1.4 Towards Statistical Description
13
active particles is likely to be more structured than thermal noise. It makes it even
more challenging to derive macroscopic dynamics from underlying microscopic
interaction rules. This has prompted sophisticated statistical theories aspiring to
explain on a deeper level the dynamics of the utterly simple and not too realistic
model central to this chapter. The motivation was that the results might be applicable
to a wider class of phenomena in “dry” active matter.
Bertin et al (2006) set the aim of building a statistical description of a modified version of the Vicsek model, wherein particles may change their propagation
direction (but not the magnitude of their velocity) either by a random “kick” or as
a result of a binary collision that aligns the velocities of the two particles to their
average direction shifted by random noise. Recall that the original model allows
interactions between many particles within a set distance, but is less realistic in another respect, as each particle adjusts its direction independently, while in the model
by Bertin et al, binary collisions conserve the momentum before being biased by
thermal noise. Restricting to binary interactions is common in molecular dynamics,
and momentum conservation, with the random factors accounted for by effective
viscosity, leads to hydrodynamic equations far better justified than those of Toner
and Tu. All coefficients, including inertial effects, linear and nonlinear viscosities,
and even a cubic term setting the magnitude of the velocity, are computed from the
microscopic parameters of the model.
Yet, the theory is problematic in one important aspect: it is based on Boltzmann’s
hypothesis of “molecular chaos”, which assumes that colliding particles are uncorrelated. This is justified when the mean free path is large compared to the radius
of interactions, something which may be true in the “gas” phase but is not true at
realistic densities when order is established. Indeed, Boltzmann’s hypothesis fails
even in equilibrium liquids, and the theory of the liquid state has to involve sophisticated approximations accounting for molecular correlations. Ihle (2011) pointed
out this drawback and put forward an alternative theory based on the weak gradient
expansion, mirroring the theory of weakly inhomogeneous media (Chapman and
Cowling, 1970). His derivations led to more complicated nonlinear hydrodynamic
equations than those of Bertin et al, but with coefficients also expressed through the
microscopic parameters of the model.
Unlike Bertin et al, Ihle took as the basis of his theory the original Vicsek model
evolving, as in standard agent-based computations, by discrete steps – indeed, by
large steps, as he relied on the assumption that the free path between collisions
was much larger than the interaction range. Such a highly discrete character of
the motion is a questionable point in Ihle’s theory, as time steps of agent-based
computations are relatively small and, of course, real flocks interact continuously.
The continuum statistical theory of flocks (or, more generally, of “dry” active matter)
remains unsettled, as reflected, in particular, by the two contending papers in a
discussion issue on active matter (Peshkov et al, 2014; Ihle, 2014), which contain
both justification of these efforts and convincing mutual criticisms of the rival
theories.
13
active particles is likely to be more structured than thermal noise. It makes it even
more challenging to derive macroscopic dynamics from underlying microscopic
interaction rules. This has prompted sophisticated statistical theories aspiring to
explain on a deeper level the dynamics of the utterly simple and not too realistic
model central to this chapter. The motivation was that the results might be applicable
to a wider class of phenomena in “dry” active matter.
Bertin et al (2006) set the aim of building a statistical description of a modified version of the Vicsek model, wherein particles may change their propagation
direction (but not the magnitude of their velocity) either by a random “kick” or as
a result of a binary collision that aligns the velocities of the two particles to their
average direction shifted by random noise. Recall that the original model allows
interactions between many particles within a set distance, but is less realistic in another respect, as each particle adjusts its direction independently, while in the model
by Bertin et al, binary collisions conserve the momentum before being biased by
thermal noise. Restricting to binary interactions is common in molecular dynamics,
and momentum conservation, with the random factors accounted for by effective
viscosity, leads to hydrodynamic equations far better justified than those of Toner
and Tu. All coefficients, including inertial effects, linear and nonlinear viscosities,
and even a cubic term setting the magnitude of the velocity, are computed from the
microscopic parameters of the model.
Yet, the theory is problematic in one important aspect: it is based on Boltzmann’s
hypothesis of “molecular chaos”, which assumes that colliding particles are uncorrelated. This is justified when the mean free path is large compared to the radius
of interactions, something which may be true in the “gas” phase but is not true at
realistic densities when order is established. Indeed, Boltzmann’s hypothesis fails
even in equilibrium liquids, and the theory of the liquid state has to involve sophisticated approximations accounting for molecular correlations. Ihle (2011) pointed
out this drawback and put forward an alternative theory based on the weak gradient
expansion, mirroring the theory of weakly inhomogeneous media (Chapman and
Cowling, 1970). His derivations led to more complicated nonlinear hydrodynamic
equations than those of Bertin et al, but with coefficients also expressed through the
microscopic parameters of the model.
Unlike Bertin et al, Ihle took as the basis of his theory the original Vicsek model
evolving, as in standard agent-based computations, by discrete steps – indeed, by
large steps, as he relied on the assumption that the free path between collisions
was much larger than the interaction range. Such a highly discrete character of
the motion is a questionable point in Ihle’s theory, as time steps of agent-based
computations are relatively small and, of course, real flocks interact continuously.
The continuum statistical theory of flocks (or, more generally, of “dry” active matter)
remains unsettled, as reflected, in particular, by the two contending papers in a
discussion issue on active matter (Peshkov et al, 2014; Ihle, 2014), which contain
both justification of these efforts and convincing mutual criticisms of the rival
theories.