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If you have a classical system (i.e obeying Newton's equations of motion) with Hamiltonian $H(x,p) = \frac{p^2}{2m} + U(x)$ then the statistical behaviour of this system is described by the probability density $e^{-\frac{\beta}{2}H}$, which can be used to find average quantities. If you want to model a system subject to friction in a hot bath without adding molecular details than the Newtonian equations of motion change to $m\ddot{x} = F - \mu \dot{x} +R(t)$ where $R(t)$ is the random forcing term and $F$ is a conservative force. The flow given by this in the phase space does not conserve energy or phase space volume. How does one describe the statistical physics of such dynamics? What is the probability density one has to use? Grand canonical ensembles seems to deal with such stuff as well but I can not see how to formulate friction as such a system. Or what happens if you ignore $R(t)$ and write the system as "Hamiltonian" derived from a non-canonical change of variables? Can you just plugin the new Hamiltonian (without $R(t)$) to the usual partition function but integrate over the new variables? This doesnt seem right since without $R(t)$ the system would halt evantually, so $R(t)$ does have a critical role in this sense. Any keywords would be also welcome. For people who need a more concerete question I could ask how would you compute time average quantities of say a stochastic ODE like $$ \dot{v} = kx - \mu v + R(t) ? $$ Is there any partitian function like methods for this? Or just write the solutin and find average time integral if you can?

Thanks.

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    $\begingroup$ I think what you might be looking for is Leggett and Caldeira's 1983 paper sciencedirect.com/science/article/pii/0378437183900134 and the field associated with it. $\endgroup$
    – Wolpertinger
    Commented Feb 19, 2016 at 17:44
  • $\begingroup$ Do I really need to use a path integral formulation if I want to find time average quantities of a simple stochastic flow like $\dot{v} = kx - \mu v + R(t)$? $\endgroup$
    – Sina
    Commented Feb 19, 2016 at 19:01
  • $\begingroup$ long story short: the friction changes nothing but timescales. Terms to google: Langevin Equation, Fokker-Planck-equation, Fluctuation-Dissipation-Theorem (Einstein-Smoluchowski-Relation) $\endgroup$
    – Bort
    Commented Feb 19, 2016 at 19:09
  • $\begingroup$ @Bort: Are you saying that if I have a system of particles each with its own friction coefficient (not necessarily overdamped), the average quantities behave in the long run as if they were obeying $m\ddot{x} = F$? $\endgroup$
    – Sina
    Commented Feb 19, 2016 at 19:38
  • $\begingroup$ no i saying that the fact that you assume a heat bath (so that equilibrium is given by boltzmann weights) necessarily introduces a friction $\endgroup$
    – Bort
    Commented Feb 19, 2016 at 19:47

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This is a very general and very hard question. As I understand it, you are asking about the steady state of a system where driving (through $R(t)$) an dissipation coexist. The answer depends on the statistics of $R(t)$. If you look up 'Einstein Relation', you will see how to relate the correlations of $R(t)$ to the dissipation coefficient $\mu$ in the case of a Gaussian white noise.

More general cases quickly become very difficult. And it is not always known if there exists a Hamiltonian formulation. A very general formalism for such stochastic differential equations is the Martin-Siggia-Rose formalism. This makes it possible to write a generating functional for the correlation functions of a stochastic classical field theory as a path integral weighted by a given action.

For example, adding such a stochastic driving to Navier-Stokes equations leads to modern research topic within the study of hydrodynamic turbulence.

I detailed how to compute averages in the case of a Gaussian white noise and when $k=0$ in this post. Generalising to $k\neq 0$ should not be hard.

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  • $\begingroup$ I assume the simplest case where $R(t)$ is a Gaussian process with $0$ mean. Is the above claim that average quantities behave as if it was usual Newtonian mechanics $$\ddot{x}=F(x)$$ in the long run? $\endgroup$
    – Sina
    Commented Feb 19, 2016 at 19:40
  • $\begingroup$ You never leave Newtonian mechanics. The dynamics are defined by Newton's equations. However if you compute average quantities, you must take into account the fluctuations of $R(t)$ carefully. $\langle x(t) \rangle'' = - k \langle x \rangle - \mu \langle x \rangle''$ as if $R(t)$ was not there, but $R(t)$ enters the computation of $\langle x(t) x(t') \rangle$ for example. Check out the link in my answer. $\endgroup$
    – Steven Mathey
    Commented Feb 19, 2016 at 19:45
  • $\begingroup$ Yes but I meant newtonian mechanics with a conservative force=field that is derived from a potential that only depends on coordinate variables. Thanks for the link and the answer I will check if it is what I want. $\endgroup$
    – Sina
    Commented Feb 19, 2016 at 23:01
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    $\begingroup$ One way to be sure that your dynamics are conservative is to choose every individual realisation of your stochastic process to be. If $\mu=0$ and the probability that $R(t)$ does not derives from a potential is zero, then your dynamics will be conservative by construction. If you are asking if effective conservative dynamics emerge once the random noise has been averaged over, then I don't know. This is a very hard problem and a modern research topic. $\endgroup$
    – Steven Mathey
    Commented Feb 20, 2016 at 9:24
  • $\begingroup$ I see I am starting to understand the correct way to pose this question now thanks to your replies. I guess I should have asked does the time average of observables for the stoachastic ODE $\ddot{x} = -\nabla U(x) - \nu \dot{x} + R(t)$ converge to the statistical average of a system with Hamiltonian H = K + U via the equipartition theorem? Can you suggest some reading material on this? $\endgroup$
    – Sina
    Commented Feb 21, 2016 at 0:36

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