Discretizing a Fluctuation Dissipation Theorem

[onkel_tuca]

Hey!

I want to discretize a fluctuation dissipation theorem for the white noise ζ of a stochastic differential equation on a 2D domain (sphere). For that I integrate over "Finite Volume" elements with area A and A' (see below).

$\begin{eqnarray*} \int_{A} d A \int_{A'} d A' \langle\zeta(\mathbf{r},t)\zeta(\mathbf{r}',t')\rangle &=& -2\int_{A} d A \int_{A'} d A'(\nabla_s^{\mathbf{r}})^2\delta(\mathbf{r}-\mathbf{r}')\delta(t-t')\\ &=& 2\int_{A} d A \int_{A'} d A'\nabla_s^{\,\mathbf{r}'}\cdot\nabla_s^{\,\mathbf{r}}\delta(\mathbf{r}-\mathbf{r}')\delta(t-t')\\ &=& 2\int_{A} d A \int_{\partial A'} d S'\left[\nabla_s^{\,\mathbf{r}}\delta(\mathbf{r}-\mathbf{r}')\cdot\mathbf{n}'\right]\delta(t-t')\\ &=& 2\int_{A} d A \nabla_s^{\,\mathbf{r}}\cdot\underbrace{\int_{\partial A'} d S'\left[\delta(\mathbf{r}-\mathbf{r}')\mathbf{n}'\right]}_{\mathbf{g}}\delta(t-t') \end{eqnarray*}$

Note that ∇_s is the 2D surface gradient (thus only w.r.t. angles on the sphere, not radius). The superscript on ∇_s indicates whether differentiation is w.r.t to r or r'. All quantities are assumed to be nondimensionalised to the same length-scale (sphere radius R): dA and dA' each have dimension 'area', each ∇_s has dimension '1/length', δ(r-r') has dimension '1/area', thus the whole thing has dimension '1'.

1.) In the first step I rewrite one of the two ∇_s w.r.t. r', which gives a negative sign due to the δ(r-r') function.

2.) In the second step I use the divergence theorem for surfaces to transform the surface integral dA' into an integral over the boundary.

3.) Next I write the remaining ∇_s operator in front of the integral g, because the operator doesn't depend on r'.

4.) The integral g depends on the postition of the two elements. Assuming the elements are different and not neighbors, the delta function δ(r-r') in g vanishes, because r and r' do never 'meet'. Now, when the elements are identical, A=A', I would assume that g=n, which allows me to use the divergence theorem again for the remaining integral. This gives me 2∫dS n⋅n δ(t-t')=2Lδ(t-t'), where L is the perimeter of the element, measured in lengthscale R. The case of neighboring elements is more complicated.

Am I doing this correct? I'm especially not sure about evaluating integral g: the integral g has dimension '1/lenght' but my assumed result in the case of A=A' is n, which has dimension 1. Actually I was expecting something like 2L²δ(t-t') in the end, which would be consistent with the way I discretize δ(t-t')...

Maybe somebody has an idea...

 

[onkel_tuca]

Just in case someone has a similar problem. I ended up with this approach (taken from a preprint):

Here, we discretize the noise ζ in a field equation into element noise vector z, with components z_i of the finite volume (FVM) elements i.
We define the correlation function between z_i and z_j by integrating
\begin{equation}
\langle\zeta(\mathbf{r},t)\zeta(\mathbf{r}',t')\rangle=-2 \nabla_s^2\delta(\mathbf{r}-\mathbf{r}')\delta(t-t')\;,
\end{equation}
over element areas A_i and A_j:
\begin{eqnarray}
&&\langle z_i(t)z_j(t')\rangle\equiv\iint\limits_{A_i}\! d A_i \iint\limits_{ A_j}\! d A_j \langle\zeta(\mathbf{r}_i,t)\zeta(\mathbf{r}_j,t')\rangle\quad\\
%
&&=2\int_{\partial A_i}\limits\! d S_i\, \mathbf{n}_i\cdot\!\int_{\partial A_j}\limits\! d S_j\mathbf{n}_j\,\delta(\mathbf{r}_i-\mathbf{r}_j)\delta(t-t')\label{eq:element_noise_vector2}\\
%
&&=2 \sum_{q} l_{iq}\sum_{p} l_{jp}\delta_{q,p}\mathbf{n}_{iq} \cdot \mathbf{n}_{jp} \delta(t-t')\;.\label{eq:element_noise_vector3}
\end{eqnarray}
In (\ref{eq:element_noise_vector2}) we used the divergence theorem and in (\ref{eq:element_noise_vector3}) we converted the line integrals into sums over the element boundaries.

Here we discretize δ(r_i-r_j) by partitioning the surface into rhombi of area A_◊ (see figure above) and defining:
\begin{equation*}
\delta_{q,p}= \begin{cases}
1/A_{\Diamond} & \mathrm{for} \enspace q=p\;, \\
0 & \mathrm{for} \enspace q\neq p\;,
\end{cases}
\end{equation*}
where q and p are the indices of the boundaries of element i and j respectively.
Three cases have to be considered: Supposed the elements i and j are neither identical nor neighbors, δ_q,p vanishes in (\ref{eq:element_noise_vector3}) for all q and p. For i=j, however,

$$\delta_{q,p}=1/A_{\Diamond}\; and\; \mathbf{n}_{iq} \cdot \mathbf{n}_{jp}=1$$

for all q and p. Finally, for neighboring elements, there is one common boundary where

$$\delta_{q,p}=1/A_{\Diamond}\; and\;\mathbf{n}_{iq} \cdot \mathbf{n}_{jp}=-1$$.

Thus, one finds:
\begin{eqnarray}
\langle\underline{z}(t)\otimes\underline{z}(t')\rangle&=&\frac {2Nl^2}{A_{\Diamond}} \left(\underline{\underline{1}}-\frac 1 N \underline{\underline{Q}}\right) \delta(t-t')\;,
\label{eq:fluc_diss_element_conti}
\end{eqnarray}
where N is the number of element boundaries. Here, Q_ij=1 if elements i and j are neighbors and zero otherwise. Note that in (\ref{eq:fluc_diss_element_conti}), we assumed constant boundary length l and number of boundaries N. This is reasonable for a refined icosahedron with 642 FVM elements. The form of (\ref{eq:fluc_diss_element_conti}) acknowledges the conservation law for the noise.