CSH Workshop: “Stochastic dynamics for complex systems”


Jun 01, 2022Jun 03, 2022

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This workshop is organized by Christian Kühn (CSH External Faculty & TU Munich) together with Maximilian Engel (FU Berlin) and Alexandra Neamtu (University of Konstanz).

The workshop takes place at the Complexity Science Hub Vienna.

 

In many complex systems modeling at a full-scale detail is impossible. Either, not all relevant processes are known precisely, or even if they are, there are uncertainties in parameters and external influences. Furthermore, microscopic models often become far too complicated for analytical or even numerical treatment. In all these cases, stochastic methods have proven to be highly beneficial to understand complex dynamical systems. Although stochastic approaches have been extremely successful, e.g., in the context of stochastic differential equations, data analysis, information processing, or dynamical systems, the methodological approaches are often surprisingly disconnected at first sight.

 

The main goal of this workshop is to foster the interplay between different approaches and to explore new unifying directions and methods to tackle challenges in complex systems via stochastic dynamics. Is there a unification of stochastic methods in dynamics?

 

Speakers

Stefanie Sonner: Random attractors for stochastic parabolic evolution equations via pathwise mild solutions


We study the longtime behavior of stochastic parabolic partial differential equations with additive noise. The differential operators in the equation depend on time and the underlying probability space. We prove the existence of global and exponential random attractors and derive estimates on their fractal dimension. To apply the framework of random dynamical systems we use the concept of pathwise mild solutions which yields pathwise representation formulas for solutions. Our approach to prove the existence of random attractors is different from the classical one where stochastic evolution equations are transformed into partial differential equations with random coefficients via the stationary Ornstein-Uhlenbeck process. This is joint work with C. Kuehn and A. Neamtu.

Jan Korbel: Thermodynamics of structure-forming systems


Structure-forming systems are ubiquitous in nature, ranging from atoms building molecules to self-assembly of colloidal amphibolic particles. The understanding of the underlying thermodynamics of such systems remains an important problem. Here, we derive the entropy for structure-forming systems that differs from Boltzmann-Gibbs entropy by a term that explicitly captures clustered states. For large systems and low concentrations, the approach is equivalent to the grand-canonical ensemble; for small systems, we find significant deviations. We derive the detailed fluctuation theorem and Crooks’ work fluctuation theorem for structure-forming systems. The connection to the theory of particle self-assembly is discussed.
We apply the results to several systems. We present the phase diagram for patchy particles described by the Kern-Frenkel potential. We show that the Curie-Weiss model with molecule structures exhibits a first-order phase transition. Finally, we show how the self-assembly of spin glass can be used for the prediction of group-size distribution in a society with homophilic interactions.

Péter Koltai: Collective variables in complex systems: from molecular dynamics to agent-based models and fluid dynamics


The identification of persistent forecastable structures in complicated or high-dimensional dynamics is vital for a robust prediction (or manipulation) of such systems in a potentially sparse-data setting. Such structures can be intimately related to so-called collective variables known for instance from statistical physics. We have recently developed a first data-driven technique to find provably good collective variables in molecular systems. Here we will show that these generalize to other applications as well, such as fluid dynamics and social dynamics.

Tommaso Rosati: Spectral gap for projective processes of hyperviscous SDPEs


We consider a linear hyperviscous SPDE with multiplicative noise as a toy model for an SPDE that does not satisfy order preservation and study the finiteness and uniqueness of its Lyapunov exponent. Under mild conditions on the noise, we can construct a Lyapunov functional for the associated projective process, based on the analysis of energy level sets. Under stronger conditions we obtain ergodicity and a spectral gap for the projective process (and hence uniqueness and Furstenberg-Khasminskii formulas for the Lyapunov exponent). This extends previous results, which in the infinite dimensional setting were essentially restricted to order preserving cases. Joint work with Martin Hairer.

Sebastian Riedel: Random dynamical systems on fields of Banach spaces


We present a framework that allows studying evolution equations on time-evolving Banach spaces with random dynamical systems. Our prime example is a stochastic delay differential equation for which the evolving Banach spaces are the spaces of controlled paths, known from rough paths theory. We formulate a multiplicative ergodic theorem and use it to prove the existence of local invariant manifolds. Eventually, we discuss further applications of our theory. Joint work with Mazyar Ghani Varzaneh (Hagen).

Alex Blumenthal: Lyapunov exponents on scales of Banach spaces


Lyapunov exponents describe the exponential rate at which trajectories of a dynamical system diverge, and have resulted in a successful abstract theory for describing the asymptotic regimes of chaotic dynamics in finite dimensions. However, many problems of interest are posed instead on infinite dimensional spaces, e.g. evolution equations, for which a variety of nonequivalent norms are often available. This leaves open the possibility that Lyapunov exponents depend on the norm. In this talk I will discuss these issues and circumstances when Lyapunov exponents are independent of the norm. Our results apply to a variety of fluid mechanics models, including incompressible passive scalar advection and linearization along trajectories of the Navier Stokes equations with either noisy or time periodic forcing.
Joint work with Sam Punshon-Smith.

Iulian Cimpean: On the path-continuity of Markov processes


Suppose that we are given a general second order integro-differential operator defined merely on a class of test functions, which corresponds to a cadlag Markov process, e.g. through the martingale problem. The aim is to present a general result which claims that if the class of test functions is rich enough (yet not necessarily a core), and if G is an open domain on which the generator has the local property expressed in a suitable way, then the Markov process has continuous paths when it passes through G. In fact, because the class of test functions is not necessarily a core, the aforementioned result holds for any Markov extension of the operator. The approach is potential theoretic and covers (possibly time-dependent) operators defined on domains in Hilbert spaces or on spaces of measures. This is a joint work with L. Beznea and M. Roeckner.

Details

Start
Jun 01, 2022
End
Jun 03, 2022

Venue

Complexity Science Hub Vienna
Josefstädter Straße 39
Vienna,1080Austria
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