Nonlinear Markov processes

Some elementary properties and examples of Markov processes are reviewed. It is shown that the definition of the Markov property naturally leads to a classification of Markov processes into linear and nonlinear ones.     

Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.     

On the way towards a generalized entropy maximization procedure

We propose a generalized entropy maximization procedure, which takes into account the generalized averaging procedures and information gain definitions underlying the generalized entropies. This novel generalized procedure is then applied to Rényi and Tsallis entropies. The generalized entropy maximization procedure for Rényi entropies results in the exponential stationary distribution asymptotically for q∈(0,1] in contrast to the stationary distribution of the inverse power law obtained through the ordinary entropy maximization procedure. Another result of the generalized entropy maximization procedure is that one can naturally obtain all the possible stationary distributions associated with the Tsallis entropies by employing either ordinary or q-generalized Fourier transforms in the averaging procedure.     

Characterizing dynamical transitions in bistable systems using a non-equilibrium measurement of work

We show how the Jarzynski relation can be exploited to analyze the nature of order-disorder, and a bifurcation type dynamical transition in terms of a response function derived on the basis of work distribution over non-equilibrium paths between two thermalized states. The validity of the response function extends over a linear as well as a nonlinear regime, and far from equilibrium situations.     

Finite-size-induced transition in ensemble of globally coupled oscillators

The collective behavior of overdamped nonlinear noise-driven oscillators coupled via mean field is investigated numerically. When a coupling constant is increased, a transition in the dynamics of the mean field is observed. This transition scales with the number of oscillators and disappears when this number tends to infinity. Analytical arguments explaining the observed scaling are presented.     

The non-equilibrium work relation: Thermodynamic analysis and microscopic foundations

We discuss the conditions for which the non-equilibrium work relation is valid by means of thermodynamic and microscopic arguments.     

Rigid rotators and diatomic molecules via Tsallis statistics

We obtain an analytic expression for the specific heat of a system of N rigid rotators exactly in the high temperature limit, and via a perturbative approach in the low temperature limit. We then evaluate the specific heat of a diatomic gas with both translational and rotational degrees of freedom, and conclude that there is a mixing between the translational and rotational degrees of freedom in nonextensive statistics.     

A definition of metastability for Markov processes with detailed balance

A definition of metastable states applicable to arbitrary finite state Markov processes satisfying detailed balance is discussed. In particular, we identify a crucial condition that distinguishes genuine metastable states from other types of slowly decaying modes and which leads to properties similar to those postulated in the restricted ensemble approach [1]. The intuitive physical meaning of this condition is simply that the total equilibrium probability of finding the system in the metastable state is negligible. As a concrete application of our formalism we present preliminary results on a 2D kinetic Ising model.     

Nonequilibrium work fluctuations in a driven Ising model

Monte Carlo simulations of a sheared Ising model are used to study nonequilibrium fluctuations of mechanical work. The validity of the transient (starting from equilibrium) and the steady state fluctuation relations is verified. A fluctuation relation has been also shown to hold for the mechanical work done on the system, during the transition between two nonequilibrium steady states corresponding to different drivings.     

Equilibration of two power-law tailed distributions in a parton cascade model

We study the process of equilibration between two non-extensive subsystems in the framework of a particular non-extensive Boltzmann equation. We have found that even subsystems with different non-extensive properties achieve a common equilibrium distribution. We extract the thermodynamic temperature from final energy distributions in a particular case.     

Phase diagram and scattering intensity of binary amphiphilic systems

A Ginzburg-Landau model with a scalar and a vector order parameter, which describe the concentration and orientation of the amphiphile, respectively, is used to study the phase diagram and the scattering intensity of binary amphiphilic systems. With increasing amphiphile concentration, the calculated phase diagram shows the typical sequence of ordered phases observed experimentally, that is micellar liquid cubic micellar hexagonal lamellar cubic bicontinuous invers hexagonal. The scattering intensity in the homogeneous phase is calculated in the oneloop approximation. In the vicinity of a phase transition to an ordered phase, the intensity is found to show a 1/q behavior for not too small wave vectorsq, followed by a small peak, and a 1/q ² decay for large wave vectors, in agreement with experimental observations in theL ₃-(or sponge-)phase.Dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Equivalence among different formalisms in the Tsallis entropy framework

In a recent paper [T. Wada, A.M. Scarfone, Phys. Lett. A 335 (2005) 351] the authors discussed the equivalence among the various probability distribution functions of a system in equilibrium in the Tsallis entropy framework. In the present letter we extend these results to a system which is out of equilibrium and evolves to a stationary state according to a nonlinear Fokker-Planck equation (NFPE). By means of time-scale conversion, it is shown that there exists a “correspondence” among the self-similar solutions of the NFPEs associated with the different Tsallis formalisms. The time-scale conversion is related to the corresponding Lyapunov functions of the respective NFPEs.     

