Moderator: Gunter M. Schuetz (Juelich)
Bernard Derrida (Paris),
Chiara Franceschini (Modena),
Patricia Gonçalves (Lisboa),
Frank Redig (Delft),
Tridib Sadhu (Mumbai),
Gunter M. Schuetz (Juelich),
Ali Zahra (Lisboa),
Gunter M. Schuetz (Juelich): From reverse duality to shock random walks in the open asymmetric simple exclusion process. pdf
A reverse duality between the ASEP with open boundary conditions and an ASEP with particle-dependent jump rates is proved for special manifolds of boundary parameters of the ASEP. The boundary conditions of the dual random process can be reflecting or absorbing, depend on the choice duality function. As a consequence of this duality the full time-dependent distribution of the open ASEP starting from a Bernoulli shock measure with $n$ shocks at microscopic position $x_i$ is given for any time $t > 0$ by a convex time-dependent combination of shock measures with $n$ shock at positions $y_i$. The $(n+1)$-dimensional representations of the matrix algebra for the stationary matrix product measure of the open ASEP are also shown to be convex combinations of Bernoulli shock measure with $n$ shocks.
Bernard Derrida (Paris): At the transition between pulled and pushed fronts. pdf
Travelling wave equations describe the large scale evolution of interacting particle systems.
Depending on the linearities, the travelling waves can be pushed or pulled has an influence on the position of the front.
The cross-over between these two regimes can be calculated.
The same cross-over can be recovered for a Fleming Viot process in one dimension.
Bernard Derrida. "Cross-Overs of Bramson's Shift at the Transition Between Pulled and Pushed Fronts." J. Stat. Phys. 2023
Chiara Franceschini (Modena): Long-range correlations via duality for (a-)symmetric particle systems. pdf
I will present some symmetric and asymmetric interacting diffusions in contact with two external reservoirs which place the system out of equilibrium and thus create a non-reversible dynamics. These processes are connected to a simpler process via a duality relation. Thanks to this dual simpler process, which turns out to be the same in the symmetric and asymmetric regime, it is possible to gather information on space correlations in the steady states.
Patricia Gonçalves (Lisboa): On the non-equilibrium fluctuations of partial exclusion with boundary. pdf
In this talk, we will recall results on non-equilibrium fluctuations for the symmetric simple exclusion process with at most one particle per site in the presence of slow/fast boundary dynamics that inject and remove particles in the system.
We will then explain how the results extend to the symmetric simple partial exclusion process that allows up to a fixed number M>1 of particles per site. The results rely on precise asymptotic results for the two-points correlation function, which now are more complicated to derive due to the fact that in this exclusion process particles can occupy the same site. We obtain the results for all the regimes of slow and also fast boundary and for general initial measures.
This is a joint work with C. Franceschini, M Jara, B. Salvador.
Frank Redig (Delft): Duality and intertwining.
Duality is a technique that permits to connect boundary driven systems to systems with absorbing boundaries. Intertwining is a strongly related notion that permits to have a ``gateway’’ from one Markov process to another one. I will discuss connections between both notions, and provide several examples of both. In particular, I will discuss the notion of ``consistency’’ which permits to understand many dualities via commutation of the semigroup with the operator which removes a randomly chosen particle. We will see how this property gives many useful identities in the study of non-equilibrium steady states, as well as universal properties of correlation functions. I will also discuss Poisson intertwining which allows to connect discrete particle systems to interacting diffusion processes and expresses the non-equilibrium steady states as a mixture of Poisson product measures.
Based on joint works with G Carinci (Modena), C Giardina (Modena), S. Floreani (Bonn), F. Sau (Trieste).
Tridib Sadhu (Mumbai): Effect of boundary on the large deviations of density and of current in the non-equilibrium stationary state of symmetric simple exclusion process. pdf
I shall discuss our exact results for the large deviation function of the density profile and of the current in the non-equilibrium stationary state of a one-dimensional symmetric simple exclusion process coupled to boundary reservoirs with varied coupling strength. These new results extend the earlier seminal works of Derrida and collaborators for the same model where rates at the boundaries were comparable to the rates in the bulk, to regimes where boundary rates are significantly slower or faster.
I shall then show how these new results can be reproduced using the fluctuating hydrodynamics description of the macroscopic fluctuation theory. In describing this approach I shall discuss a "derivation" of the fluctuating hydrodynamics for the model and solutions of the variational problems that give the optimal paths. An advantage of the hydrodynamics formulation is its generality. I shall conclude by presenting the results of large deviations for a class of diffusive systems for which the variational problems are solvable.
Ali Zahra (Lisboa): Steady-state selection in multi-species driven diffusive systems. pdf
We introduce a general method to determine the large-scale non-equilibrium steady-state properties of one-dimensional multi-species driven diffusive systems with open boundaries, generalizing thus the max-min current principle known for systems with a single type of particles. This method is based on the solution of the Riemann problem of the associated system of conservation laws. We demonstrate that the effective density of a reservoir depends not only on the corresponding boundary hopping rates but also on the dynamics of the entire system, emphasizing the interplay between bulk and reservoirs. We highlight the role of Riemann variables in establishing the phase diagram of such systems. We apply our method to three models of multi-species interacting particle systems and compare the theoretical predictions with numerical simulations. This is joint work with Luigi Cantini [arxiv:2309.06231]