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I was wondering if there are any other means of obtaining exact (or analytical approximations) of the phase space probability density for a system evolving according to Langevin dynamics. The typical approach seems to be to pass to the Fokker-Planck equation corresponding to the underlying Langevin equation and then solve for its steady state. However, for some types of noise (general coloured noise for example) this approach seems to give rise to insurmountable difficulties. I'm not familiar with any other means of getting to the steady-state probability distribution but wondered if there are any out there?

aQuarkyName
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  • Interesting question! Do you have a reference for the "insurmountable difficulties" when there is coloured noise? Related https://physics.stackexchange.com/q/258089/226902 – Quillo Feb 25 '24 at 11:33
  • Hi, thanks a lot for the comment and the link - very interesting and seems like it might give me some hints regarding where to look. – aQuarkyName Mar 02 '24 at 18:06
  • Regarding the difficulties involved in solving the Fokker-Planck equation, I looked around a bit to see what references people suggest when looking to tackle and it would appear from skimming Risken (https://link.springer.com/book/10.1007/978-3-642-61544-3) and some papers introducing novel approaches to the FP equation (https://iopscience.iop.org/article/10.1088/0305-4470/35/9/301) that the methods out there need to make quite considerable approximations/are limited in the systems to which they can be applied. – aQuarkyName Mar 02 '24 at 18:06
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    An other formulation (that you are probably aware off) of the langevin equation (or corresponding fokker plank equation) is the path integral representation of the langevin equation. Either through the Onsager Machlup action or through the more convoluted (but more powerful) Martin Siggia Rose Janssen De Dominicis action. From which you can extract the probability of the stationary state by simply looking at the path integral average of $\delta(x-x(t))$. Since you can use the tools of QFT you can more easily deal with convoluted problems in a pertubative way. – Syrocco Mar 08 '24 at 22:25
  • Hi, thanks a lot for your comment. It does seem like the MSR formalism could be extremely useful. I was wondering if you know of a good reference that considers the underdamped Langevin equation $M\ddot{x}=-\frac{\partial V}{\partial x} -\gamma \dot{x} + \xi(t) $ in the MSR formalism. I found https://inordinatum.wordpress.com/2012/09/27/a-quick-introduction-to-the-martin-siggia-rose-formalism/ which deals with the overdamped Langevin equation and gives the corresponding path integral. Is it possible to do something similar with the underdamped case? – aQuarkyName Mar 27 '24 at 09:19

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