Hydrodynamics without Boost-Invariance from Kinetic Theory: From Perfect Fluids to Active Flocks

TL;DR

This work derives hydrodynamics without boost invariance directly from kinetic theory by formulating a Vlasov hierarchy with an axiomatic collision functional. Central to the framework are the kinetic mass density $\rho$, its equation of motion $\partial_t \rho + \nabla \cdot (\rho \mathbf{v}) = g \rho$, and an influence kernel that encodes inter-particle forces, enabling a derivative expansion that yields diffusion terms without explicit noise. The analysis recovers a generalized Euler equation for a boost-noninvariant fluid and, for flocking systems, reproduces Toner-Tu-type hydrodynamics with hierarchy relations among coefficients governed by the equation of kinetic state and kernel properties. The approach clarifies why certain Toner-Tu coefficients are of order one while others are small and provides a kinetic-theory foundation for active matter hydrodynamics, with potential extensions to retarded kernels and dual descriptions.

Abstract

We derive the hydrodynamic equations of perfect fluids without boost invariance [1] from kinetic theory. Our approach is to follow the standard derivation of the Vlasov hierarchy based on an a-priori unknown collision functional satisfying certain axiomatic properties consistent with the absence of boost invariance. The kinetic theory treatment allows us to identify various transport coefficients in the hydrodynamic regime. We identify a drift term that effects a relaxation to an equilibrium where detailed balance with the environment with respect to momentum transfer is obtained. We then show how the derivative expansion of the hydrodynamics of flocks can be recovered from boost non-invariant kinetic theory and hydrodynamics. We identify how various coefficients of the former relate to a parameterization of the so-called equation of kinetic state that yields relations between different coefficients, arriving at a symmetry-based understanding as to why certain coefficients in hydrodynamic descriptions of active flocks are naturally of order one, and others, naturally small. When inter-particle forces are expressed in terms of a kinetic theory influence kernel, a coarse-graining scale and resulting derivative expansion emerge in the hydrodynamic limit, allowing us to derive diffusion terms as infrared-relevant operators distilling different parameterizations of microscopic interactions. We conclude by highlighting possible applications.