A question on inelastic relativistic collisions-extension

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding some comments about the mean molecular translational kinetic
energy of a system as observed in a frame in which the system's center of
mass and it connection to the temperature of a system culled from
snippets from some posts by Ed Schweber:

> ...
>   What about Kelvin temperature. Is that absolute or relativistic? Kelvin
> temperature is proportional to the average KE per molecule and would not
> that average KE be relative to the motion of the observer?

also

> ...                But from the rest frame of the fluid would not the
> solid object now have a different average kinetic energy per molecule?
>
>  Would there not now be a transfer of this molecular KE between the solid
> and the fluid -that is a transfer of heat between objects at the same
> temperature - in violation of the Second Law of Thermo?

and

> ...                                                      My point is
> that due to the motion of the solid, its molecules now have a different
> vibratory KE with respect to the frame of the fluid (although the still and
> fluid still have the same temp - because temperature is correlated to
> internal KE only in the center of mass frame of the object) ....

and from Miguel Santos:

> ...
> I got this asuming the equipartition theorem and applying a naive
> rescaling of mass and velocity within the average of KE, when
> both observers move with relative velocity v along the x direction.
>
> As KE is different for both observers so will be the temperature. ... .

The idea that the mean molecular translational kinetic energy in the
center-of-mass-at-rest frame for an object is directly proportional to
the system's temperature and that the equipartition theorem holds for
those translational degrees of freedom is a consequence of *classical*,
i.e. nonquantum, statistical mechanics, *and* the *Newtonian
approximation* for the relationship between a particle's momentum and
its kinetic energy.  To the extent that the particle's kinetic energy is
proportional to the square of the particle's momentum magnitude to *that
extent only* is the (absolute, thermodynamic) temperature T related to
the translational kinetic energy U_trans via the usual equipartition
form: U_trans = (3/2)*k*T.  Now I realize that Ed, Miguel and others
were not necessarily considering a system whose temperature in the
center-of-mass-at-rest frame was so hot the internal interparticle
motions were themselves mutually relavitistic--just frames where the
center of mass of the system was observed as relativistic.  But, any
expression for the motion of a particle as observed in a frame moving
relativistically w.r.t. the object's center of mass will *still* require
a relativistic expression for each particle's kinetic energy as a
function of its momentum *in that lab frame*.  So in the spirit of
completeness I discuss relativistic effects of high temperatures even in
a frame where the center of mass of the system is at rest.

In relativistic (thermo)dynamics the kinetic energy U_trans of a particle
is *not* proportional to p^2; rather it is only proportional to this
quantity at temperatures so low that k*T << m*c^2.  But at temperatures
so high that typical particle motions tend to be ultrarelativistic, i.e.
k*T >> m*c^2  (quark-antiquark soup temperatures) then the relationship
changes to: U_trans = 3*k*T (note 3 not 3/2) as a consequence of the
kinetic energy now being effectively proportional to the *first* power of
the momentum magnitude (much like ordinary massless light is at normal
temperatures).  As long as U_trans is proportional to some power b of
momentum magnitude p, then in equilibrium we have the modified
equipartition result: U_trans = (3/b)*k*T.

At intermediate temperatures of the order k*T ~ m*c^2 the relationship
between U_trans and T is *much* more complicated.  This is because in the
intermediate case the kinetic energy is a more complicated function of
particle momentum, i.e. U_trans = sqrt((m*c^2)^2 + (p*c)^2) - m*c^2.
In this case the correct formula relating these quantities is given by:
U_trans = 3*k*T - (m*c^2)*(1 - K_1(z)/K_2(z)) where K_1(z) and K_2(z) are
modified Hankel functions of order 1 & 2 respectively, and
z == (m*c^2)/(k*T).  In the nonrelativistic limit of z >> 1 the ratio of
modified Hankel functions becomes 1 - 3/(2*z) and this makes the
complicated formula boil down to the usual equipartition result.  In the
opposite ultra-relativistic temperature result then the first term
completely dominates the second one and the value approaches the massless
3*k*T formula.

It should be noted that it is not possible to merely differentiate the
above formula for U_trans w.r.t. T to find the constant volume specific
heat c_v.  This is because at such high temperatures that the particles
are becoming relativistic means that the probability of spontaneous pair
creation/destruction processes out of the vacuum start turning on and
are becoming relevant.  This means that the system does not any longer
have a conserved number of particles, and the particle consentration (as
well as the antiparticle concentration) increases with temperature.  In
this case the system is more properly described by a Grand Ensemble than
the Canonical Ensemble.  Here the mix of particle and antiparticle
concentrations found in equilibrium is determined by a single chemical
potential type parameter (rather than by the former single conserved
particle number parameter supplemented with the requirement that the
antiparticle concentration was zero).  This is because for each particle
species we have the constraint on the particle and antiparticle chemical
potentials: [mu]_particle + [mu]_antiparticle = - 2*m*c^2 by virtue of
the pair creation/destruction reactions that become possible at such
high temperatures.  So any calculation of the specific heat c_v would
need to take into account not only the heat needed to heat up the
particles already present but also account for the increasing
concentrations of the particles with temperature as well. The c_v
function would also include a large contribution from the ubiquitous EM
radiation present in the system's volume as well.  (The pair creation/
destruction processes typically are accompanied by photon destruction
creation processes as well).

David Bowman
dbowman@georgetowncollege.edu