Feynman-Kac formula points out an SDE corresponds to a FPE. Given an SDE, I can simulate numerically by following the equation, then I can plot the histogram of the simulated single particle. However, FPE provides probabilistic evolution which bases on large amount of particles. Do these two results match? Or whether or not the ergodicity guarantees the matching?
Yes, they match. The SDE gives you a particular solution corresponding to a given realization of noise, but the FP equation gives you the probability distribution of the solutions to the SDE.
Thanks. If they match, doesn't it imply the ergodicity? Because for one single particle, statistical law comes from temporal sampling and for many particles from spatial sampling.
Shu-Chang, I don't quite follow your argument. Ergodicity (which has not been rigorously proven in general) means that time averages are ensemble averages in equilibrium systems, as you probably imply? However, the SDE/FPE description in not limited to equilibrium or Boltzmann distributed systems. The SDE and FPE descriptions are just two sides of the same coin. I don't see how this is connected to ergodicity.
Yes, I regard ergodicity as two averages consistent. I just thought when simulating a SDE, only one single particle is considered. But I see the point now. You're right.
Would you mind my asking for related references about equivalence between SDE and FPE?
Reichl's book on Stat. Mech. is a pretty complete reference book on most things in stat. mech. (it's a bad textbook, though :)). There's also Risken's classic book on the FP equation.
Let an SDE be driven by a white noise process. Its solution is a Markov process that is completely characterized by its transition probability density function. This function is the solution of the PDE Fokker-Planck-Kolmogorov, whose terms are derived by the drift and diffusion terms of the SDE. Thus, the two things are strictly related or in other words they are two complementary aspects of the same problem.
check out Gardiner's handbook. it helped me a lot with understanding this link
Crispin W. Gardiner, Handbook of stochastic methods: For Physics, Chemistry and the Natural Sciences
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