Power-Law Tails from Multiplicative Noise

Abstract

We show that the well-known linear Langevin equation, modeling the Brownian motion and leading to a Gaussian stationary distribution of the corresponding Fokker-Planck equation, is changed by the smallest multiplicative noise. This leads to a power-law tail of the distribution for large enough momenta. At finite ratio of the correlation strength for the multiplicative and additive noise the stationary energy distribution becomes exactly the Tsallis distribution.Power-law tails are present in numerous distributions studied in physics or elsewhere when dealing with complex systems [1]. They are of generic interest, regarded as to signal long range order, non-vanishing correlations or scale invariance in complex systems with strong dynamics, which's details are mostly unknown.These are often contrasted to the traditional statistical system, showing the Gibbs distribution (exp(−E/T )) in energy, which is Gaussian in the momenta of free, massive particles (exp(−p 2 /2mT )). The latter is considered as the generic case for thermal equilibrium of non-correlated or shortrange correlated systems. This concept has been carried far beyond of its original field describing monoatomic ideal gas (Maxwell-Boltzmann statistics), by applying the Gibbs distribution in thermal equilibrium to areas such as particle physics and field theory. It serves as starting point of high-temperature field theory calculations both with analytical and numerical methods. Lattice gauge theory is based on the formal similarity between Euclidian path integrals and the canonical partition sum.A very simple and elegant, microdynamical explanation for the Maxwell-Boltzmann statistics is offered by the Langevin equation, describing a free particle moving under the influence of a deterministic damping force and a stochastic drive [2]. The latter accelerates the particle in a short time, changing its momenta randomly and uncorrelated. The stationary solution of this stochastic equation follows the Gaussian statistics, compatible to Gibbs' principle. It seems that in many statistical considerations of complex physical models from that on it is tacitly assumed that the presence of this additive noise is a dominant effect: the equilibrium distribution follows Gibbs' formula (the Gaussian distribution in momentum for a free, massive particle). Since the harmonic oscillator is just the extension of this free motion Langevin equation into the phase space, also for free quantum systems the above picture is generally accepted.We shared this expectations, thinking that any nonGaussian (or in the energy non-exponential) distribution, especially the power-law tail observed in many phenomena including particle spectra in high energy physics, may only come from non-thermalized or in an other way nonequilibrium situation. In this note we would like to share our deep astonishment about that this is not so: we found out that treating the damping constant in the Langevin equation also stochastically (considering this way both multiplicative and additive noise) the...

Non-extensive equilibration in relativistic matter

“…There are several speculations on possible causes for such deformations, measured in a summerized way by a parameter used in Tsallis' non-extensive [8][9][10][11][12] entropy formula, or Rényi's extensive one [13]. Fluctuations of temperature or energy equipartition [14][15][16], anomalous diffusion [17] or multiplicative noise [18,19], or alternatively an altering in the two-body energy composition rule [20][21][22][23][24] were investigated in theoretical approaches. In order to solidify the motivation for such an approach we sketch some arguments about possible occurences of non-extensivity in thermodynamically treateble, large systems (the size is measured not by a volume, but by the number of particles produced and analyzed).…”

Nonextensive perfect hydrodynamics — a model of dissipative relativistic hydrodynamics?

“…In particular, the entropy vector is quadratic in the fluxes, containing terms characterizing the deviation from local equilibrium. This situation, plus our experience with the nonextensive formalism, [29][30][31][32][33][34][35][36][37][42][43][44][45][46][47][48] prompted us to investigate the simple nonextensive formulation of the perfect hydrodynamic model, a perfect -hydrodynamics [88][89][90]. It turned out that this describes the experimental data fairly well.…”

“…Roughly speaking, all observed effects amount to a broadening of the respective spectra of the observed secondaries (both in transverse momentum space and in rapidity space), they take the form of -exponents instead of the naively expected usual exponents: exp(−X /T ) =⇒ exp (−X /T ) = [1 − (1 − )X /T ] 1/ (1− ) . From these studies emerged a commonly accepted interpretation of the nonextensivity parameter (in fact, | − 1|), as the measure of some intrinsic fluctuations characteristic of the hadronizing systems under consideration [29][30][31]. For > 1, in transverse momentum space, could represent fluctuation of the temperature, T , corresponding to some specific heat parameter C .…”

“…The only works discussing some general features of the nonextensive hydrodynamics, which we are aware of [19][20][21], use a nonrelativistic approach and are therefore not suitable for the applications we are interested in. On the other hand it is known that an approach based on non-extensive statistical mechanics (used mainly in the form proposed by Tsallis [22][23][24][25][26][27][28] with only one new parameter, the nonextensivity parameter ) describes different sets of data in a better way than the usual statistical models based on BG statistics, cf., [22][23][24][25][26][27][28] for general examples and [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] for applications to multi-particle production processes. Roughly speaking, all observed effects amount to a broadening of the respective spectra of the observed secondaries (both in transverse momentum space and in rapidity space), they take the form of -exponents instead of the naively expected usual exponents: exp(−X /T ) =⇒ exp (−X /T ) = [1 − (1 − )X /T ] 1/ (1− ) .…”

Lévy-statistics for partially equilibrated systems

“…Beck and Cohen [15] introduced a generalized Boltzmann-Gibbs factor leading to the concept of "superstatistics" and Tsallis and Souza [16] constructed a statistical mechanics for such superstatistics. Recently studies on multiplicative noice revealed the presence of power-law tails [17]. In this article the notion of non-extensivity is examined for a class of partially equilibrated systems.…”