# Séminaire du LPTMS: Alexander Hartmann

## 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.

**Date/Time : ** 02/05/2017 - *11:00 - 12:00*

**Location : ** LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay

#### Upcoming seminars

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