Weak Galilean invariance as a selection principle for coarse-grained diffusive models

TL;DR

This paper introduces the concept of weak Galilean invariance as a fundamental principle for selecting physically consistent coarse-grained diffusive models, which inherently violate classical Galilean invariance.

Contribution

It proposes weak Galilean invariance as a new selection criterion for coarse-grained stochastic models, ensuring their physical consistency despite violations of classical invariance.

Findings

Coarse-grained models must satisfy weak Galilean invariance.

Traditional models can violate Galilean invariance but still be physically consistent.

Weak Galilean invariance constrains model formulation for accurate physical representation.

Abstract

Galilean invariance is a cornerstone of classical mechanics. It states that for closed systems the equations of motion of the microscopic degrees of freedom do not change under Galilean transformations to different inertial frames. However, the description of real world systems usually requires coarse-grained models integrating complex microscopic interactions indistinguishably as friction and stochastic forces, which intrinsically violate Galilean invariance. By studying the coarse-graining procedure in different frames, we show that alternative rules -- denoted as "weak Galilean invariance" -- need to be satisfied by these stochastic models. Our results highlight that diffusive models in general can not be chosen arbitrarily based on the agreement with data alone but have to satisfy weak Galilean invariance for physical consistency.