid even for simple chaotic systems (i.e. the hyperbolic ones), which acquire special interest for large systems: the Chaotic Hypothesis leads to the Fluctuation Theorem which provides general properties of certain very large fluctuations and reflects the time-reversal symmetry. Implications on Fluids and Quantum systems are briefly hinted. The physical meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation Theorem is discussed in the context of their interpretation and relevance in terms of Coarse Grained Partitions of phase space. This review is written taking some care that each section and appendix is readable either independently of the rest or with only few cross references.     

Diffusion in flashing periodic potentials

The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential modulated by: (i) an external white Gaussian noise and (ii) a Markovian dichotomous noise. For both cases the exact expressions for the effective diffusion coefficient are derived. We obtain acceleration of diffusion in comparison with the free diffusion case for fast fluctuating potentials with arbitrary profile and for sawtooth potential in case (ii). In this case the parameter region where this effect can be observed is given. We obtain also a finite net diffusion in the absence of thermal noise. For rectangular potential the diffusion slows down, for all parameters of noise and of potential, in comparison with the case when particles diffuse freely.     

Finite-size effects in the self-organized critical forest-fire model

We study finite-size effects in the self-organized critical forest-fire model by numerically evaluating the tree density and the fire size distribution. The results show that this model does not display the finite-size scaling seen in conventional critical systems. Rather, the system is composed of relatively homogeneous patches of different tree densities, leading to two qualitatively different types of fires: those that span an entire patch and those that do not. As the system size becomes smaller, the system contains less patches, and finally becomes homogeneous, with large density fluctuations in time. Received 24 April 1999 and Received in final form 26 October 1999     

Onsager-Machlup Theory and Work Fluctuation Theorem for a Harmonically Driven Brownian Particle

We extend Tooru-Cohen analysis for nonequilibrium steady state (NSS) of a Brownian particle to nonequilibrium oscillatory state (NOS) of Brownian particle by considering time dependent external drive protocol. We consider an unbounded charged Brownian particle in the presence of oscillating electric field and prove work fluctuation theorem, which is valid for any initial distribution and at all times. For harmonically bounded and constantly dragged Brownian particle considered by Tooru and Cohen, work fluctuation theorem is valid for any initial condition (also NSS), but only in large time limit. We use Onsager-Machlup Lagrangian with a constraint to obtain frequency dependent work distribution function, and describe entropy production rate and properties of dissipation functions for the present system using Onsager-Machlup functional.     

Slow coarsening in a class of driven systems

The coarsening process in a class of driven systems is studied. These systems have previously been shown to exhibit phase separation and slow coarsening in one dimension. We consider generalizations of this class of models to higher dimensions. In particular we study a system of three types of particles that diffuse under local conserving dynamics in two dimensions. Arguments and numerical studies are presented indicating that the coarsening process in any number of dimensions is logarithmically slow in time. A key feature of this behavior is that the interfaces separating the various growing domains are macroscopically smooth (well approximated by a Fermi function). This implies that the coarsening mechanism in one dimension is readily extendible to higher dimensions. Received 3 April 2000     

Exact propagator of the Fokker-Planck equation with a space-dependent diffusion coefficient and a time-dependent mean-reverting force

We have investigated the algebraic structure of the Fokker-Planck equation with a variable diffusion coefficient and a time-dependent mean-reverting force. Such a model could be useful to study the general problem of a Brownian walker with a space-dependent diffusion coefficient. We also show that this model is related to the Fokker-Planck equation with a constant diffusion coefficient and a time-dependent anharmonic potential of the form V(x, t) = ?a(t)x ² + b ln x, which has been widely applied to model different physical and biological phenomena, e.g. the study of neuron models and stochastic resonance in monostable nonlinear oscillators. Using the Lie algebraic approach we have derived the exact diffusion propagators for the Fokker-Planck equations associated with different boundary conditions, namely (i) the case of a single absorbing barrier, and (ii) the case of two absorbing barriers. These exact diffusion propagators enable us to study the time evolution of the corresponding stochastic systems. Received 23 October 2001 and Received in final form 24 December 2001     

Lie algebraic approach for Fokker-Planck dynamics with space-dependent diffusion and mean-reverting drift

Using the Lie algebraic approach we have derived the exact diffusion propagator of the Fokker-Planck equation with a time-dependent variable diffusion coefficient and a time-dependent mean-reverting force between two absorbing boundaries. The exact diffusion propagator not only enables us to study the time evolution of the corresponding stochastic system, but the knowledge of the propagator can also provide a benchmark for testing approximate numerical or analytical procedures. Furthermore, the Lie algebraic method is very simple and could be easily extended to the more general Fokker-Planck equations with well-defined algebraic structures. Received 18 December 2002 / Received in final form 3 March 2003 Published online 24 April 2003     

A sharp transition between a trivial 1D BTW model and self-organized critical rice-pile model

A one-dimensional model of a rice-pile is numerically studied for different driving mechanisms. We found that for a sufficiently large system, there is a sharp transition between the trivial behaviour of a 1D BTW model and self-organized critical (SOC) behaviour. Depending on the driving mechanism, the self-organized critical rice-pile model belongs to two different universality classes. Received 18 December 1998     

