May 2001 - 010502
FRACALMO PRE-PRINT 0102 by F. Mainardi and F. Tampieri (to
We are pleased to offer to the interested visitor of the WEB site
www.fracalmo.org the possibility of down-loading the FRACALMO
pre-print (pp 25, 12 figures) by F. Mainardi and F. Tampieri
"Diffusion regimes in Brownian motion induced by the Basset history
Technical Report No 1 ISAO - TR -1/99, ISAO-CNR Institute, Bologna 1999.
the pdf file: fmisao02.pdf (277 kb)
The velocity autocorrelation and the displacement variance of a Brownian
particle moving in an incompressible viscous fluid are calculated
taking into account the effects of added mass and both
Stokes and Basset hydrodynamic forces.
These forces are known to describe the friction effects in a viscous
fluid, respectively in the steady state and in the transient state
of the motion, in the limit of vanishing Reynolds number.
The explicit expressions of these functions
versus time are provided in terms of Mittag-Leffler functions
and compared with the respective ones for the classical Brownian motion.
The effect of added mass is only to modify the time scale,
that is the characteristic relaxation time induced by the Stokes force.
The effect of the Basset force, which is of hereditary type
namely history-dependent, is to perturb the white noise of the random
force and change the decay character of the velocity
autocorrelation function from pure exponential to power law.
Furthermore, the displacement variance is shown to maintain, for
sufficiently long times, the linear behaviour which is typical of normal
diffusion, with the same diffusion coefficient of the classical case.
However, for light particles, the Basset history force
induces a long retarding effect in the establishing of the linear
behaviour, allowing for a regime of fast anomalous diffusion.
In conclusion, if an observer investigates the time evolution
of a cloud of light Brownian particles, he recognizes
that the normal diffusion is preceeded by a regime of fast anomalous
diffusion, which lasts for long time.
If the observation interval is not sufficiently long, he may be
induced to trust in the occurring of fast anomalous diffusion.