26 2 Models of Dynamics
transition (or a binary fluid near its consolute point) or the superfluid fluctuations
(amplitude and phase) near the lambda point or the fluctuations in the staggered
magnetization near the antiferromagnetic transition.
Our interest, as we emphasized in the last chapter is in systems which are
described in terms of macroscopic fields. In this chapter, we will describe the
kind of dynamics that one expects for these systems. We begin with a system
near the critical point that we have discussed in the last chapter. If we are to
describe dynamics of this system, then the equation of motion is not so obvious.
Forone thing, those degreesof freedom which have been averagedover, are going to
influence the dynamics and this cannot be in a deterministic fashion. Consequently,
the dynamics will be described in terms of stochastic rather than deterministic
differential equations. The dynamics of the fluctuations is generally expected to be
relaxational as we expect a small fluctuation from the equilibrium state to disappear
in time. This relaxation generally is the linear part of the equation of motion -
the expectation is that the equation of motion will be first order in time since
specification of only the initial value of the order parameter should be enough to
specify its future dynamics. Thus the expected dynamics is of Langevin variety.
A different example is provided by crystal growth by ballistic deposition of
atoms. The depositing atoms diffuse across the surface to settle down at places
where the local energy is a minimum. This smooths out the growing surface. On
the other hand, there is intrinsic noise in the deposition process and this causes
the surface to be rough. The natural variable to describe the growth of this surface
is the local height variable h(r,t). Here r is the coordinate in the substrate. The
dynamics of the field h(r,t) has two clear cut parts:
• i) a surface diffusion which helps smooth out any fluctuation in h and
• ii) a noise part corresponding to the fluctuation in the deposition process
Thus, once more the dynamics is of the Langevin variety.
Yet another example of a mesoscopic system is the dynamics of polymer chains.
Consider a polymer chain put in a solvent. If the polymer is hydrophobic, then to
prevent contact with the water molecules the polymer tends to fold into a ball. The
entropyterm in the thermodynamic free energy would likethe polymer to spread out
and hence there is a competition between the energy and the entropy effects. As the
temperature is lowered, the chain is expected to undergo a transition into a compact
structure. The compact structure is typically of the order of microns (examples
of these compact structures are the enzyme like proteins) and constitute another
example of a mesoscopic system. The dynamics of a polymer chain is governed
again by a Langevin equation. For a randomly hydrophobic and hydrophillic chain,
thus the dynamics can be of some relevance to the interesting problem of protein
folding.
Finally, we mention the problem of turbulence. Here the randomness is gener-
ated by the non-linear term in the Navier-Stokes’ equation. However, to maintain
the turbulence we need to have a maintained mean flow and energy transfer has
to occur from the mean flow to the fluctuations. There would also be the question