- #1
onkel_tuca
- 6
- 0
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).
[itex]
\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*}
[/itex]
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...
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).
[itex]
\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*}
[/itex]
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...