Effective energy, interactions and out of equilibrium nature of scalar active matter

Estimating the effective energy, Eeff of a stationary probability distribution is a challenge for non-equilibrium steady states. Its solution could offer a novel framework for describing and analyzing non-equilibrium systems. In this work, we address this issue within the context of scalar active matter, focusing on the continuum field theory of Active Model B+. We show that the Wavelet Conditional Renormalization Group (WCRG) method allows us to estimate the effective energy of active model B+ from samples obtained by numerical simulations. We investigate the qualitative changes of Eeff as the activity level increases. Our key finding is that in the regimes corresponding to low activity and to standard phase separation the interactions in Eeff are short-ranged, whereas for strong activity, the interactions become long-ranged and lead to micro-phase separation. By analyzing the violation of Fluctuation-Dissipation theorem and entropy production patterns, which are directly accessible within the WCRG framework, we connect the emergence of these long-range interactions to the non-equilibrium nature of the steady state. This connection highlights the interplay between activity, range of the interactions and the fundamental properties of non-equilibrium systems. Estimating the effective energy of a stationary probability distribution is a challenge for non-equilibrium steady states, and its solution could provide insights into understanding such systems. This study employs the Wavelet Conditional Renormalization Group method, a data-driven approach, to infer the effective energy, which is used to quantify the connection between the emergence of long-ranged interactions and the non-equilibrium properties "—such as entropy production and violation of the fluctuation-dissipation theorem—" of scalar active matter.