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The stationary solution of your Fokker-Planck equation is an equilibrium distribution. Here, assuming that $v$ is the variable for the velocity, that mean a Maxwell-Boltzmann distribution. So if your initial condition for the distribution $W(v,t)$ is not an equilibrium distribution, your system will not be at equilibrium. However the evolution of your ...

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So you could start with the propagator, chain them together and integrate over the intermediate values and you'll end with a discretized version of the path integral. This is, however, rather tedious, so let's just instead write the stochastic difference equation: $$\Delta X_n = -\lambda \Delta t + \sqrt{2} \Delta W_n$$ Here $\Delta W_n$ are independent ...

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This so-called Langevin equation is just the Newtons second Law with an extra random force: $$\frac{d\mathbf{p}(t)}{dt} = -\mathbf{\nabla}V(x) + \mathbf{\eta}(t)$$ together with $\mathbf{p}=\frac{d\mathbf{x}(t)}{dt}$ obviously. It is better behaved mathematically if you write this in the integral form: $$\mathbf{p}(t) = \int dt ~ \big[-\mathbf{\nabla}V(x) +... 2 Define the inner product \langle f, g\rangle = \int fg \,\mathrm{d}x. Define also the adjoint \mathcal{A}^\dagger of \mathcal{A} as one satisfying \langle \mathcal{A}f,g\rangle = \langle f, \mathcal{A}^\dagger g\rangle for all f, g. Obviously (\mathcal{A}^\dagger)^\dagger = \mathcal{A}. Given the forward FPE:$$\partial_t p = \mathcal{L}_t p, \...

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