Random Gaussian fields applied to fluctuating polymers and biopolymers, eg protein chains and DNA. First, non-conducting knotted charged strings or polymers in a non-conducting viscous fluid leads to the concept of knot energy functionals. The electrosatic energy will minimise without intersection due to Coulomb forces. One can then define a knot-energy funcional. Let (K,\bar{K}) be two such knots in a domain \mathbb{D} such that (K,\bar{K})\subset\mathbb{D}\subset\mathbb{R}^{3} . Let x^{i}\subset K and  y^{j}\subset \bar{K}. A regulated knot energy functional \mathbf{\mathcal{K}ef}(K,\bar{K}) is defined as

 \mathbf{\mathcal{K}ef}(K,\bar{K}) = - \delta^{ij}lim_{\epsilon \rightarrow 0} \bigg({\displaystyle \int_{K}}{\displaystyle \int_{\bar{K}}} {\displaystyle \frac{dx^{i} dy^{j} } {|x-y|^{2}}}-\frac{2}{\epsilon}\bigg)  

Similary, two knots carrying unit currents leads to the Bio-Savart law

 \mathbf{\mathcal{K}f}(K,\bar{K}) = \delta_{ij}\bigg({\displaystyle \int_{K}}{\displaystyle \int_{\bar{K}}} {\displaystyle \frac{dx^{i} dy^{j} } {|x-y|} }\bigg)  

\bf{Theorem}

Let \widehat{\mathcal{Q}}_{i}(x,\omega) be a random Gaussian vector field defined for all x\in\mathbb{D}\subset\mathbb{R}^{3} and with respect to a triplet \Omega,\mathcal{F},\mathbf{P}). Then if:

\mathbf{E}(\widehat{\mathcal{Q}}_{i}(x,\omega)={\displaystyle\int}_{\Omega}d\mathbf{P}(\omega)\widehat{\mathcal{Q}}_{i}(x,\omega) =0

\mathbf{E}(\widehat{\mathcal{Q}}_{i}(x,\omega)\otimes \widehat{\mathcal{Q}}_{j}(y,\omega)

={\displaystyle\int}_{\Omega}d\mathbf{P}{\displaystyle\int}_{\Omega}d\mathbf{P}(\omega)\widehat{\mathcal{Q}}_{i}(x,\omega)(\omega)\widehat{\mathcal{Q}}_{j}(y,\eta))=  {\displaystyle \frac{\zeta^{2}}{|x-y|^{2}}}

The expectation of the stochastic Wilson line of \mathcal{Q}_{i}(x,\omega) gives

\mathbf{E}(\widehat{W}[K])={\displaystyle \int_{\Omega}}d\mathbf{P}(\omega)exp\bigg({\displaystyle\int_{K}}  \widehat{\mathcal{Q}}_{i}(x,\omega)dx^{i}\bigg)  exp \bigg({\displaystyle \int_{K}\int_{\bar{K}}\delta_{ij}\zeta^{2} \frac{dx^{i}dy^{j}}{|x-y|^{2}}}\bigg)

\mathbf{E}(\widehat{W}[K])={\displaystyle \int_{\Omega}}d\mathbf{P}(\omega)exp  \bigg(i{\displaystyle\int_{K}}  \widehat{\mathcal{Q}}_{i}(x,\omega)dx^{i}\bigg)  exp \bigg(-{\displaystyle \int_{K}\int_{\bar{K}}\delta_{ij}\zeta^{2} \frac{dx^{i}dy^{j}}{|x-y|^{2}}}\bigg)

if:

\mathbf{E}(\widehat{\mathcal{Q}}_{i}(x,\omega)={\displaystyle \int}_{\Omega}d\mathbf{P}(\omega)\widehat{\mathcal{Q}}_{i}(x,\omega) =0

\mathbf{E}(\widehat{\mathcal{Q}}_{i}(x,\omega)\otimes \widehat{\mathcal{Q}}_{j}(y,\omega)

={\displaystyle\int}_{\Omega}d\mathbf{P}{\displaystyle\int}_{\Omega}d\mathbf{P}(\omega)\widehat{\mathcal{Q}}_{i}(x,\omega)(\omega)\widehat{\mathcal{Q}}_{j}(y,\eta))=  {\displaystyle \frac{\zeta}{|x-y|}}

The expectation of the stochastic Wilson line of  \widehat{\mathcal{Q}}_{i}(x,\omega) gives

\mathbf{E}(\widehat{W}[K])={\displaystyle \int_{\Omega}}d\mathbf{P}(\omega)exp\bigg({\displaystyle\int_{K}}\widehat{\mathcal{Q}}_{i}(x,\omega)dx^{i}\bigg)  exp \bigg({\displaystyle \int_{K}\int_{\bar{K}}\delta_{ij}\zeta \frac{dx^{i}dy^{j}}{|x-y|}}\bigg)

\mathbf{E}(\widehat{W}[K])={\displaystyle \int_{\Omega}}d\mathbf{P}(\omega)exp\bigg(i{\displaystyle\int_{K}}\widehat{\mathcal{Q}}_{i}(x,\omega)dx^{i}\bigg)  exp \bigg(-{\displaystyle \int_{K}\int_{\bar{K}}\delta_{ij}\zeta \frac{dx^{i}dy^{j}}{|x-y|}}\bigg)

 

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