Multi-species interacting particle systems: duality, integrability and scaling limits

Abstract

Interacting particle systems (IPS) are paradigmatic models for non-equilibrium statistical me chanics. Indeed, they allow to keep the computations manageable without losing the fundamen tal features of the systems they aim to reproduce. Usually, two central questions are investigated: the non-equilibrium stationary probability distribution, that is reached in the long-time horizon, and the hydrodynamic limit, which allows us to derive macroscopic laws that the IPS satisfies, once rescaled to the continuum. In the literature many results and techniques have been developed regarding single species IPS, which are models where indistinguishable particles move on a discrete geometry occupying the available vacancies. At the microscopic level, duality and integrability allow to obtain closed expressions for the non-equilibrium stationary distribution, while, the scaling limits theory has made rigorous the passage from micro to macro dynamics. At present, there is a growing interest in systems with multiple conservation laws, therefore, from the point of view of statistical mechanics, research focuses on multi-species interacting particles systems. In these IPS, there is the presence of multiple different species (or colours) of particles which perform the dynamic on a discrete geometry. The difference with respect to the single species set up is that, now, in addition to the occupation of the available vacancies, the different species of particles interact with each other, for instance by exchanging their positions and mutating species. This dynamics, is also reflected in the scaling limits, giving rise to systems of partial differential equations (PDE’s) describing multi-component transport models. In this realm, new phenomena may arise, such as multi-component uphill diffusion, which is a situation where the flux (of mass, particle, energy, charges...) has the same sign as the difference between boundary densities, causing the current to go uphill and violating the Fick’s law of diffusion. In this thesis we focus on boundary driven multi-species IPS where each vertex of the discrete geometry can host at most a finite number of particles, which can be chosen among the different species. For this reason, these IPS are said to have compact state space. First we analyse the so-called multi-species stirring process, which consists of a generalization of the symmetric exclusion process (SEP), when many colours of particles are considered. After describing the model without boundary interaction, we derive its scaling limits, both the hydrodynamic limit and the equilibrium fluctuations from it. Then, we put the system out of equilibrium by adding boundary reservoirs. We describe the non-equilibrium generator via a Lie algebra and we use this property to find an absorbing duality result. This absorbing duality, allows us to characterized the non-equilibrium steady state by means of absorption probabilities, which are the probabilities that dual particles are absorbed at extra-sites. These extra-sites replace, in the dual process, the original boundary reservoirs. While this duality result is available regardless of the geometry and maximal occupancy of the process, the complete characterization of the non-equilibrium steady state is accomplished for the integrable version of multi-species stirring process via the combination of duality and matrix product ansatz (MPA). This allows us to write a closed formula for the multi-point correlations. Finally, as a still partially open problem, we construct, for this integrable chain, the so-called mapping of non-equilibrium onto equilibrium by the use of quantum inverse scattering method. After having analyzed the stirring process we add a further transition mechanism: the re action. Starting from a system of linear PDE’s showing uphill diffusion for one of the species (partial uphill), we construct a multi-species reaction-diffusion process with the feature of having evolution equations for the average occupation variable (thought as a proxy for the true density) given by the discretization on a finite lattice of the previously introduced linear system of PDE’s. Then, the hydrodynamic limit is obtained. We observe that at this last level the uphill diffusion is lost. Furthermore, we prove the absorbing duality for this boundary driven reaction diffusion model and we derive the equilibrium density fluctuation from its hydrodynamic limit. Finally, we report two perspective for future research. The first concerns the study of multi-species processes with non-compact state space, i.e. having the possibility of hosting an unbounded number of particles at each site. In this context we introduce the multi-species independent random walker where, exploiting the lack of interaction we manage to completely characterize the non-equilibrium steady state via duality. We then introduce the multi-species harmonic process and the multi-species simple inclusion process. For both we define the genera tor with boundary driving and we find the reversible (equilibrium) measure. This is the starting point for the proof of absorbing duality, an goal of future research. As a second perspective we report a result regarding a single species asymmetric energy trans port model with boundary driving. The reservoirs have been designed to maintain the absorbing duality property. Furthermore, using this duality it is possible to characterize the exponential current in the non-equilibrium steady state. We believe that this single species result can be a starting point for the generalization of this bulk-boundary driven model to the multi-species set-up. Moreover, the scaling limits of this bulk-boundary driven process could also be used to describe the macroscopic equations of multi-component transport models with drift.