Chain statistics as a path integral

Consider a single polymer chain, defined by a space-curve representation. The probability distribution (or Green function) \(G(b|a)\) of the chain extending from \(a\) to \(b\) is

\[G(b|a) = \sum_{all \; paths} \phi [x(s)]\]

where the state \(a\) denotes location \(x_{a}\) at arclength \(s_{a}\) and state \(b\) denotes location \(x_{b}\) at arclength \(s_{b}\). The summation over emph{all paths} indicates a sum of all possible chain conformations (or paths) that begin at state \(a\) and end at state \(b\). This concept of a path integral is extensively developed by Feynman and Hibbs~cite{feynman} to describe the statistical treatment of quantum mechanical particles. For polymer statistics, the weighting of a path (or chain conformation) is governed by statistical mechanics. The statistical weight \(\phi[x(s)]\) of a path \(x(s)\) is given by the integration of the Boltzmann weighting of the path. Define the Action \(S\) for the particular path to be

\[S[x(s)] = \int_{s_{a}}^{s_{b}} \mathcal{H}(x, x', x'', ...;s) ds\]

Thus, the statistical weight of a path is given by

\[\phi[x(s)] = \frac{1}{A} \exp (-\beta S[x(s)]),\]

where \(A\) is a normalization constant to be discussed later.

The Hamiltonian \(\mathcal{H}\) includes differential elastic deformation energy \(e_{elas}(x',x'',...;s)\) and a potential energy per unit length \(V(x,s)\). The Green function \(G(b|a)\) is written in %the form path-integral form

\[G(b|a) = \int_{a}^{b} \exp (-\beta S[x(s)]) \mathcal{D} x(s)\]

Currently the derivation of the path model is in one dimension; however, the ideas are easily extended to higher dimensions.

Schematic of path summation in the space-arclength plane

Gaussian Chain model Green function—explicit path integration

Consider a polymer chain defined by the Gaussian Chain model in 1-D with no external field (\(V=0\)). The action is given by

\[S[x(n)]= \frac{k_{B}T}{2 b^{2}} \int_{n_{a}}^{n_{b}} \! dn \, \left( \frac{\partial x(n)}{\partial n} \right)^{\! 2}\]

where \(n=s/b\) is a dimensionless arclength parameter. This gives a statistical weight

\[\phi[x(n)] = \frac{1}{A} \exp \! \left[ -\frac{1}{2 b^{2}} \int_{n_{a}}^{n_{b}} \! dn \, \left( \frac{\partial x(n)}{\partial n} \right)^{\! 2} \right]\]

For this treatment, convert the path integral into a discrete-chain representation. This procedure provides a practical method to perform the summation over all possible chain conformations. Break the chain into \(M\) segments of length \(\delta = (n_{b}-n_{a})/M\). The position of the \(m\), and the endpoints are \(x_{0}=x_{a}\) and \(x_{M}=x_{b}\). The limit \(M \rightarrow \infty\) recovers our continuous chain representation

Discrete-chain representation converts the chain into \(M\) segments (in this case \(M=6\)) of length \(\delta=(n_{b}-n_{a})/M\).

The statistical weight is now approximated as

\[\phi[x(n)] \approx \frac{1}{A} \exp \! \left[ -\frac{1}{2 b^{2} \delta} %\int_{n_{a}}^{n_{b}} \! dn \, \left( \frac{\partial x(n)}{\partial n} \right)^{\! 2} \sum_{m=1}^{M} \left( x_{m}-x_{m-1} \right)^{2} \right]\]

and the Green function is approximated as

\[G(b|a) = \frac{1}{A} \int_{-\infty}^{\infty} \! \! \! \! \! d x_{1} \int_{-\infty}^{\infty} \! \! \! \! \! d x_{2} \ldots \int_{-\infty}^{\infty} \! \! \! \! \! d x_{M-1} %\left( \prod_{m'=1}^{M-1} d x_{m'} \right) \exp \! \left[ -\frac{1}{2 b^{2} \delta} %\int_{n_{a}}^{n_{b}} \! dn \, \left( \frac{\partial x(n)}{\partial n} \right)^{\! 2} \sum_{m=1}^{M} \left( x_{m}-x_{m-1} \right)^{2} \right].\]

The \(x_{1}\) integral is given by

\[\begin{split}\normalsize & & \int_{-\infty}^{\infty} \! \! \! \! \! d x_{1} \exp \! \left\{ -\frac{1}{2 b^{2} \delta} \left[ \left( x_{2}-x_{1} \right)^{2}+ \left( x_{1}-x_{0} \right)^{2} \right] \right\}= \nonumber \\ \normalsize & & \int_{-\infty}^{\infty} \! \! \! \! \! d x_{1} \exp \! \left\{ -\frac{1}{2 b^{2} \delta} \left[ 2 \left( x_{1}-x_{1}^{\star} \right)^{2}+ \frac{1}{2} \left( x_{2}-x_{0} \right)^{2} \right] \right\}= \nonumber \\ & & \sqrt{\pi b^{2} \delta} \exp \! \left[ - \frac{1}{2 b^{2} \delta} \frac{1}{2}(x_{2}-x_{0})^{2} \right],\end{split}\]

where \(x_{1}^{\star}=\frac{x_{0}+x_{2}}{2}\).

