Séminaire du LPTMS: Alexander Hartmann


11:00 - 12:00

LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay
15 Rue Georges Clemenceau, Orsay, 91405

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Large deviations for equilibrium and non-equilibrium processes

Alexander Hartmann (Universität Oldenburg, Germany)

Large deviations and rare events play an ever increasing role in science, economy and society. Large deviations play a crucial role for example for the estimation of impacts of storms, the calculation of  probabilities of stock-market crashes or the sampling of transition  paths for conformation change of  proteins. More fundamentally, when  studying any random process, only the full probability distribution,  including the large-deviation tails, gives a complete information about  the underlying system.

The basic principal to study large deviations using numerical  simulations is simple: make unlikely events more probable and correct in  the end for the bias. Here, we present a very general black-box method,  based on  sampling vectors of random numbers within an artificial  finite-temperature (Boltzmann). This allows to access rare events and  large deviation for almost arbitrary equilibrium and non-equilibrium  processes. In this way, we obtain probabilities as  small as $10^{-500}$    and smaller, hence rare events and large deviation properties  can be  easily obtained.

The method can be applied to equilibrium/static sampling problems, e.g., the distribution of the number and size of connected components of  random graphs. Here, applications from different fields are presented:

1. Distribution of work performed for a  critical (T=2.269) two-dimensional  Ising system of size LxL=128×128 upon rapidly changing  the external magnetic field(also applying  theorems of Jarzynski and  Crooks to obtain the free energy difference  of such a large system)

2. Distribution of perimeters and area of convex hulls of two- and  higher dimensional single and multiple random walks.

3. Distribution of the flow for the Nagel-Schreckenberg traffic model.

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