I'm reading this paper by Starobinsky and Yokoyama where they give the coarse-grained equation of motion, $$ \dot{\bar{\phi}}({\bf x},t ) = -\frac{1}{3H}V'(\bar{\phi}) + f({\bf x},t) $$ where $f({\bf x},t)$ is given by $$ f({\bf x},t) = \epsilon a(t)H^2 \int\frac{d^3 k}{(2\pi)^{3/2}}\delta(k - \epsilon a(t)H) \left[ a_{\bf k}\phi_{\bf k}(t) e^{-i {\bf k}\cdot {\bf x}} + a_{\bf k}^\dagger\phi_{\bf k}^*(t) e^{i {\bf k}\cdot {\bf x}} \right]. $$ This equation can be considered to be a Langevin equation for the stochastic quantity $\bar{\phi}$ with a stochastic noise term $f({\bf x},t)$ with correlation properties given by $$ \left\langle f({\bf x}_1, t_1) f({\bf x}_2, t_2) \right\rangle = \frac{H^3}{4\pi^2}\delta(t_1 - t_2) j_0(\epsilon a(t)H|{\bf x}_1 - {\bf x}_2|), \qquad j_0(z) = \frac{\sin z}{z}. $$ The authors then write down the Fokker-Planck equation corresponding to this Langevin equation, $$ \frac{\partial \rho[\bar{\phi}({\bf x},t)]}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\bar{\phi}} \left\{ V'[\bar{\phi}({\bf x},t)]\rho[\bar{\phi}({\bf x},t)] \right\} + \frac{H^3}{8\pi^2} \frac{\partial^2 \rho[\bar{\phi}({\bf x},t)]}{\partial \bar{\phi}^2}. $$

Question: How do you derive this equation?

I've attempted to derive this using the Kramer-Moyel expansion. For a Langevin equation of the form, $$ \dot{\xi}(t) = h(\xi, t) + g(\xi, t)\Gamma(t) $$ where, $$ \left\langle \Gamma(t) \right\rangle = 0, \qquad \left\langle \Gamma(t)\Gamma(t') \right\rangle = 2\delta(t - t') $$

The corresponding Fokker-Planck equation is given by $$ \frac{\partial W(\xi,t)}{\partial t} = -\frac{\partial}{\partial \xi}[D^{(1)}(\xi,t )W(\xi, t)] + \frac{\partial^2}{\partial \xi^2}[D^{(2)}(\xi,t )W(\xi, t)] $$ where $D^{(i)}$ are the Kramer-Moyel expansion coefficients (see Risken, The Fokker-Planck Equation) given by, $$ D^{(1)}(\xi,t) = h(\xi,t) + \frac{\partial g(\xi,t)}{\partial \xi} g(\xi,t), \qquad D^{(2)}(\xi,t) = g^2(\xi,t) $$

However, this result doesn't seem to work because in the problem of interest the stochastic force is dependent on both ${\bf x}$ and $t$. Additionally, the correlation $\left\langle f({\bf x}_1, t_1) f({\bf x}_2, t_2) \right\rangle$ is far more complicated in this case.

Is there a more general formalism that could help with more general force $f({\bf x},t)$?

Any help or resources about this particular derivation would be greatly appreciated. Thanks in advance!


1 Answer 1


For the purpose of deriving the Fokker-Planck equation, we only need correlation between two different times $t_1$ and $t_2$ for the stochastic force, not for two different spatial locations. In other words, you can set $x_1=x_2 \equiv x$ in the correlation relation when deriving the Fokker-Planck equation. In the paper, the correlation relation is given in that more general form with $x_1$ and $x_2$ for other purposes than for derivation of the Fokker-Planck equation. From setting $x_1=x_2 \equiv x$ in the correlation relation follows $j_0(...)=1$ and thus we are left with a delta-correlation. The Fokker-Planck equation follows directly from the Kramers-Moyal expansion you have written there when you identify $\Gamma=f/\sqrt{H^3/8 \pi^2}$ and $g=\sqrt{H^3/8 \pi^2}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.