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. Thanks.

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.

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 without adding molecular details than the Newtonian equations of motion change to $m\ddot{x} = F - \mu \dot{x}$ where energy is not conserved. 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 write the system as "Hamiltonian" derived from a non-canonical change of variables? Can you just plugin the new Hamiltonian to the usual partition function but integrate over the new variables? Thanks.

p.s: Actually I am interested in the case also when there is a gaussian random force term $R(t)$ acting on the system but I guess that changes the average quantities of the system in the long run.

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. Thanks.