Four-photon processes stimulated by a fluctuating pump

Four-photon parametric light amplification stimulated by noisy pump in a dispersive medium, is theoretically analyzed. The model of the fluctuating phase is used. The increments in case of temporal as well as of spatial laser pump modulation are considered. It is shown that the efficiency of the four-photon process due to a broad-band pump is comparable to that of monochromatic pump. Quantitative estimates for the medium of metal vapor are given, too.     

A generalized Kolmogorov-Sinai-like entropy under Markov shifts in symbolic dynamics

In this paper, a generalized Kolmogorov-Sinai-like entropy ( entropy) in the sense of Tsallis is proposed with a nonextensive parameter q under Markov shifts, which contains the classical Kolmogorov-Sinai (KS) entropy and the Rényi entropy as well as Bernoulli shifts as special cases. To verify the formula of this entropy, a one-dimensional iterative system is chosen as an example of Markov shifts, and its entropy is evaluated by a new refinement method of symbolic dynamics called symbolic refinement which differs from the conventional numerical method. The numerical results show that this entropy is monotonically decreasing as q increases.     

Absence of hyperscaling violations for phase transitions with positive specific heat exponent

Finite size scaling theory and hyperscaling are analyzed in the ensemble limit which differs from the finite size scaling limit. Different scaling limits are discussed. Hyperscaling relations are related to the identification of thermodynamics as the infinite volume limit of statistical mechanics. This identification combined with finite ensemble scaling leads to the conclusion that hyperscaling relations cannot be violated for phase transitions with strictly positive specific heat exponent. The ensemble limit allows to derive analytical expressions for the universal part of the finite size scaling functions at the critical point. The analytical expressions are given in terms of generalH-functions, scaling dimensions and a new universal shape parameter. The universal shape parameter is found to characterize the type of boundary conditions, symmetry and other universal influences on critical behaviour. The critical finite size scaling functions for the order parameter distribution are evaluated numerically for the cases =3, =5 and =15 where is the equation of state exponent. Using a tentative assignment of periodic boundary conditions to the universal shape parameter yields good agreement between the analytical prediction and Monte-Carlo simulations for the two dimensional Ising model. Analytical expressions for critical amplitude ratios are derived in terms of critical exponents and the universal shape parameters. The paper offers an explanation for the numerical discrepancies and the pathological behaviour of the renormalized coupling constant in mean field theory. Low order moment ratios of difference variables are proposed and calculated which are independent of boundary conditions, and allow to extract estimates for a critical exponent.     

Damage-spreading in the parallel Bak-Sneppen model

We study the behavior under perturbations of the Parallel Bak-Sneppen model (PBS) in 1+1 dimension, which has been shown to belong to the universality class of Directed Percolation (DP) in 1+1 dimensions [#!SD96!#]. We focus our attention on the damage-spreading features of the PBS model with both random and deterministic updating, which are studied and compared to the known results for the extremal Bak-Sneppen model (BS) and for DP. For both random and deterministic updating, we observe a power law growth of the Hamming distance. In addition, we compute analytically the asymptotic plateau reached by the distance after the growing phase. Received: 24 September 1998 / Revised: 17 November 1998 / Accepted: 19 November 1998     

Sensitivity to initial conditions in the Bak-Sneppen model of biological evolution

We consider biological evolution as described within the Bak and Sneppen 1993 model. We exhibit, at the self-organized critical state, a power-law sensitivity to the initial conditions, calculate the associated exponent, and relate it to the recently introduced nonextensive thermostatistics. The scenario which here emerges without tuning strongly reminds of that of the tuned onset of chaos in say logistic-like one-dimensional maps. We also calculate the dynamical exponent z. Received: 5 November 1997 / Received in final form: 11 November 1997 / Accepted: 19 November 1997     

Dynamics of Symmetric Conserved Mass Aggregation Model on Complex Networks

We investigate the dynamical behaviour of the aggregation process in the symmetric conserved mass aggregation model under three different topological structures. The dispersion a（t, L） = （∑i（mi - ρo ） ^2 / L ）1/2 is defined to describe the dynamical behaviour where Po is the density of particle and mi is the particle number on a site. It is found numerically that for a regular lattice and a scale-free network, σ（t, L） follows a power-law scaling σ（ t, L） ~ t^δ1 and σ（ t, L） ~ t^δ4 from a random initial condition to the stationary states, respectively. However, for a small-world network, there are two power-law scaling regimes, σ（t, L） ~ t^δ2 when t 〈 T and 〈（t, L） ~ t^δ3 when t 〉 T. Moreover, it is found numerically that 62 is near to 61 for small rewiring probability q, and 63 hardly changes with varying q and it is almost the same as 64. We speculate that the aggregation of the connection degree accelerates the mass aggregation in the initial relaxation stage and the existence of the long-distance interactions in the complex networks results in the acceleration of the mass aggregation when t 〉 T for the small-world networks. We also show that the relaxation time r follows a power-law scaling τ ~ L^z and σ（t, L） in the stationary state follows a power-law Gs（L） - L^α for three different structures.