Two types of path intergrals in statistical physics - difference?

Question

In non-equilibrium statistical physics as far as I can tell there are two types of path integrals to find conditional probabilities:

• Path integrals over the noise in the Langevin equation, $\vec u(t)$: $$P(\vec x,t\mid \vec x_0,0)=\langle\delta^3(\vec x-\vec r(t))\rangle$$ where $$\langle F[\vec u(t)]\rangle\propto \int \mathcal{D}\vec u(t) F[u(t)] e^{-\frac{1}{4D} \int dt \vec u^2(t)}$$
• Or path integrals over trajectories e.g.: $$P(\vec x,t\mid \vec x_0,0)\propto \int^{r(t)=\vec x}_{r(t_0)=\vec x_0} \mathcal{D} \vec r(t) \exp \left(-\int^t_{t_0} dt'\frac{1}{4D}((\dot{\vec r}-\vec v(\vec r))^2+\Theta(0) \nabla \cdot \vec v(\vec r) ) \right)$$

My question is what is the difference between these two? Are they identical in every situation? Does the first approach work if we have a net drift?

Answer 1

^{_{This is my attempt for an answer. I am not 100% sure it is correct. If someone who knows could confirm/deny that would be great.}}

Imagine we have a Langevin equation of the form: $$\frac{d\vec r}{dt}=\vec v+\vec u$$ where $\vec v$ is a deterministic force (for now independent of position etc.) and $\vec u$ is the noise term. Now in such a situation the integral needed to perform a path integral over the noise term could indeed by done.

But now consider when the deterministic force $\vec v$ has some position dependence and/or we have multiplicative noise: $$\frac{d\vec r}{dt}=\vec v(\vec r)+f(\vec r)\vec u$$ Now ask could this integral be done using the a path integral over the noise. The answer is yes in theory but no in practice. The $\vec r$ dependency would be a pain to deal with.

The solution: Do what is essentially equivalent to a change of variable so that the path integral is over the trajectory $\vec r$ rather then the noise term. In which case the path integral may be doable.

An example of a situation which can be done using the first approach is free particle diffusion whilst an example of the second is the harmonic potential.

Reference

(1) CAP-NSERC Summer Institute in Theoretical Physics, Edmonton, Alberta, 10-25 July 1987, F. C. Khanna (pg146); I could only see a google snippet of this book but it mentions the change of basis discussed here