Effective Energy, Interactions And Out Of Equilibrium Nature Of Scalar Active Matter

Estimating the effective energy, $E_\text{eff}$ 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 method allows us to estimate the effective energy of active model B+ from samples obtained by numerical simulations. We investigate the qualitative changes of $E_\text{eff}$ 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 $E_\text{eff}$ 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.

💡 Research Summary

The paper tackles the long‑standing problem of characterizing the stationary probability distribution of non‑equilibrium systems by constructing an effective energy functional (E_{\text{eff}}=-\ln p_{\text{ss}}). Because detailed balance is broken in most driven systems, there is no general principle that links the microscopic forces to a static energy landscape, and analytical expressions for (E_{\text{eff}}) are rarely available. The authors address this gap by employing the Wavelet Conditional Renormalization Group (WCRG) – a data‑driven, multiscale inference scheme that reverses the usual RG flow: it starts from coarse‑grained fields and proceeds to finer scales, estimating at each level the conditional probability of the wavelet coefficients given the coarser field.

WCRG leverages the sparsity of high‑frequency wavelet components, making the estimation of conditional distributions numerically stable even when long‑range correlations are present. The conditional probabilities are modeled with low‑dimensional, physics‑informed parametric forms (often close to Gaussian). By minimizing the score (the gradient of the log‑probability) rather than the full KL divergence, the method remains computationally efficient. Once all conditional distributions are learned, the full joint distribution of the microscopic field (\phi) can be reconstructed, yielding an explicit expression for the effective energy:
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