Advanced Topics in Equilibrium Statistical
Mechanics
Glenn Fredrickson
6. Statistical Mechanics of Classical Fields
Up to this point we have talked about how to set up and perform equilibrium
statistical mechanical calculations of systems with a finite
set of phase space
coordinates, e.g.,
{
r
N
,
p
N
}
.
However, often we are faced with objects, e.g.,
polymers, that are better described by continuous fields
and have an infinite
number of degrees of freedom. We must learn how to do statistical mechanics
for such objects.
A. Transverse Oscillations of a Stretched String
To illustrate the approach, consider a string of a musical instrument that is
stretched between two supports:
On a nanoscale
, the same situation can be created by pulling on a DNA
molecule with optical tweezers. One obvious question is the following:
What is the equilibrium distribution of string shapes if the string is
equilibrated with its environment at temperature
T
?
There are two ways to tackle this problem. The first
would be to approximate
a continuous elastic string by a beadspring discrete chain:
Here the bead positions
r
N
are used to approximate the string shape and
1
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each bead has a mass
m
=
σ
·
a
where
a
is the average bead spacing
and
σ
is the mass per length of string.
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 Spring '09
 Calculus, Fourier Series, Energy, Trigraph, Equilibrium Statistical Mechanics, Functional

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