Question

Given the heat equation:

$$\partial_{t}{\varPhi(x,t)}=k^2\partial_{xx}{\varPhi(x,t)}$$

with the boundary conditions:

$$\Phi(x,0)=\Phi_0$$

and a Neumann boundary condition of the kind:

$${\partial_{x}}{\Phi(0,t)=\nu(t)+C}$$

where $\nu(t)$ is a stochastic variable with gaussian distribution ${\sigma=\sigma_0,\mu=0}$ and $C$ a constant, what is the distribution of the $\Phi(L,t)$?

Thanks in advance

Answer 1

In this case $\Phi(x,t)$ is itself a stochastic process and this equation should be rewritten in a proper way. There is some literature as this and a more general theory of stochastic pde due to John Walsh (a tutorial can be found here). In this case, a general solution can be written down using the fundamental solution of the heat equation given by

$$\Delta(x,t)=\frac{1}{\sqrt{4\pi k^2 t}}e^{-\frac{x^2}{4k^2 t}}$$

and then one has

$$\Phi(x,t)=\int dx'\Phi_0(x')\Delta(x-x',t)+k^2\int dt'[\nu(t')+C]\Delta(x,t-t')$$

and we can easily compute

$$\langle\Phi(x,t)\rangle = \int dx'\Phi_0(x')\Delta(x-x',t)+k^2C\int dt'\Delta(x,t-t')$$

$$\langle\Phi(x,t)\Phi(y,s)\rangle=\int dx'\Phi_0(x')\Delta(x-x',t)\int dy'\Phi_0(y')\Delta(y-y',s)+$$ $$k^2C\int dx'\Phi_0(x')\Delta(x-x',t)\int ds'\Delta(y,s-s')+$$ $$k^2C\int dt'\Delta(x,t-t')\int dy'\Phi_0(y')\Delta(y-y',s)+$$ $$k^4C^2\int dt'\Delta(x,t-t')\int ds'\Delta(x,s-s')+k^4\sigma^2_0\int dt'\Delta(x,t-t')\Delta(y,s-t')$$

where I used the fact that $\langle\nu(s)\nu(t)\rangle=\sigma^2_0\delta(t-s)$. It is interesting to note the simplest case $\Phi_0=0$ and $C=0$ producing immediately

$$\langle\Phi(x,t)\rangle = 0$$

and

$$\langle\Phi(x,t)\Phi(y,s)\rangle=k^4\sigma^2_0\int dt'\Delta(x,t-t')\Delta(y,s-t').$$

So, all higher order even correlation functions are given by products of the fundamental solution of the heat equation properly integrate in intermediate times.