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So if I write the action for a gas:

$$ S = \int (T -U) dt \tag{1}$$ where $T$ is the kinetic energy and $U$ is potential energy.

I suspect there is a constant term (which does not affect the equations of motion) for the gas (in classical mechanics). I was hoping to verify my paper's idea (There is now an updated version see here) up to equation $(19)$.The summary of ideas section wise are as follows:

$(1.1)$ We consider an arbitrary potential which $0$ everywhere until both the gas molecules collide.

$(1.2)$ Assuming a non-ideal collision (finite collision time) I get a non-zero action

$(1.3)$ We use Newton's second law to find the finite collision time and the average force is considered as pressure.

$(1.4)$ The deformed face of the molecule has an area which is also considered.

$(1.5)$ Since we are considering $2$ spheres colliding which is a function of their relative velocity we can find the average momentum change assuming the collision is not a special event.

$(1.6)$ The collision per unit 4-volume is mentioned as derived by Einstein.

Combining all the above subsections we derive a constant action term:

$$S_c = \int \frac{N R \tilde V'(2R)}{\pi} \frac{ \langle \mu \rangle \lambda }{k_B } dt \tag{2}$$

where $N$ is the number of particles, $R$ is the radius of the particle, $ \tilde V'(x)$ is the spatial derivative potential and $ \frac{ \langle \mu \rangle \lambda }{k_B } $ are constants.

Is this expression correct?

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    $\begingroup$ (by linking to your paper you are now Less Anonymous :) $\endgroup$ Commented Jan 7, 2022 at 16:53

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