### Next seminar : Séminaire du LPTMS: Chikashi Arita

##### Tuesday, February 20 2018 at 11:00:00

## Variational calculation of diffusion coefficients in stochastic lattice gases

### Chikashi Arita (Universität des Saarlandes, Saarbrücken)

Deriving macroscopic behaviors from microscopic dynamics of particles is a fundamental problem. In stochastic lattice gases one tries to demonstrate this hydrodynamic limit. The evolution of a stochastic lattice gas with symmetric hopping rules is described by a diffusion equation with density-dependent diffusion coefficient. In practice, even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient cannot be explicitly computed, except when a lattice gas additionally satisfies the "gradient condition", e.g. the diffusion coefficients of the simple exclusion process and non-interacting random walks are exactly identical to their hopping rates. We develop a procedure to obtain systematic analytical approximations for the diffusion coefficient in non-gradient lattice gases with known equilibrium. The method relies on a variational formula found by Varadhan and Spohn. Restriction on test functions to finite-dimensional sub-spaces allows one to perform the minimization and gives upper bounds for the diffusion coefficient. We apply the procedure to the following two models; one-dimensional generalized exclusion processes, where each site can accommodate at most two particles (2-GEPs) [1], and the Kob-Andersen (KA) model on the square lattice, which is classified into kinetically-constrained gas [2]. The prediction of the diffusion coefficient depends on the domain ("shape") of test functions. The smallest shapes give approximations which coincide with the mean-field theory, but the larger shapes, the more precise upper bounds we obtain. For the 2-GEPs, our analytical predictions provide upper bounds which are very close to simulation results throughout the entire density range. For the KA model, we also find improved upper bounds when the density is small. By combining the variational method with a perturbation approach, we discuss the asymptotic behavior of the diffusion coefficient in the high density limit.- [1] C. Arita, P. L. Krapivsky and K. Mallick,
*Variational calculation of transport coefficients in diffusive lattice gases*, Phys. Rev. E**95**, 032121 (2017) - [2] C. Arita, P. L. Krapivsky and K. Mallick,
*Bulk diffusion in a kinetically constrained lattice gas*, preprint cond-mat arXiv:1711.10616

### Last Highlight : Protein aggregation: a matter of frustration?

**Two physicists from LPTMS and the University of Chicago propose a new point of view on protein fibers associated with Alzheimer’s disease: to interpret their formation as the assembly of a jigsaw puzzle with mismatched pieces. This study is published in the journal Nature Physics.**

The cells of our body contains numerous biochemical machines with diverse roles known as proteins. Just like other machines, proteins can unfortunately malfunction and cause damage in their surroundings. This is what happens in a number of neurodegenerative diseases, but also in some forms of diabetes and anemia, where proteins meant to float freely within the cellular environment start aggregating with each other in the shape of long fibers which then interfere with other vital processes.

It is this tendency to form fibers that has caught the attention of two theoretical physicists, Martin Lenz from LPTMS (CNRS and Université Paris-Sud), and Thomas Witten, from the James Franck Institute of the University of Chicago. Sidestepping the traditional approach of observing the aggregation of a specific protein type using sophisticated experimental techniques, the researchers wondered whether the ubiquity of fibers across a very diverse range of aggregating proteins could be manifestation of a yet undiscovered general physical principle. By reflecting on the difference between a protein and the relatively symmetrical objects that physicists usually consider, they proposed that it is precisely the very irregular surface of proteins that prevents them from fitting cleanly together and forming a regular three-dimensional aggregate similar to the crystals that atoms for when they pile up.

To demonstrate that the mere presence of irregularities can induce a fibrous morphology, the physicists studied objects that were as simple as possible, but nevertheless incapable of fitting cleanly together to form a crystal. In contrast with a collection of cubes of the pieces of a jigsaw puzzle, such particles form so-called “frustrated” aggregates. Using mathematical calculations as well as computers, the researchers simulated the formation of aggregates from flexible polygons such as pentagons, irregular hexagons and octagons. The results are striking: whatever the type of particle used, fibers always form in the expected parameter regime, suggesting that identical particles will always tend to form fibers provided they are complex enough. “Our hope is to allow specialists in the field to discern a form a simplicity in these otherwise hugely complex systems, and to guide them towards a better understanding of a number of diseases.”, says Lenz. This new principle could also inspire methods to manufacture new materials from irregular nano-objects – proving that ideas from fundamental physics can fuel very diverse fields of application!

*Examples of frustrated particles and of the fibers they form when aggregated in a computer simulation. The particles in each fiber are deformed, and each new particle attaches to the tip of the aggregate to avoid deforming even more the ones that are already present.*

**Reference**: https://www.nature.com/articles/nphys4184

See also https://www.nature.com/articles/nphys4201