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of constitutive equation for the suspension's stress tensor on the basis of symmetry
consideration. The coupling of flow and polar order is described via an equation for
the local polarization of the suspension. This model has been used to identify the
nonequilibrium defect structures that can occur in the polar state [45, 46] and to
analyze the behavior of an active polar suspension in specific geometries [52, 47].
In particular, it was shown that the interplay between order and activity can yield
a spontaneous flowing state for a solution near a wall [47]. Closely related hydro-
dynamic models have been used to describe generically the collective dynamics of
self-propelled particles in solution, such as swimming bacterial colonies, in both ne-
matic and polar states [49, 50, 51]. This work builds on earlier work by Toner and
Tu on hydrodynamic models of flocking, where it was shown that the nonequilib-
rium nature of internally driven systems allows for novel symmetry breaking phase
transitions that are forbidden in equilibrium systems with continuous symmetry in
one and two dimensions [53, 54, 55].
The main objective of our work has been to establish the connection between
microscopic single-polymer dynamics and the phenomenological hydrodynamic mod-
els by deriving the hydrodynamic equations from a mean-field kinetic equation of
filament dynamics. In the phenomenological approach, the system is described in
terms of a few coarse-grained fields (conserved densities and broken symmetry vari-
ables) whose dynamics is inferred from symmetry considerations. The strength of
this method is its generality. Its drawback is that for systems that are far from
thermal equilibrium, and therefore lack constraints such as those provided by the
fluctuation-dissipation theorem or the Onsager relations, all the parameters in the
equations are undetermined. We have bridged the gap between microscopic models
and continuum theories by deriving the hydrodynamic equations through a sys-
tematic coarse-graining of the microscopic dynamics. This derivation provides an
estimate of the various parameters in the equations in terms of experimentally ac-
cessible quantities. We start with a Smoluchowski equation for filaments in solution,
where motor proteins are described as active cross-linkers capable of exchanging
forces and torques between filaments. The active currents arising from such motor-
mediated exchange of forces and torques are obtained by considering the kinematics
of two filaments cross-linked by a single active protein cluster that can rotate and
translate at prescribed rates as a rigid object relative to the filaments. The hydrody-
namic equations are then obtained by suitable coarse-graining of the Smoluchowski
equation. This method yields a general form of the hydrodynamic equations that
incorporates all terms allowed by symmetry, yet it provides a connection between
the coarse-grained and the microscopic dynamics. In a series of earlier publications,
we described in detail the derivation of the hydrodynamic equations for filaments in
a quiescent solvent [33, 34, 56, 35, 36]. Here we generalize this work by incorporating
the flow of the solvent. This is essential for describing the rheological properties of
the solution. A brief account of some of the results presented here have been given
elsewhere [57].
7.3 Hydrodynamics of a Solution of Polar Filaments
We consider a collection of rigid polar filaments in a viscous solvent. The solution
forms a quasi two-dimensional film, of a thickness much smaller than the length of
the filaments. Our goal is to study the interplay of order and flow in controlling the
