Review of Some Promising Fractional Physical Models

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties by using integrations and differentiation of non-integer orders, i.e., by methods of the fractional calculus. This paper is a review of physical models that look very promising for future development of fractional dynamics. We suggest a short introduction to fractional calculus as a theory of integration and differentiation of non-integer order. Some applications of integro-differentiations of fractional orders in physics are discussed. Models of discrete systems with memory, lattice with long-range inter-particle interaction, dynamics of fractal media are presented. Quantum analogs of fractional derivatives and model of open nano-system systems with memory are also discussed.Comment: 38 pages, LaTe

Stochastic Theory of Relativistic Particles Moving in a Quantum Field: II. Scalar Abraham-Lorentz-Dirac-Langevin Equation, Radiation Reaction and Vacuum Fluctuations

We apply the open systems concept and the influence functional formalism introduced in Paper I to establish a stochastic theory of relativistic moving spinless particles in a quantum scalar field. The stochastic regime resting between the quantum and semi-classical captures the statistical mechanical attributes of the full theory. Applying the particle-centric world-line quantization formulation to the quantum field theory of scalar QED we derive a time-dependent (scalar) Abraham-Lorentz-Dirac (ALD) equation and show that it is the correct semiclassical limit for nonlinear particle-field systems without the need of making the dipole or non-relativistic approximations. Progressing to the stochastic regime, we derive multiparticle ALD-Langevin equations for nonlinearly coupled particle-field systems. With these equations we show how to address time-dependent dissipation/noise/renormalization in the semiclassical and stochastic limits of QED. We clarify the the relation of radiation reaction, quantum dissipation and vacuum fluctuations and the role that initial conditions may play in producing non-Lorentz invariant noise. We emphasize the fundamental role of decoherence in reaching the semiclassical limit, which also suggests the correct way to think about the issues of runaway solutions and preacceleration from the presence of third derivative terms in the ALD equation. We show that the semiclassical self-consistent solutions obtained in this way are ``paradox'' and pathology free both technically and conceptually. This self-consistent treatment serves as a new platform for investigations into problems related to relativistic moving charges.Comment: RevTex; 20 pages, 3 figures, Replaced version has corrected typos, slightly modified derivation, improved discussion including new section with comparisons to related work, and expanded reference

A new non-perturbative approach to Quantum Brownian Motion

Starting from the Caldeira-Leggett (CL) model, we derive the equation describing the Quantum Brownian motion, which has been originally proposed by Dekker purely from phenomenological basis containing extra anomalous diffusion terms. Explicit analytical expressions for the temperature dependence of the diffusion constants are derived. At high temperatures, additional momentum diffusion terms are suppressed and classical Langivin equation can be recovered and at the same time positivity of the density matrix(DM) is satisfied. At low temperatures, the diffusion constants have a finite positive value, however, below a certain critical temperature, the Master Equation(ME) does not satisfy the positivity condition as proposed by Dekker.Comment: 5 page