# Contribution in action from collisions for a gas?

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?

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