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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) +...


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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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