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## Memory effects in dynamical processes: theory and computational implementation

#### Erwin Schroedinger International Institute for Mathematics and Physics (ESI)

#### Organisers

**Attention: **The workshop needed to be postponed.

In statistical physics, one frequently deals with processes that involve a very large number of degrees of freedom, implying significant computational cost if one needs to resolve all microscopic trajectories. However, there is a wide class of dynamical phenomena that can be described by the evolution of a reduced set of variables. The most direct way to get to a description of this kind is to apply coarse-graining procedures [1, 2]. These consist in defining effective particles as a subset of the original particles. Each of the newly defined particles interacts with the others via effective interactions. This upscaling method allows to study systems on larger length- and timescales. It is widely used in polymer physics and biophysics, because there the molecules of interest are composed of a very large number of atoms. Studying the collective behaviour of an ensemble of these molecules requires such coarse-graining methods [3].

Another way to simplify many-body problems is to go even beyond coarse-grained interactions

and to study the dynamics of a restricted set of observables. One often refers to such variables

as ‘reaction coordinates’, and one demands that their value at a certain time yields a good description of the global state of the system and captures its main features. As an example, the

dynamics of phase-transitions can be studied in terms of the evolution of order parameter fields. In biophysics, the behaviour of complex molecules such as proteins is often described using structural motifs [4]. Similar examples can be found in different fields; thus it is relevant to be able to directly describe the evolution of the reaction coordinates instead of needing to solve all the microscopic equations of motion. But the link between the dynamics of a reaction coordinate and the microscopic dynamics is rarely trivial. As a generalization of Langevin’s pioneering works, Robert Zwanzig has shown in the early 1960s that the equations governing the dynamics of coarse-grained variables and reaction coordinates must necessarily be non-local in time [5]. The consequence of which are so-called ‘non-Markovian’ or ‘memory’ effects: not only its current state but also the past history of the system impacts its subsequent evolution. These effects arise in a broad class of processes and they can be formalized in various ways. However, their structure often causes them to be difficult to treat mathematically as well as numerically. In many situations, physicists assume that memory effects are negligible (to simplify the numerical modelling procedures) although no proof of this is given. Another strategy to avoid dealing with memory consists in constructing a reaction coordinate that is expected to have a Markovian evolution [6]. Unfortunately, such constructed observables are not always experimentally relevant or physically intuitive. Another route to deal with coarse-grained/reaction-coordinate dynamics is to take memory effects serious. In fact, it is often preferable to describe a process in terms of a variable that can be easily measured, even if the equation of motion that controls it is complicated. Memory effects can appear in various forms and thus are treated numerically in different ways. The common feature of the underlying equations is the presence of so-called memory kernels, which are the functions that control the extent of the memory. In this workshop, we aim at gathering experts on memory effects from different subfields in order to share experience and knowledge about these problems.

## References

**Austria**

Christoph Dellago (University of Vienna) - Organiser

**Germany**

Anja Kuhnhold (Albert-Ludwigs-Universitaet) - Organiser

Hugues Meyer (Saarland University) - Organiser & speaker

Tanja Schilling (University of Freiburg) - Organiser