To develop a model for non-equilibrium statistical mechanics, the system is typically brought into contact with two thermodynamic reservoirs, known as boundary reservoirs. These reservoirs impose their own particle density at the system's boundary, thereby inducing a current. Over time, a non-equilibrium steady state emerges, characterized by a stationary current value.

Recently, there has been increasing interest in multi-component systems, where various particle species (sometimes referred to as colors) coexist. In such setups, interactions between diferent species are possible alongside the occupation of available sites.

This work focuses on the boundary-driven multi-species stirring process on a one-dimensional lattice. This process extends naturally from the symmetric exclusion process (SEP) when multiple particle species are considered. Its dynamics involve particles exchanging positions with holes or with particles of diferent colors, each occurring at a rate of 1. Additionally, the system interacts with boundary reservoirs that inject, remove, and exchange types of particles.

After defining the process's generator using an appropriate representation of the gl(N) Lie algebra, we establish the existence of an absorbing dual process defined on an extended chain, where the two boundary reservoir are replaced by absorbing extra sites. This dual process shares bulk dynamics with the original but includes extra sites that absorb particles over extended time periods.

This multi-species stirring process can be mapped onto a higher rank open XXX-Heisenberg spin chain, therefore we employ absorbing duality and the matrix product ansatz to derive closed-form expressions for the non-equilibrium steady-state multi-point correlations of the process. This result is reported in [1].

Next, scaling limits of the process are examined, particularly the behavior of the properly scaled empirical density of the process. First, hydrodynamic equations are derived, illustrating typical system behavior (in the spirit of the law of large numbers). Second, fluctuations from this hydrodynamic limit are investigated, revealing a set of Gaussian processes coupled through noise, resembling aspects of the central limit theorem. Finally, large deviation results are reported, describing the probability of rare trajectories deviating from typical behaviors. An additional outcome of this analysis is the identification of a system of hydrodynamic equations featuring a drift due to interaction with an external field. These scaling limit results are reported in [2] and [3].

References:

[1] F. Casini, R. Frassek, C. Giardinà, Duality for the multispecies stirring process with open boundaries. (2024) J. Phys. A: Math. Theor. 57 295001

[2] F. Casini, C. Giardinà, F. Redig, Density Fluctuations for the Multi-Species Stirring Process. (2024) J Theor Probab 37, 33173354.

[3] F. Casini, F. Redig, H. van Wiechen, A large deviation principle for the multispecies stirring process. (2024), Arxiv: 2410.20857.

In this work, we deal with the symmetric exclusion process with k slow bonds equally spaced in the torus with kn sites, where the strength of a slow bond is $\alpha n^{-\beta}$, where $\beta>1$. For k fixed, it was known (T. Franco, P. Gonçalves, A. Neumann, AIHP'13) that the hydrodynamic limit in the diffusive scaling of this process is given by the heat equation with Neumann boundary conditions, meaning that the system does not allow flux through a slow bond in the limit. In this joint work with Tiecheng Xu and Dirk Erhard, we obtain another three superdiffusive scalings for this system. If $k$ is fixed and the (time) scaling is $n^{\theta}$, where $2< \theta<1+\beta$, the system reaches equilibrium instantaneously at each box between two consecutive slow bonds, being constant in time and space. If k is fixed and $\theta=1+\beta$, the density is spatially constant inside each box, and evolves as the discrete heat equation. And if $\theta=1+\beta$ and $k$ goes to infinity, we recover the continuous heat equation (on the torus).