Question
I was developing the following ideas for a gas and was wondering if these ideas were already in the literature?
The main ideas are:
-
Using the action in thermodynamics
- Including the collision
- Using the mean free path to define temperature
I'm not sure how one would derive the intended formula at the end.
Background
Usually, the Hamiltonian of a gas at thermal equilibrium does not include a collision term which would imply at a collision:
⋅⏟A→←⋅⏟B
⋅⏟B←→⋅⏟A
They actually go through each other.
My attempt
Consider a relativistic gas (point) particles with a 2 particles A and B in a box and the only collide once.
The line element of the Aand B before a collision is given by ds2i
where i=A or i=B. Similarly, the action is given by:
Si=−mic∫Pdsi
Where P is the world line before the collision sμA=sμB where sμi is the four position vector. After the collision, we know the momentum pμ is conserved:
pμA+pμB=p′μA+p′μB
where p′μi denotes the momentum after the collision. Differentiating with respect to ddsi and using
dpμidsi=0 then:
dpμjdsi=dp′μAdsi+dp′μBdsi
with j≠i
After the collision the action is given by:
S′i=−mic∫P′ds′i
where P′ is the world line after the collision and ds′i is defined by dp′ids′i=0.
Let us write dsA in terms of ds′A. We proceed with:
dpμAdsi+dpμBdsi=dp′μjdsi
Using the chain rule:
dpμAdsi+dpμBdsi=dp′μjds′jds′jdsi=dp′μjds′j(ds′jdsi)−1
And as the L.H.S above is finite:
dp′μjds′j→0⟹(ds′jdsi)−1→0
Using L' Hopital Rule:
dp′μjdsi=dpμAdsi+dpμBdsi=d2p′μjds′j2dds′j(ds′jdsi)−1
Or:
dds′j(ds′jdsi)−1dp′μjdsi=d2p′μjds′j2
The above should be solvable as we have 2 boundary conditions (conservation of momentum and si=sj). Hence:
dsi=ds′j∫d2p′μjds′j2(dp′μjdsi)−1ds′j
Hence, the action of particle i is:
Si=−mic∫Pds′j(∫d2p′μjds′j2(dp′μjdsi)−1ds′j)−mic∫P′ds′i
One can take sum over i to find the net action.
General case and more work
Now, there are an infinite number of collisions for N particles and the is temperature T1. We know the mean free path is a function of temperature. Hence,
S′i=−mic∫Pdsi−mic∫P′ds′i−…
T=f(Most Probable(Length(P″k))
Where Length(P) is the length of an arbitrary worldline P″k, Most Probable tells one the most probable function in the series of the action Si and f is an unknown function.