Hydrodynamic Equations for Active Matter: Consequences of Galilean Transformations

Abstract

The continuum mechanics description of dry active matter consisting of self-propelled elements has been obtained through plausible extensions of the corresponding hydrodynamic equations for a passive fluid. In the generalized-hydrodynamic description of the active fluid, the advective term of the equation of motion for the momentum density gα(x,t) is modified with factor λ(ρ0) where ρ0 is the average density of the fluid. λ ≠ 1 implies that the hydrodynamic equations are not invariant under Galilean transformations. In the present work, we propose a microscopic level interpretation for the factor λ(ρ0) in terms of a set of activity-indices {fi}, introduced to define the comoving frame in which the fluid is locally at rest. If fi = 1, for all i, then λ(ρ0) = 1, and λ is different from unity if the fi’s are treated as stochastic variables. The present analysis is valid even if we consider only the reversible equations of Eulerian-hydrodynamics. We also obtain the factor κ(ρ0), similar to λ(ρ0), that appear in the corresponding dissipative parts of the equation for the momentum density gα(x,t) .