Two boxes 1 and 2 are in a potential $V$ such as $V_1=0$ and $V_2=V_0$. The boxes contain an ideal gas, temperature and volume are constant and set to 1. Hence $N_i=P_i$ where $N_i$ is the number of particles in box $i$ and $P_i$ is the pressure in box $i$.
The boxes are connected by a pipe and particles can thermally jump from one box to another. I would like to write the transition rates between the boxes. Note $R_{ij}$ the rate from box i to box j.
1rst idea: don't include pressure in the rates $$R_{12} = N_1e^{-V_0}$$ $$R_{21} = N_2e^{V_0}$$
2nd idea: include the contribution of the pressure to the energy of the particles $$R_{12} = N_1e^{-V_0+P_1-P_2}$$ $$R_{21} = N_2e^{V_0-P_1+P_2}$$
What is the right way to do ? Should I include the contribution of the pressure in the transition rates?
