Eur. Phys. J. Special Topics 228, 93–109 (2019) c  EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature, 2019 https://doi.org/10.1140/epjst/e2019-800111-4 THE E UROPEAN PHYSICAL J OURNAL SPECIAL TOPICS Regular Article From simple lattice models to systems of interacting particles: the role of stochastic regularity in transport models Antonio Brasiello 1 , Davide Cocco 2 , Fabio Garofalo 3 , and Massimiliano Giona 2, a 1 Dipartimento di Ingegneria Industriale, Universit` a degli Studi di Salerno, via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy 2 Dipartimento di Ingegneria Chimica DICMA Facolt` a di Ingegneria, La Sapienza Universit`a di Roma via Eudossiana 18, 00184 Roma, Italy 3 Department of Biomedical Engineering, Lund University, 221 00 Lund, Sweden Received 18 July 2018 / Received in final form 5 October 2018 Published online 30 May 2019 Abstract. The concept of stochastic regularity in lattice models corre- sponds to the physical constraint that the lattice parameters defining particle stochastic motion (specifically, the lattice spacing and the hop- ping time) attain finite values. This assumption, that is physically well posed, as it corresponds to the existence of bounded mean free path and root mean square velocity, modifies the formulation of the classical hydrodynamic limit for lattice models of particle dynamics, transform- ing the resulting balance equations for the probability density function from parabolic to hyperbolic. Starting from simple, but non trivial, lat- tice models of non interacting particles, the article analyzes the role of stochastic regularity in the formulation of the hydrodynamic equations. Specifically, the case of multiphase lattice models is considered both in regular and disordered structures, and the way of including interaction potential within the hyperbolic transport formalism analyzed. 1 Introduction Lattice models of interacting particles constitute an extremely fruitful theoretical and computational framework for addressing the emergent properties of thermodynamic systems both in equilibrium and in non-equilibrium conditions [1,2]. Recently, Colangeli et al. [3] have considered a one-dimensional lattice particle model satisfying an exclusion principle in the presence of an interparticle Kac poten- tial and found interesting dynamic phenomena similar to the occurrence of uphill diffusion [4]. For further details see also [5]. Uphill phenomena correspond to the occurrence, both in time and space, of local conditions where the flux of the trans- ported entity is oriented in the direction of positive concentration gradients, and not the opposite, as in the classical Fickian case [6]. The analysis of uphill phenomena has a long history, from the early work by Darken [7], accounting for experimental results on carbon transport in austenitic a e-mail: massimiliano.giona@uniroma1.it