Consider the probability distribution function 
of a chain consisting of N freely jointed units of length b, with the
end-to-end vector 
. By introducing the auxiliary field
, it is rather trivial [21] to show:
Here 
is the characteristic function
for the freely jointed chain. Its successive derivatives with respect to
give the chain moments, from which the
can be inferred very accurately [5, 6]. However, we shall present
a perturbative method here for simplicity.
Approximating the characteristic function to 
gives rise to the usual Gaussian distribution function, but more accurately
we can write it as a power series expansion in 
:
which gives the probability distribution function in the form of:
This integration can be performed with the saddle point approximation to give the leading correction to the Gaussian probability. The saddle point is found to be at:
To evaluate the full correction, we write the standard Taylor
expansion for 
about the saddle point 
:
where 
measures
the distance away from the saddle point, and 
denotes that all derivatives are to be evaluated at the same saddle
point . The leading Gaussian behaviour [1]
dictates the likely scaling relation: 
; and this guides our expansion in orders of 1/N
throughout this paper.
Some details of this expansion are
provided in the appendix.
The Eq.(3) can be rewritten as:
with summations over repeated indices implied here.
The first exponential in the integrand describes a Gaussian average which,
on its own, gives the prefactor in the
saddle point approximation; the second exponential
can be expanded into a power series in k, and their average with respect
to the Gaussian distribution is straightforward thanks to Wick's
theorem
.
The resulting probability distribution can be written in the following
exponential form:
Note that this probability distribution only depends on the magnitude of
, as the system is isotropic. The exponent is
a power series in 
, and for large N, should
be well approximated by the first few terms. In principle, this probability
distribution function can be written to a higer accuracy by the inclusion
of higher order terms; but in practice the expansion becomes difficult
for short chains.
The probability distribution function given by
the inverse Langevin approximation [3] is
equivalent to taking the saddle point value 
:
the exponent of which agrees with that of Eq.(8) only partially.
In comparison, the multiplicative factors in the exponent, such as
, are new and necessary for a self-consistent
approximation in orders of 1/N.
Therefore, expanding the inverse Langevin function in a power series, e.g.
in Refs.[20, 22], will give incomplete correction to
Gaussian result at order 1/N. The error in the inverse
Langevin approximation comes mainly from the use of Stirling's
approximation, which can be viewed as a saddle point approximation of
a Gamma function integral.
In Eq.(8,9) we have non-Gaussian probability distributions,
which contains the leading order (in 1/N) corrections due to
the finite extensibility.
However, the exact expectation values are available
for a 3-dimensional freely jointed chain: 
and
, which account for the finite
extensibility of the chains. These expectation values can be readily
verified with the probability distribution of Eq.(8),
but not with that of Eq.(9).
