- #1
Ravi Mohan
- 196
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I am studying an article which involves stochastic variables http://www.rmki.kfki.hu/~diosi/prints/1985pla112p288.pdf.
The author defines a probability distribution of a stochastic potential [itex]V[/itex] by a generator functional
[tex]
G[h] = \left<exp\left(i\int V(\vec{r},t)h(\vec{r},t)d\vec{r}dt\right)\right>,
[/tex]
where [itex]h[/itex] is an arbitrary function and [itex]\langle\rangle[/itex] stands for expectation values evaluated by means of the probabil-
ity distribution of [itex]V[/itex].
He, then equates it to (equation 1 in the article)
[tex]
G[h] = exp\left(-\frac{1}{2}\iint h(\vec{r},t)h(\vec{r}^{\prime},t)f(\vec{r}-\vec{r}^{\prime})d\vec{r}d\vec{r}^{\prime}dt\right).
[/tex]
How do we mathematically work out the steps? Any relevant reference or hint will be of great help. Thanks.
The author defines a probability distribution of a stochastic potential [itex]V[/itex] by a generator functional
[tex]
G[h] = \left<exp\left(i\int V(\vec{r},t)h(\vec{r},t)d\vec{r}dt\right)\right>,
[/tex]
where [itex]h[/itex] is an arbitrary function and [itex]\langle\rangle[/itex] stands for expectation values evaluated by means of the probabil-
ity distribution of [itex]V[/itex].
He, then equates it to (equation 1 in the article)
[tex]
G[h] = exp\left(-\frac{1}{2}\iint h(\vec{r},t)h(\vec{r}^{\prime},t)f(\vec{r}-\vec{r}^{\prime})d\vec{r}d\vec{r}^{\prime}dt\right).
[/tex]
How do we mathematically work out the steps? Any relevant reference or hint will be of great help. Thanks.