Anomalous Diffusion in Deterministic Systems
Lead Research Organisation:
University of Surrey
Department Name: Mathematics
Abstract
Certain gas models (planar periodic Lorentz flows) were introduced as deterministic models for Brownian motion. A statistical law called the almost sure invariance principle (ASIP) makes precise the connection with Brownian motion. Recent work established the ASIP for the Lorentz flow.Brownian motion has a characteristic growth rate of order sqrt t (square root of t ) where t denotes time. Currently, there is a great deal of interest in anomalous diffusion with a different growth rate. The aim of this project is the systematic study of anomalous diffusion (both subdiffusive and superdiffusive) in deterministic systems. In the superdiffusive (fast) case, this means proving the ASIP with Brownian motion replaced by a Levy process. In the subdiffusive (slow) case, we will focus on Sinai diffusion where the growth rate is of order (log t)^2.The results will apply to infinite horizon Lorentz gases and Bunimovich stadia. In addition, there are applications in dynamical systems with symmetry, including spiral waves in excitable media, and travelling waves.
Organisations
People |
ORCID iD |
| Ian Melbourne (Principal Investigator) |
Publications
BRUIN H
(2011)
ON YOUNG TOWERS ASSOCIATED WITH INFINITE MEASURE PRESERVING TRANSFORMATIONS
in Stochastics and Dynamics
Gottwald G
(2016)
Central limit theorems and suppression of anomalous diffusion for systems with symmetry
in Nonlinearity
Gottwald G
(2013)
Homogenization for deterministic maps and multiplicative noise
in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Gottwald G
(2013)
A Huygens principle for diffusion and anomalous diffusion in spatially extended systems
in Proceedings of the National Academy of Sciences
Ian Melbourne (Author)
(2006)
Decay of correlations for flows with unbounded roof function, including the infinite horizon planar periodic Lorentz gas
Ian Melbourne (Author)
(2011)
Statistical properties and decay of correlations for interval maps with critical points and singularities
Luzzatto S
(2013)
Statistical Properties and Decay of Correlations for Interval Maps with Critical Points and Singularities
in Communications in Mathematical Physics
Melbourne I
(2011)
Operator renewal theory and mixing rates for dynamical systems with infinite measure
in Inventiones mathematicae
| Description | Certain gas models (planar periodic Lorentz flows) were introduced as deterministic models for diffusive behaviour in the form of Brownian motion. A statistical law called the almost sure invariance principle (ASIP) makes this precise. In previous work we established the ASIP for the finite horizon Lorentz flow. The general aim of this project was the systematic study of diffusion and anomalous in deterministic systems. In the superdiffusive case, this means proving convergence to a Levy |
| Exploitation Route | Mathematics research |
| Sectors | Other |
| URL | http://homepages.warwick.ac.uk/~maslaq/publications.html |
| Description | Visiting Professorship - Professor Jonathan Aaronson |
| Amount | £23,600 (GBP) |
| Organisation | The Leverhulme Trust |
| Sector | Charity/Non Profit |
| Country | United Kingdom |
| Start | 08/2011 |
| End | 06/2012 |
| Description | Workshop on "Infinite Ergodic Theory" at Surrey |
| Amount | £14,555 (GBP) |
| Funding ID | EP/J02130X/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 04/2012 |
| End | 05/2012 |
| Description | Workshop on Infinite Ergodic Theory |
| Amount | £5,725 (GBP) |
| Funding ID | 11102 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 04/2012 |
| End | 05/2012 |