Research
Homepage Barbara Gentz	Overview
My research interest is in probability theory, in particular in stochastic processes and their applications in the sciences. I am interested in the statistical mechanics of disordered spin systems as well as in the effect of noise on dynamical systems.
List of publications
The Hopfield Model as a Spin System with Random Interactions
The Hopfield model became popular in the beginning of the Eighties as a model for an associative memory in the context of artificial neural networks. It is also an interesting example of a spin system with random interactions. We studied the fluctuations of its order parameter, the so-called overlap, for non-critical and critical temperature, the latter in collaboration with Matthias Löwe.
Slowly Time-Dependent Dynamical Systems in the Presence of Noise
Consider a system modelled by an ordinary differential equation where is a parameter describing some exterior influence, for instance a control parameter in an experiment or the incoming solar radiation on Earth as in the examples below. We are interested in the combined effect of a slow deterministic forcing and noise on this system:
The deterministic forcing is modelled by allowing to vary slowly in time. Noise is modelled by additive white noise.
Thus we want to investigate non-autonomous stochastic differential equations of the form
where and are small positive parameters and is a standard Wiener process.
In collaboration with Nils Berglund, we have developed a new, constructive approach to the small-noise case which allows to characterize the behaviour of sample paths by the means of concentration results for typical paths in suitably defined space–time sets.
Examples to which we applied this approach so far, include:
Bifurcation delay   Consider a deterministic system undergoing a pitchfork bifurcation and assume that the control parameter is slowly swept through the bifurcation point. Then the deterministic system exhibits a macroscopic bifurcation delay. Naturally, this is an undesirable effect when determining a bifurcation diagram experimentally. In such a situation, noise of suitable strength can have a constructive effect and reduce the bifrucation delay significantly.   More on bifurcation delay (to come)
Stochastic resonance   Consider a bistable system where transitions between stable states are not possible. Small-amplitude periodic forcing alone will not be sufficient to allow for transitions, while arbitrarily small additive noise will cause interwell transitions at random times. However, when both kinds of perturbation are applied simultaneously, with suitably tuned amplitude and noise intensity, almost periodic transitions may occur. Such a system was first introduced as a model of the periodic appearance of the ice ages. In this context, the exterior forcing modelled the incoming solar radiation as a function of the eccentricity of the Earth's orbit. Noise-induced synchronization in a modulated double-well potential: A typical sample path   More on stochastic resonance
Hysteresis cycles   Consider the case where derives from a periodically forced double-well potential. In the deterministic case, hysteresis is observed: For small-amplitude forcing, the hysteresis cycles enclose a microscopic area, while for larger amplitudes, the area is of order one. For small, but positive noise intensities, with overwhelming probability, the enclosed area is still close to the corresponding deterministic value, while for noise intensities exceeding a certain threshold value, the area is typically close to a deterministic value which depends on the noise intensity but not on frequency or amplitude of the periodic forcing.   More on the effect of noise on hysteresis cycles
The Effect of Noise on General Slow–Fast Systems
With Nils Berglund, we have extended this approach, developed for slowly driven systems, to fully coupled slow–fast stochastic differential equations of the form
Results include concentration of the random fast variables near asymptotically stable slow manifolds or centre manifolds as well as the reduction to lower dimensional systems.
Stochastic Dynamics in Spatially Extended Systems
In collaboration with Nils Berglund and Bastien Fernandez, we currently investigate metastability in a system of coupled bistable oscillators, subject to noise. For sufficiently strong coupling, the system is synchronised . We study the sequence of desynchronisation transitions the system passes through as the coupling strength decreases.
Last modified: 16 September 2009, Barbara Gentz