From Microscopic Interactions to Macroscopic Laws: Duality, Algebraic Structures, and Universality in Interacting Particle Systems

Markov processes such as interacting particle systems, diffusion processes and redistribution models provide powerful mathematical tools for describing the collective behavior of a large number of interacting components. In this context, particles represent energies or masses, and the models are applied to study several phenomena such as mass transport, magnetism and heat conduction. The primary goal of this project is to investigate and characterize these models from two complementary perspectives: the microscopic scale, which focuses on the dynamics of the particles and their local interactions and the macroscopic scale, where the global evolution of the system emerges. The rigorous connections between these two scales is established through scaling limits that lead to descriptions of the dynamics where different universality classes emerge. At the microscopic level, the main question we are interested in comes from statistical physics: once the models are placed in contact with external reservoirs operating under different parameters, non-equilibrium phenomena arise. Roughly speaking, a system is out of equilibrium if there is a current (of particles, charge or energy) flowing through the system. In contrast, in equilibrium, i.e. when the systems are isolated from the environment, they admit a stationary and reversible measure in simple form. Reversibility is lost out of equilibrium and the stationary microscopic distributions are unknown, with very few exceptions. To gather information about the such steady state, the duality property of the models plays a crucial role, allowing to collect information on the long-range correlations. At the macroscopic level, the core of the project is to derive the large-scale laws governing the evolution of the system from the underlying particle interactions. This is achieved by performing suitable scaling limits, which lead to deterministic or stochastic partial differential equations with boundary conditions reflecting the strength of the external reservoirs. In the existing literature, there are numerous results for several systems which establish hydrodynamic limits and the corresponding fluctuations around it. However, the proofs are often tailored to specific models and rely heavily on model-dependent techniques. Therefore, we plan to focus on classes of models that share natural and basic assumptions, such as the gradient condition and the duality property, aiming to develop a general approach that is not tied to a particular model. The project therefore aims to unveil the phenomenological laws governing the dynamics of the system. This is pursued through a microscopic description of different models, which, in turn, leads to a unified macroscopic treatment, namely by obtaining partial differential equations whose solution describes the space/time evolution of the systems. Indeed, up to now, a complete rigorous derivation of the phenomenological laws of non-equilibrium from the underlying microscopic dynamics is lacking, and so this research would be a further step in that direction.