The subsequent \(x_{2}\) integral is given by

\[\begin{split}\normalsize & & \int_{-\infty}^{\infty} \! \! \! \! \! d x_{2} \exp \! \left\{ -\frac{1}{2 b^{2} \delta} \left[ \left( x_{3}-x_{2} \right)^{2}+ \frac{1}{2} \left( x_{2}-x_{0} \right)^{2} \right] \right\}= \nonumber \\ \normalsize & & \int_{-\infty}^{\infty} \! \! \! \! \! d x_{2} \exp \! \left\{ -\frac{1}{2 b^{2} \delta} \left[ \frac{3}{2} \left( x_{2}-x_{2}^{\star} \right)^{2}+ \frac{1}{3} \left( x_{3}-x_{0} \right)^{2} \right] \right\}= \nonumber \\ & & \sqrt{\frac{4\pi b^{2} \delta}{3}} \exp \! \left[ - \frac{1}{2 b^{2} \delta} \frac{1}{3}(x_{3}-x_{0})^{2} \right]\end{split}\]

where \(x_{2}^{\star}=\frac{x_{0}+2x_{3}}{3}\). The \(x_{m}\) integral after performing integrals \(x_{1}\), \(x_{2}\),ldots, \(x_{m-1}\) is

\[\begin{split}& & \int_{-\infty}^{\infty} \! \! \! \! \! d x_{m} \exp \! \left\{ -\frac{1}{2 b^{2} \delta} \left[ \left( x_{m+1}-x_{m} \right)^{2}+ \frac{1}{m} \left( x_{m}-x_{0} \right)^{2} \right] \right\}= \nonumber \\ & & \sqrt{\frac{2\pi b^{2} \delta}{1+\frac{1}{m}}} \exp \! \left[ - \frac{1}{2 b^{2} \delta} \frac{1}{m+1}(x_{m+1}-x_{0})^{2} \right]\end{split}\]

This gives the Green function

\[\begin{split}G(b|a) & = & \frac{1}{A} \prod_{m=1}^{M-1} \sqrt{\frac{2 \pi b^{2} \delta}{1+\frac{1}{m}}} \exp \! \left[ - \frac{1}{2 b^{2} \delta} \frac{1}{M} ( x_{M}-x_{0} )^{2} \right] \nonumber \\ & = & \frac{1}{A} \prod_{m=1}^{M-1} \sqrt{\frac{2 \pi b^{2} \delta}{1+\frac{1}{m}}} \exp \! \left[ - \frac{( x_{b}-x_{a} )^{2}}{2 (n_{b}-n_{a}) b^{2}} \right] \nonumber \\ & = & \frac{1}{\sqrt{2 \pi (n_{b}-n_{a}) b^{2} }} \exp \! \left[ - \frac{( x_{b}-x_{a} )^{2}}{2 (n_{b}-n_{a}) b^{2}} \right]\end{split}\]

where \(A\) is set to ensure \(\int dx_{b} G(b|a)=1\)

Gaussian Chain model Green function—Schrödinger equation

Generally, it is more convenient to develop the path integral into a diffusion equation (or Schr"{o}dinger equation) to solve for the chain statistics. Consider the Gaussian Chain model in 1-D with an external potential, giving an action

\[S[x(n)]= \int_{0}^{N} \! dn \, \left[ \frac{k_{B}T}{2 b^{2}} \left( \frac{\partial x(n)}{\partial n} \right)^{\! 2} + V[x(n)] \right]\]

where we set \(n_{a}=0\) and \(n_{b}=N\) (without loss of generality), and assume \(V=V(x)\) only. The chain obeys Markovian statistics; thus, the Green function for a chain that starts at \(A\) (emph{i.e.} position \(x_{A}\) and arclength \(n_{A}\)) to state \(C\) (emph{i.e.} \(x_{C}, n_{C}\)) through intermediate state \(B\) (emph{i.e.} \(x_{B}, n_{B}\)) is

\[G(x_{C},n_{C} |x_{A},n_{A}) = \int_{-\infty}^{\infty} \! \! \! dx_{B} G(x_{C}, n_{C} | x_{B}, n_{B}) G(x_{B},n_{B} |x_{A},n_{A}).\]

Schematic of a chain that obeys Markovian statistics

Take the state \(C\) to differ an infinitesimal length from state \(B\) by setting \(n_{C}=N+\delta\) and \(n_{B}=N\) (\(n_{A}=0\)). For simplicity, set \(x_{A}=x_{0}\), \(x_{B}=y\), \(x_{C}=x\), giving

\[G(x,N+\delta |x_{0},0) = \int_{-\infty}^{\infty} \! \! \! dy G(x, N+\delta | y,N) G(y,N |x_{0},0).\]

