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The well-posedness of a multi-population dynamical system with an entropy regularization and its convergence to a suitable mean-field approximation are proved, under a general set of assumptions. Under further assumptions on the evolution…

Entropic dynamics is a framework in which quantum theory is derived as an application of entropic methods of inference. Entropic dynamics on flat spaces has been extensively studied. The objective of this paper is to extend the entropic…

We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and…

The nonperturbative real-time evolution of quantum fields out of equilibrium is often solved using a mean-field or Hartree approximation or by applying effective action methods. In order to investigate the validity of these truncations, we…

We describe the application of methods from the study of discrete dynanmical systems to the problem of the continuum limit of evolving spin networks. These have been found to describe the small scale structure of quantum general relativity…

Important gaps remain in our understanding of the thermodynamics and statistical physics of self-gravitating systems. Using mean field theory, here we investigate the equilibrium properties of several spherically symmetric model systems…

We analyze the continuum limit of a thresholding algorithm for motion by mean curvature of one dimensional interfaces in various space-time discrete regimes. The algorithm can be viewed as a time-splitting scheme for the Allen-Cahn equation…

Perceptrons are the building blocks of many theoretical approaches to a wide range of complex systems, ranging from neural networks and deep learning machines, to constraint satisfaction problems, glasses and ecosystems. Despite their…

For algorithms based on interacting particle systems that admit a mean-field description, convergence analysis is often more accessible at the mean-field level. In order to transfer convergence results obtained at the mean-field level to…

In this paper we establish new simple local geometric criteria for discrete entropic curvature introduced in [47] that are powerful enough to capture many geometric properties of complex models arising in mathematical physics. These results…

These lectures notes are aimed at introducing the reader to some recent mathematical tools and results for the mean-field limit in statistical dynamics. As a warm-up, lecture 1 reviews the approach to the mean-field limit in classical…

We establish the convergence of threshold dynamics-type approximation schemes to propagating fronts evolving according to an anisotropic mean curvature motion in the presence of a forcing term depending on both time and position, thus…

We investigate the role of entropic concepts for the relaxation dynamics in granular systems. In these systems the existence of a geometrical frustration induces a drastic modification of the allowed phase space, which in its turn induces a…

This work addresses the mean-field limit of inertial particle systems with singular interactions in a perturbative regime around Gibbs equilibrium. We prove that small fluctuations around equilibrium are asymptotically governed by the…

In this paper we consider global $\theta$-curvatures of finite Markov chains with associated means $\theta$ in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-\'Emery…

We establish a general perturbative method to prove entropic Ricci curvature bounds for interacting stochastic particle systems. We apply this method to obtain curvature bounds in several examples, namely: Glauber dynamics for a class of…

We study the full out-of-thermal-equilibrium dynamics of a relativistic classical scalar field through a symmetry breaking phase transition. In these circumstances we determine the evolution of the ensemble averages of the correlation…

Exponential quintessence models motivated by string compactifications naturally involve both a dilatonic scalar and its axionic partner evolving on a curved field space, while spatial curvature enlarges the cosmological phase space and may…

Fluctuation dynamics of an experimentally measured observable offer a primary signal for nonequilibrium systems, along with dynamics of the mean. While universal speed limits for the mean have actively been studied recently, constraints for…

We study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on…