336 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
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9.1.1′
- The idea of reducing the multitude of molecular details of a material down to just a few parameters
may seem too ad hoc. How do we know that the viscous force rule (Equation 5.9 on page 149),
which we essentially pulled from a hat, is complete? Why can’t we add more terms, likef/A=
−ηddxv+η 2 dd^2 xv 2 +η (^3) ddx^3 v 3 +···?Itturns out that the number of relevant parameters is kept small by
dimensional analysisand bysymmetriesinherited from the microscopic world.
Consider for example the constantη 3 just mentioned. It is supposed to be an intrinsic property
of the fluid, completely independent of the sizeaof its pipe. Clearly it has dimensionsL^2 times those
of the ordinary viscosity. The only intrinsic length scale of a simple (Newtonian) fluid, however,
is the average distancedbetween molecules. (Recall that the macroscopic parameters of a simple
Newtonian fluid don’t determine any length scale; see Section 5.2.1 on page 146.) Thus we can
expect thatη 3 ,ifpresent at all, must turn out to be roughlyd^2 as large asη.Since the gradient
of the velocity is roughlya−^1 as large asvitself (see Section 5.2.2), we see that theη 3 term is less
important than the usualηterm by roughly a factor of (d/a)^2 ,atinynumber.
Turning to theη 2 term above, it turns out that an even stronger result forbids it altogether:
This term cannot be written in a way that is invariant under rotations. Thus it cannot arise in the
description of an isotropic Newtonian fluid (Section 5.2.1 on page 146), which by assumption is the
same in every direction. In other words, symmetries of the molecular world restrict the number and
typesofeffective parameters of a fluid. For more discussion of these points, see Landau & Lifshitz,
1987; Landau & Lifshitz, 1986.
The conclusions just given are not universal—hence the qualification that they apply to isotropic
Newtonian fluids. They get replaced by more complicated rules in the case of non-Newtonian or
complex fluids(for example the viscoelastic ones mentioned in Section 5.2.3′on page 167, or the
liquid crystals in a wristwatch display).
- Some authors refer to the systematic exploitation of Idea 9.1 as “generalized elasticity and
hydrodynamics.” Certainly there is some art involved in implementing the idea in a given situation,
for example in determining the appropriate list of effective degrees of freedom. Roughly speaking, a
collective variable gets onto the list if it describes a disturbance to the system that costs very little
energy, or that relaxes very slowly, in the limit of long length scales. Such disturbances in turn
correspond to broken symmetries of the lowest-energy state, or to conservation rules. For example,- The centerline of a rod defines a line in space. Placing this line somewhere breaks two
translation symmetries, singling out two corresponding collective modes, namely the
twodirections in which we can bend the rod in its normal plane. Section 9.1.2 shows
that indeed bending costs very little elastic energy on long length scales. - In diffusion, we assumed that the diffusing particles could not be created or destroyed—
they’re conserved. The corresponding collective variable of the system is the particle
density, which indeed changes very slowly on long length scales according to the
diffusion equation (Equation 4.19).
- The centerline of a rod defines a line in space. Placing this line somewhere breaks two
Forexamples of this approach in the context of soft condensed matter physics, see Chaikin &
Lubensky, 1995.