1.4 Towards Statistical Description
11
mechanics, since the particles move with a constant velocity defined in a specific
(“laboratory”) frame of reference. An easy way to forget these objections is to tell
oneself that the Vicsek model is after all a model rather than a law of nature, and the
continuous equations of Toner and Tu can be viewed as just another model, related
in some respects but different in others.
Once a system of differential equations, justified or not, is in place, it can be
studied in a standard way. One can test the linear stability of its “trivial” homogeneous solutions propagating with a certain speed, and find their symmetry-breaking
bifurcations leading to a family of propagating bands. Solon, Caussin, et al (2015)
worked in this way with a modified system, simplified in some respects compared
with that of Toner and Tu but including a nonlinear term that tends to keeps the absolute value of the velocity constant, as assumed in the Vicsek model. They obtained
what Chaté (2020) calls an embarrassingly large family of linearly stable solutions:
periodic patterns, solitary bands, phase-separated domains. All this variety disappears when a noise term, missing in the simplified system, is reintroduced, causing a
unique solution to be selected. This is consistent with the role of noise in equilibrium
systems where noise facilitates transition from metastable states to a state with the
minimal energy, but energy is neither well defined nor conserved in active matter.
Chaté (2020) lists this among current riddles, asking: “How do we understand the
selection of a unique solution observed at microscopic and fluctuating hydrodynamic
levels but which is not present at the deterministic hydrodynamic level?”
Clearly, noise is an essential component, and formulating phenomenological hydrodynamic equations (even if fully justified) is only the beginning of the road.
Theories of phase transitions (Landau et al, 1980) have to account not just for
fluctuations but for their correlations in time and space. This cannot be done in a
straightforward way, since pair correlation functions depend on triple correlation
functions, and so on, necessitating a cut-off under some assumptions. A powerful
method in the theory of critical phenomena is the renormalization group, based on
invariance to scale transformations (Goldenfeld, 1992). The theory is precise in 4D,
and applications to physical dimensions are commonly based on the 4 − expansion,
where is assumed to be small, even though 4 − 3 = 1 is not what would normally
be treated as a small parameter. Toner and Tu (1998) boldly applied the renormalization group further down, in 2D, even though admitting that some parameters of
their theory are not scale-invariant, and in spite of all above mentioned shortcomings, claimed that their predictions (or retrodictions?) are in agreement with the
computations by Vicsek et al (1995).
1.4 Towards Statistical Description
The central theoretical question is understanding both similarities to and distinctions
from the behavior of passive matter obeying thermodynamic laws. The phenomenology of the Viscek model, as reflected in Fig. 1.3, looks at first sight to be not so
very different from that of ordinary fluids: the liquid and gas phase differ by density,
11
mechanics, since the particles move with a constant velocity defined in a specific
(“laboratory”) frame of reference. An easy way to forget these objections is to tell
oneself that the Vicsek model is after all a model rather than a law of nature, and the
continuous equations of Toner and Tu can be viewed as just another model, related
in some respects but different in others.
Once a system of differential equations, justified or not, is in place, it can be
studied in a standard way. One can test the linear stability of its “trivial” homogeneous solutions propagating with a certain speed, and find their symmetry-breaking
bifurcations leading to a family of propagating bands. Solon, Caussin, et al (2015)
worked in this way with a modified system, simplified in some respects compared
with that of Toner and Tu but including a nonlinear term that tends to keeps the absolute value of the velocity constant, as assumed in the Vicsek model. They obtained
what Chaté (2020) calls an embarrassingly large family of linearly stable solutions:
periodic patterns, solitary bands, phase-separated domains. All this variety disappears when a noise term, missing in the simplified system, is reintroduced, causing a
unique solution to be selected. This is consistent with the role of noise in equilibrium
systems where noise facilitates transition from metastable states to a state with the
minimal energy, but energy is neither well defined nor conserved in active matter.
Chaté (2020) lists this among current riddles, asking: “How do we understand the
selection of a unique solution observed at microscopic and fluctuating hydrodynamic
levels but which is not present at the deterministic hydrodynamic level?”
Clearly, noise is an essential component, and formulating phenomenological hydrodynamic equations (even if fully justified) is only the beginning of the road.
Theories of phase transitions (Landau et al, 1980) have to account not just for
fluctuations but for their correlations in time and space. This cannot be done in a
straightforward way, since pair correlation functions depend on triple correlation
functions, and so on, necessitating a cut-off under some assumptions. A powerful
method in the theory of critical phenomena is the renormalization group, based on
invariance to scale transformations (Goldenfeld, 1992). The theory is precise in 4D,
and applications to physical dimensions are commonly based on the 4 − expansion,
where is assumed to be small, even though 4 − 3 = 1 is not what would normally
be treated as a small parameter. Toner and Tu (1998) boldly applied the renormalization group further down, in 2D, even though admitting that some parameters of
their theory are not scale-invariant, and in spite of all above mentioned shortcomings, claimed that their predictions (or retrodictions?) are in agreement with the
computations by Vicsek et al (1995).
1.4 Towards Statistical Description
The central theoretical question is understanding both similarities to and distinctions
from the behavior of passive matter obeying thermodynamic laws. The phenomenology of the Viscek model, as reflected in Fig. 1.3, looks at first sight to be not so
very different from that of ordinary fluids: the liquid and gas phase differ by density,