Origins and diagnostics of the nonequilibrium character of active systems

Abstract

We present in detail a Langevin formalism for constructing stochastic dynamical equations for active-matter systems coupled to a thermal bath. We apply the formalism to clarify issues of principle regarding the sources and signatures of nonequilibrium behaviour in a variety of polar and apolar single-particle systems and polar flocks. We show that distance from thermal equilibrium depends on how time-reversal is implemented and hence on the reference equilibrium state. We predict characteristic forms for the frequencyresolved entropy production for an active polar particle in a harmonic potential, which should be testable in experiments.

Nonmutual torques and the unimportance of motility for long-range order in two-dimensional flocks

“…Equations ( 10) and ( 11) are exactly the equations that one would have obtained for a uniaxial polar liquid crystal on a substrate with a vector orientational order parameter v. That the system is intrinsically nonequilibrium enters only at one point: the forcing term ζ v in (12). If one integrated out s, by taking its dynamics to be fast, the two terms with the coefficients u and u would yield the "weathercock" term [27] familiar from theories of polar liquid crystals on substrates [26,35,36]. Further, ω and D together would yield the flow alignment term familiar from theories of nematic liquid crystal [21,37,38] in this limit.…”

“…In the Toner-Tu [2] equation the velocity or the polarization field is not advected at the same rate as the density. While it is tempting to attribute this asymmetry simply to the absence of Galilean invariance in the theory, both Galilean invariance and detailed balance have to be absent for this effect to emerge, as was demonstrated in [26]. For a stable flock, i.e, A < A c , the dynamics implied by our theory on long timescales is equivalent to the Toner-Tu equations, with the coefficient of the advective term shifted by a contribution proportional to A.…”

“…However, the continuum equations that we arrived at also describe the collective dynamics of aligning motile polar rods on a substrate, such as those in [27] or multiple other systems described using the Vicsek model in which the microscopic polar, aligning interaction is purely due to contact mechanics and, therefore, symmetric [33]. In other words, the breaking of time-reversal invariance through motility [26] generates an asymmetric effective interaction even when the microscopic interaction of the constituents is purely symmetric as it arises from a pair potential. To see this, we start with a theory of polar liquid crystal hydrodynamics [34] modified to include the dynamics of spin angular momentum [21] and a momentum sink.…”

“…Interestingly, the antisymmetry parameter A provides a natural mechanism for a coefficient different from unity for the advection term in ( 14), a feature of the Toner-Tu formulation [2], often simply ascribed to the lack of Galilean invariance in the model. However, losing Galilean invariance alone cannot generate such a term: the absence of detailed balance, as signalled by A, is crucial [26]. In an equilibrium fluid moving in contact with a momentum sink the coefficients in the momentum and density equations must agree.…”

Chiral active matter: microscopic ‘torque dipoles’ have more than one hydrodynamic description

“…and the last term in (C.2) is the so-called 'spurious' drift [28,36,37,45]. It is, of course, not spurious, except in the sense that it is absent from the the noiseless equations despite being a deterministic contribution (albeit proportional to the noise variance via k T B and varying in form depending on the interpretation rule for the Langevin equation [36]).…”