In the limit \(\delta \rightarrow 0\), the Green function \(G(x, N+\delta | y,N)\) will be dominated by a single path whose action is

\[S \approx \frac{k_{B} T}{2 b^{2}} \delta \left( \frac{x-y}{\delta} \right)^{2} + \delta \, V \! \! \left( \frac{x+y}{2} \right).\]

Assume the contribution to the integral is mainly near \(y = x + \eta\) where \(\eta \ll 1\). The Green function is thus given by

\[G(x, N + \delta|x_{0},0) = \int_{-\infty}^{\infty} \! \! d \eta \frac{1}{A} \exp \left( - \frac{1}{2 b^{2}} \frac{ \eta^{2} }{ \delta } \right) \exp [- \beta \delta V( x + \eta/2)] G(x + \eta, N | x_{0},0).\]

The integrand decays to zero for \(\eta > \sqrt{2 b^{2} \delta }\) thus the integrand must be expanded to linear order in \(\delta\) and quadratic order in \(\eta\) in order to capture the lowest order behavior of the Green function. The expansion of the Green function yields the solvable integral

\[\begin{split}& & G(x, N|x_{0},0) + \delta \frac{\partial G(x,N|x_{0},0)}{\partial N} = \int_{-\infty}^{\infty} \! \! d \eta \frac{1}{A} \exp \left( - \frac{1}{2 b^{2}} \frac{ \eta^{2} }{ \delta } \right) \nonumber \\ & & \hspace{0.4in} \times [1 - \beta \delta V(x)] \left( G (x,N|x_{0},0) + \eta \frac{\partial G(x,N|x_{0},0)}{\partial x} + \frac{\eta^{2}}{2} \frac{\partial^{2} G(x,N|x_{0},0}{\partial x^{2}} \right).\end{split}\]

The solution for the normalization constant \(A\) is determined by solution of the leading term on the right hand side:

\[\frac{1}{A} \int_{-\infty}^{\infty} \exp \left(- \frac{1}{2 b^{2}} \frac{ \eta^{2} }{ \delta } \right) d \eta = 1\]

thus \(A = b \sqrt{ 2 \pi \delta}\). The two additional integrals are given by

\[\begin{split}\frac{1}{A} \int_{-\infty}^{\infty} \eta \exp \left(- \frac{1}{2 b^{2}} \frac{ \eta^{2} }{ \delta } \right) d \eta & = & 0 \\ \frac{1}{A} \int_{-\infty}^{\infty} \eta^{2} \exp \left(- \frac{1}{2 b^{2}} \frac{ \eta^{2} }{ \delta } \right) d \eta & = & b^{2} \delta.\end{split}\]

The Green function is given by

\[\begin{split}& & G(x,N|x_{0},0) + \delta \frac{\partial G(x,N|x_{0},0)}{\partial N} = \nonumber \\ & & G(x,N|x_{0},0) - \delta \beta V G(x,N|x_{0},0) + \delta \frac{b^{2}}{2} \frac{ \partial^{2} G(x,N|x_{0},0)}{\partial x^{2}}.\end{split}\]

Thus, the Green function for the 1-D Gaussian Chain obeys

\[\frac{\partial G(x,N|x_{0},0)}{\partial N} = -\beta V G(x,N|x_{0},0) + \frac{b^{2}}{2} \frac{ \partial^{2} G(x,N|x_{0},0)}{\partial x^{2}},\]

with initial condition \(G(x,0|x_{0},0)=\delta(x-x_{0})\). For \(V(x)=0\), the solution for the Green function is

\[G(x,N|x_{0},0) = \frac{1}{\sqrt{2 \pi N b^{2} }} \exp \! \left[ - \frac{( x-x_{0} )^{2}}{2 N b^{2}} \right]\]

which agrees with our previous derivation (Sec.~ref{chap:gaussian-explicit}).

The governing differential equation for the Gaussian Chain in 3D is written as an extension to our 1D equation as

\[\frac{\partial G(\vec{R},N|\vec{R}_{0},0)}{\partial N} = -\beta V G(\vec{R},N|\vec{R}_{0},0) + \frac{b^{2}}{6} \vec{\nabla}^{2} G(\vec{R},N|\vec{R}_{0},0)\]

with initial condition \(G(\vec{R},0|\vec{R}_{0},0)=\delta(\vec{R}-\vec{R}_{0})\). The solution of this differential equation subject to a given potential \(V\) gives the equilibrium probability distribution for the Gaussian Chain model. For \(V(\vec{R})=0\), the solution for the Green function for a Gaussian Chain in 3-D is

\[G(\vec{R},N|\vec{R}_{0},0) = \left( \frac{2\pi N b^{2}}{3} \right)^{\! \! \! -3/2} \exp \! \left[ - \frac{3 (\vec{R} - \vec{R}_{0} )^{2}}{2Nb^{2}} \right]\]