Response theory and phase transitions for the thermodynamic limit of interacting identical systems

Abstract

We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers–Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker–Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai–Zwanzig model and of the Bonilla–Casado–Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.

Reduced Markovian Descriptions of Brownian Dynamics: Toward an Exact Theory

“…which describes the structure of the drift coefficient in the Smoluchowski equation (21). We also observe that the relation…”

“…Note that Eqs. ( 35) and (45) recover the drift and the diffusion coefficients appearing in the Smoluchowski equation (21).…”

“…We envisage further developments of similar model reduction ideas in the direction of coarse-graining of interacting particle systems as well as of partial differential equations with randomly fluctuating coefficients. Such research line might possibly connect this work with periodic and/or random homogenization questions; see [3,30,12,21,28] and [10] for recent applications of reduction schemes to epidemiological models. We also refer the reader to [32,26,16] for related matters, as well as to [23,15] for applications of similar methods to the reduction of complex dynamics arising in climate science, where the need of developing innovative reduction techniques is growing.…”

In Silico Prediction of Food Properties: A Multiscale Perspective

“…Similarly, the corresponding driving forces (e.g., osmotic or swelling pressures) are well described for systems with low TL values (i.e., random mixtures of molecules). However, they may be more difficult to derive for systems presenting an organization at nanoscale or memory effects (i.e., dislocation, plasticity, glassy state) (Lucarini et al, 2020). Two examples easily illustrate the loss of details of the continuum approach at the food scale.…”

Predictors and predictands of linear response in spatially extended systems

“…Recently, LRT has been extended in such a way that explicit formulas are given for describing how adding a forcing to a system changes its n−point correlations [44]. Another recent application of LRT has focused on detecting and characterising phase transitions in a network of coupled identical agents undergoing a stochastic evolution [45]. A recently published special issue showcases several emerging areas of applications for LRT [46].…”