Charge density plateaux and insulating phases in the $\mathsf{t-J}$ model with ladder geometry

We discuss the occurrence and the stability of charge density plateaux in ladder-like t-J systems (at zero magnetization M = 0) for the cases of 2- and 3-leg ladders. Starting from isolated rungs at zero leg coupling, we study the behaviour of plateaux-related phase transitions by means of first order perturbation theory and compare our results with Lanczos diagonalizations for t-J ladders (N = 2 × 8) with increasing leg couplings. Furthermore we discuss the regimes of rung and leg couplings that should be favoured for the appearance of the charge density plateaux.Received: 28 July 2003, Published online: 8 December 2003PACS: 71.10.Fd Lattice fermion models (Hubbard model, etc.) - 71.27. + a Strongly correlated electron systems; heavy fermions - 75.10.-b General theory and models of magnetic ordering - 75.10.Jm Quantized spin models     

Emergence of bimodality in noisy systems with single-well potential

We study the stationary probability density of a Brownian particle in a potential with a single-well subject to the purely additive thermal and dichotomous noise sources. We find situations where bimodality of stationary densities emerges due to presence of dichotomous noise. The solutions are constructed using stochastic dynamics (Langevin equation) or by discretization of the corresponding Fokker-Planck equations. We find that in models with both noises being additive the potential has to grow faster than |x| in order to obtain bimodality. For potentials ∝|x| stationary solutions are always of the double exponential form.     

Wealth distributions in asset exchange models

A model for the evolution of the wealth distribution in an economically interacting population is introduced, in which a specified amount of assets are exchanged between two individuals when they interact. The resulting wealth distributions are determined for a variety of exchange rules. For “random” exchange, either individual is equally likely to gain in a trade, while “greedy” exchange, the richer individual gains. When the amount of asset traded is fixed, random exchange leads to a Gaussian wealth distribution, while greedy exchange gives a Fermi-like scaled wealth distribution in the long-time limit. Multiplicative processes are also investigated, where the amount of asset exchanged is a finite fraction of the wealth of one of the traders. For random multiplicative exchange, a steady state occurs, while in greedy multiplicative exchange a continuously evolving power law wealth distribution arises. Received: 13 August 1997 / Revised: 31 December 1997 / Accepted: 26 January 1998     

The role of pinning and instability in a class of non-equilibrium growth models

We study the dynamics of a growing crystalline facet where the growth mechanism is controlled by the geometry of the local curvature. A continuum model, in (2+1) dimensions, is developed in analogy with the Kardar-Parisi-Zhang (KPZ) model is considered for the purpose. Following standard coarse graining procedures, it is shown that in the large time, long distance limit, the continuum model predicts a curvature independent KPZ phase, thereby suppressing all explicit effects of curvature and local pinning in the system, in the “perturbative” limit. A direct numerical integration of this growth equation, in 1+1 dimensions, supports this observation below a critical parametric range, above which generic instabilities, in the form of isolated pillared structures lead to deviations from standard scaling behaviour. Possibilities of controlling this instability by introducing statistically “irrelevant" (in the sense of renormalisation groups) higher ordered nonlinearities have also been discussed. Received 23 April 2002 / Received in final form 24 July 2002 Published online 31 October 2002 RID="a" ID="a"e-mail: akc@mpipks-dresden.mpg.de     

Scaling behavior of a nonlinear oscillator with additive noise,white and colored

We study analytically and numerically the problem of a nonlinear mechanical oscillator with additive noise in the absence of damping. We show that the amplitude, the velocity and the energy of the oscillator grow algebraically with time. For Gaussian white noise, an analytical expression for the probability distribution function of the energy is obtained in the long-time limit. In the case of colored, Ornstein-Uhlenbeck noise, a self-consistent calculation leads to (different) anomalous diffusion exponents. Dimensional analysis yields the qualitative behavior of the prefactors (generalized diffusion constants) as a function of the correlation time. Received 10 October 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: mallick@spht.saclay.cea.fr     

Fractional Langevin equation and Riemann-Liouville fractional derivative

In this present work we consider a fractional Langevin equation with Riemann-Liouville fractional time derivative which modifies the classical Newtonian force, nonlocal dissipative force, and long-time correlation. We investigate the first two moments, variances and position and velocity correlation functions of this system. We also compare them with the results obtained from the same fractional Langevin equation which uses the Caputo fractional derivative.     

Stability of a nonlinear oscillator with random damping

A noisy damping parameter in the equation of motion of a nonlinear oscillator renders the fixed point of the system unstable when the amplitude of the noise is sufficiently large. However, the stability diagram of the system can not be predicted from the analysis of the moments of the linearized equation. In the case of a white noise, an exact formula for the Lyapunov exponent of the system is derived. We then calculate the critical damping for which the nonlinear system becomes unstable. We also characterize the intermittent structure of the bifurcated state above threshold and address the effect of temporal correlations of the noise by considering an Ornstein-Uhlenbeck noise.     

Exactly solvable reaction diffusion models on a Cayley tree

The most general reaction-diffusion model on a Cayley tree with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.