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]. 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. But the link between the dynamics of a reaction coordinate and the (actual) 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 [3]. 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. 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 sub-fields (hydrodynamics [4–6], biophysics [7, 8], materials science [9, 10], statistical physics [11–18], glasses [19]) in order to share experience and knowledge about these problems. The current challenges are multiple but we can summarize them as follows.
There is as of yet no deep understanding, general consensus and widely accepted methods on:
1. How to systematically derive equations of motion for reaction coordinates or coarse-grained degrees of freedom, and models for memory functions for systems that are out of thermal equilibrium.
There are currently multiple available methods to formalize memory effects, going from ad hoc formulations to formal projection operator derivations [20, 21]. As a result, the equations take very different shapes: first- or second-order integro-differential equations [22,23], memory kernels that can be space-dependent or even dependent on other variables [24], etc. We aim at discussing the various forms that non-Markovian effects can be written in and their potential strengths and weaknesses.
2. How to numerically deal with complex integro-differential equations of motion in an efficient way.
When the memory kernel has a long time extent, as for instance is frequently encountered in hydrodynamics [25, 26], the well-known numerical solvers quickly become obsolete. Fourier-Laplace analysis is often a useful tool to deconvolute the memory terms [27], but it is unfortunately not possible to apply such methods out of equilibrium. We plan to review the recent advancements on this issue.
3. How to consistently establish a link between a Langevin description, on the level of the single experiment, and a Fokker-Planck description for the probability density of the reaction coordinates.
In equilibrium and Markovian processes, the correspondence is well established, but it becomes less clear for more complex processes. One option to create a link to generalized Langevin equations is the use of fractional Fokker-Planck equations [28, 29], but there are also other possible ways [30, 31].
References
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