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Consider a non-interacting many particle Hamiltonian system constrained to a volume $ V $ and sitting in a heat bath. The equations of motions would look something like, \begin{align} \dot{q}_i = \frac{\partial \mathcal{H}}{\partial p_i} + F_q(q_i, p_i) \\ \dot{p}_i = -\frac{\partial \mathcal{H}}{\partial q_i} + F_p(q_i, p_i) \end{align}

$ F_p, F_q $ are added to mediate the exchange of energy between the system and the heat bath. What form would these take? This might be really hard, so feel free to answer for a gas or something if needed.

Naively I would expect a form like,

\begin{align} F_q &= 0 \\ F_p(q_i, p_i)~dt &= - f(T)~\delta\{q_i(t) \in \partial V\}~ \hat{P}_{\partial V(q_i)}~p_i(t) \end{align} where $ f(T) $ is a function of temperature, $ \hat{P}_{\partial V(q_i)} $ denotes a projection matrix onto the perpendicular of the heat bath surface, and $ \delta\{q_i(t) \in \partial V\} $ denotes that the particle is on the surface at that given time.

A valid forcing should reproduce thermodynamics if simulated for a large enough number of particles.

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