Past exam of the stochastic modelling of biological processes course of the University of Oxford / 2023 / 3 / c

Write $c=c_1=c_3$, so the friction memory kernel is ${\kappa }_c(t)=c{e}^{-ct}$ and has area one for every $c>0$. Its Laplace transform is ${\tilde{\kappa }}_c(s)=c/(s+c)$. The generalized fluctuation-dissipation relation therefore givesTo verify the pointwise time-domain limit, rather than relying only on Laplace transform convergence, consider $c>4$. The two rootsare negative and satisfy ${\lambda }_{+}\to -1$, ${\lambda }_{-}\to -\infty$. Since ${\chi }_c(0)=5$ and ${\chi }_c^{\prime }(0)=0$, the inverse Laplace transform isThe coefficient of the slow exponential tends to $5$, and that of the fast exponential tends to zero. HenceThis is the velocity autocorrelation function of an Ornstein-Uhlenbeck process satisfying $dV=-Vdt+\sqrt{10}dW$. The friction memory kernel becomes local in time. Consistently, the even noise autocorrelation $5c{e}^{-c|\tau |}$ tends to $10\delta (\tau )$ in the sense of distributions, fixing the white noise normalization. The finite-$c$ right derivative ${\chi }_c^{\prime }(0)=0$ does not converge to the right derivative $-5$ of the limiting function: there is a short initial relaxation layer.

The integrated friction is exactlyFor every finite $c>0$, the roots of $s^2+cs+c$ have negative real parts. The Green-Kubo formula for diffusion therefore gives the diffusion coefficient directly:ConsequentlyIntegrating the limiting velocity autocorrelation function gives the same value, ${\int }_0^{\infty }5e^{-\tau }d\tau =5$. Computing $D_c$ before taking the limit also avoids having to justify an interchange of the infinite-time integral and the parameter limit.