67 research outputs found
Fractional diffusion in periodic potentials
Fractional, anomalous diffusion in space-periodic potentials is investigated.
The analytical solution for the effective, fractional diffusion coefficient in
an arbitrary periodic potential is obtained in closed form in terms of two
quadratures. This theoretical result is corroborated by numerical simulations
for different shapes of the periodic potential. Normal and fractional spreading
processes are contrasted via their time evolution of the corresponding
probability densities in state space. While there are distinct differences
occurring at small evolution times, a re-scaling of time yields a mutual
matching between the long-time behaviors of normal and fractional diffusion
Relaxation in statistical many-agent economy models
We review some statistical many-agent models of economic and social systems
inspired by microscopic molecular models and discuss their stochastic
interpretation. We apply these models to wealth exchange in economics and study
how the relaxation process depends on the parameters of the system, in
particular on the saving propensities that define and diversify the agent
profiles.Comment: Revised final version. 6 pages, 5 figure
Fractional Fokker-Planck dynamics: Numerical algorithm and simulations
Anomalous transport in a tilted periodic potential is investigated
numerically within the framework of the fractional Fokker-Planck dynamics via
the underlying CTRW. An efficient numerical algorithm is developed which is
applicable for an arbitrary potential. This algorithm is then applied to
investigate the fractional current and the corresponding nonlinear mobility in
different washboard potentials. Normal and fractional diffusion are compared
through their time evolution of the probability density in state space.
Moreover, we discuss the stationary probability density of the fractional
current values.Comment: 10 pages, 9 figure
Basic kinetic wealth-exchange models: common features and open problems
We review the basic kinetic wealth-exchange models of Angle [J. Angle, Social
Forces 65 (1986) 293; J. Math. Sociol. 26 (2002) 217], Bennati [E. Bennati,
Rivista Internazionale di Scienze Economiche e Commerciali 35 (1988) 735],
Chakraborti and Chakrabarti [A. Chakraborti, B. K. Chakrabarti, Eur. Phys. J. B
17 (2000) 167], and of Dragulescu and Yakovenko [A. Dragulescu, V. M.
Yakovenko, Eur. Phys. J. B 17 (2000) 723]. Analytical fitting forms for the
equilibrium wealth distributions are proposed. The influence of heterogeneity
is investigated, the appearance of the fat tail in the wealth distribution and
the relaxation to equilibrium are discussed. A unified reformulation of the
models considered is suggested.Comment: Updated version; 9 pages, 5 figures, 2 table
Corrections to Einstein's relation for Brownian motion in a tilted periodic potential
In this paper we revisit the problem of Brownian motion in a tilted periodic
potential. We use homogenization theory to derive general formulas for the
effective velocity and the effective diffusion tensor that are valid for
arbitrary tilts. Furthermore, we obtain power series expansions for the
velocity and the diffusion coefficient as functions of the external forcing.
Thus, we provide systematic corrections to Einstein's formula and to linear
response theory. Our theoretical results are supported by extensive numerical
simulations. For our numerical experiments we use a novel spectral numerical
method that leads to a very efficient and accurate calculation of the effective
velocity and the effective diffusion tensor.Comment: 29 pages, 7 figures, submitted to the Journal of Statistical Physic
Anomalous Behavior of the Diffusion Coefficient in Interacting Adsorbates
Langevin simulations provide an effective way to study collective effects of
Brownian particles immersed in a two-dimensional periodic potential. In this
paper, we concentrate essentially on the behaviour of the tracer (DTr) and bulk
(DB) diffusion coefficients as function of friction. Our simulations show that
in the high friction limit, the two physical quantities DTr and DB present
qualitatively the same behaviour, for both coupled and decoupled substrate
potentials. However, for the low friction regime, and especially for the
coupled potential case, an anomalous diffusion behaviour is found. We also
found that in the case of weak dynamical coupling between the ad-particles and
the substrate, the exponents are not universal and rather depend on the
potentials. Moreover, changes in the inter-particle potentials may reverse the
behaviour to a normal one
Weakly non-ergodic Statistical Physics
We find a general formula for the distribution of time averaged observables
for weakly non-ergodic systems. Such type of ergodicity breaking is known to
describe certain systems which exhibit anomalous fluctuations, e.g. blinking
quantum dots and the sub-diffusive continuous time random walk model. When the
fluctuations become normal we recover usual ergodic statistical mechanics.
Examples of a particle undergoing fractional dynamics in a binding force field
are worked out in detail. We briefly discuss possible physical applications in
single particle experiments
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