- fractional calculus modelling


back to NEWS page

May 2001 - 010502
FRACALMO PRE-PRINT 0102 by F. Mainardi and F. Tampieri (to download)

We are pleased to offer to the interested visitor of the WEB site  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 force",
Technical Report No 1 ISAO - TR -1/99, ISAO-CNR Institute, Bologna 1999.

dowload 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.