Memory Effects and Macroscopic Manifestation of Randomness

Abstract

It is shown that due to memory effects the complex behaviour of components in a stochastic system can be transmitted to macroscopic evolution of the system as a whole. Within the Markov approximation widely using in ordinary statistical mechanics, memory effects are neglected. As a result, a time-scale separation between the macroscopic and the microscopic level of description exists, the macroscopic differential picture is no a consequence of microscopic non-differentiable dynamics. On the other hand, the presence of complete memory in a system means that all its components have the same behaviour. If the memory function has no characteristic time scales, the correct description of the macroscopic evolution of such systems have to be in terms of the fractional calculus.

LaTeX, 18 pages, 1 postscript figure. New extended version, new paragraphs and references added

References in corpus (3)

• Fractional Calculus and the Evolution of Fractal Phenomena
• Fractional Calculus as a Macroscopic Manifestation of Randomness
• Chaotic dynamics, fluctuations, nonequilibrium ensembles

Cited by in corpus (10)

• Continuous time random walk and parametric subordination in fractional diffusion
• Fractional oscillator
• Fractional dynamics from the ordinary Langevin equation
• Black-Scholes model under subordination
• The Stochastic Nature of Complexity Evolution in the Fractional Systems
• Origin of the Curie-von Schweidler law and the fractional capacitor from time-varying capacitance
• Diffusion with preferential relocation in a confining potential
• On Fractals and Fractional Calculus Motivated by Complex Systems
• On stability, persistence and Hopf bifurcation of fractional order dynamical system
• On differences between fractional and integer order differential equations for dynamical games