.. _gausstheory: Chain statistics as a path integral =================================== Consider a single polymer chain, defined by a space-curve representation. The probability distribution (or Green function) :math:`G(b|a)` of the chain extending from :math:`a` to :math:`b` is .. math:: G(b|a) = \sum_{all \; paths} \phi [x(s)] where the state :math:`a` denotes location :math:`x_{a}` at arclength :math:`s_{a}` and state :math:`b` denotes location :math:`x_{b}` at arclength :math:`s_{b}`. The summation over \emph{all paths} indicates a sum of all possible chain conformations (or paths) that begin at state :math:`a` and end at state :math:`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 :math:`\phi[x(s)]` of a path :math:`x(s)` is given by the integration of the Boltzmann weighting of the path. Define the Action :math:`S` for the particular path to be .. math:: 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 .. math:: \phi[x(s)] = \frac{1}{A} \exp (-\beta S[x(s)]), where :math:`A` is a normalization constant to be discussed later. The Hamiltonian :math:`\mathcal{H}` includes differential elastic deformation energy :math:`e_{elas}(x',x'',...;s)` and a potential energy per unit length :math:`V(x,s)`. The Green function :math:`G(b|a)` is written in %the form path-integral form .. math:: 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. .. figure:: figures/path-int.pdf :width: 600 :align: center :alt: Schematic of path summation in the space-arclength plane 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 (:math:`V=0`). The action is given by .. math:: 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 :math:`n=s/b` is a dimensionless arclength parameter. This gives a statistical weight .. math:: \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 :math:`M` segments of length :math:`\delta = (n_{b}-n_{a})/M`. The position of the :math:`m`th point in the chain is :math:`x_{m}`, and the endpoints are :math:`x_{0}=x_{a}` and :math:`x_{M}=x_{b}`. The limit :math:`M \rightarrow \infty` recovers our continuous chain representation .. figure:: figures/discrete-path.pdf :width: 600 :align: center :alt: Discrete-chain representation converts the chain into :math:`M` segments (in this case :math:`M=6`) of length :math:`\delta=(n_{b}-n_{a})/M`. Discrete-chain representation converts the chain into :math:`M` segments (in this case :math:`M=6`) of length :math:`\delta=(n_{b}-n_{a})/M`. The statistical weight is now approximated as .. math:: \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 .. math:: 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 :math:`x_{1}` integral is given by .. math:: \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], where :math:`x_{1}^{\star}=\frac{x_{0}+x_{2}}{2}`. The subsequent :math:`x_{2}` integral is given by .. math:: \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] where :math:`x_{2}^{\star}=\frac{x_{0}+2x_{3}}{3}`. The :math:`x_{m}` integral after performing integrals :math:`x_{1}`, :math:`x_{2}`,\ldots, :math:`x_{m-1}` is .. math:: & & \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] This gives the Green function .. math:: 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] where :math:`A` is set to ensure :math:`\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 .. math:: 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 :math:`n_{a}=0` and :math:`n_{b}=N` (without loss of generality), and assume :math:`V=V(x)` only. The chain obeys Markovian statistics; thus, the Green function for a chain that starts at :math:`A` (\emph{i.e.} position :math:`x_{A}` and arclength :math:`n_{A}`) to state :math:`C` (\emph{i.e.} :math:`x_{C}, n_{C}`) through intermediate state :math:`B` (\emph{i.e.} :math:`x_{B}, n_{B}`) is .. math:: 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}). .. figure:: figures/markovchain.pdf :width: 600 :align: center :alt: Schematic of a chain that obeys Markovian statistics Schematic of a chain that obeys Markovian statistics Take the state :math:`C` to differ an infinitesimal length from state :math:`B` by setting :math:`n_{C}=N+\delta` and :math:`n_{B}=N` (:math:`n_{A}=0`). For simplicity, set :math:`x_{A}=x_{0}`, :math:`x_{B}=y`, :math:`x_{C}=x`, giving .. math:: 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 :math:`\delta \rightarrow 0`, the Green function :math:`G(x, N+\delta | y,N)` will be dominated by a single path whose action is .. math:: 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 :math:`y = x + \eta` where :math:`\eta \ll 1`. The Green function is thus given by .. math:: 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 :math:`\eta > \sqrt{2 b^{2} \delta }` thus the integrand must be expanded to linear order in :math:`\delta` and quadratic order in :math:`\eta` in order to capture the lowest order behavior of the Green function. The expansion of the Green function yields the solvable integral .. math:: & & 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). The solution for the normalization constant :math:`A` is determined by solution of the leading term on the right hand side: .. math:: \frac{1}{A} \int_{-\infty}^{\infty} \exp \left(- \frac{1}{2 b^{2}} \frac{ \eta^{2} }{ \delta } \right) d \eta = 1 thus :math:`A = b \sqrt{ 2 \pi \delta}`. The two additional integrals are given by .. math:: \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. The Green function is given by .. math:: & & 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}}. Thus, the Green function for the 1-D Gaussian Chain obeys .. math:: \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 :math:`G(x,0|x_{0},0)=\delta(x-x_{0})`. For :math:`V(x)=0`, the solution for the Green function is .. math:: 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 .. math:: \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 :math:`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 :math:`V` gives the equilibrium probability distribution for the Gaussian Chain model. For :math:`V(\vec{R})=0`, the solution for the Green function for a Gaussian Chain in 3-D is .. math:: 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]