Re: KE & temperature (was: Newton's 3rd law? ...)

[Joel Rauber <Joel_Rauber@SDSTATE.EDU>]

David wrote in part:


> In particular, the equation: E_avg_kinetic =
> (3/2)*k*T for
> a classical particle thermally equilibrated with its
> environment has the
> coefficient 3/2 . . .
<snip>
> If, more
> generally, the
> particle was allowed to move in D dimensions and its translational
> kinetic energy was proportional to the n-th power of its momentum
> magnitude, then the equation would become: E_avg_kinetic =(D/n)*k*T,
> which is *still* a proportionality.  The proportionality only
> breaks down
> for: A) nonclassical behavior of the particles due to quantum
> mechanical
> or quantum statistical effects, and/or B) relativisitic
> effects where the
> simple quadratic power law relationship: E_kinetic = p^2/(2*m) is
> replaced by: E_kinetic = m*c^2*(sqrt(1 + (p/(m*c))^2) - 1)
> which is not
> a power law except at low enough momentum so that the particle is
> effectively Newtonian, or at high enough momentum so the particle is
> effectively an ultrarelativistic particle of negligible mass.

It should be pointed out that the above proportionality

E_avg_kinetic =(D/n)*k*T

applies only to an object without structure; i.e. one for which there are no
rotational degrees of freedom or potential energy degrees of freedom that
can "store" energy.  The modifications necessary follow the lines of
discussion that David outlined.  Any degree of freedom proportional to the
square of a generalized coordinate or its conjugate momenta contribute
(1/2)*k*T to the average energy.  So the typical rotational degrees of
freedom, of the form (1/2)*I_1*omega_1^2, do so; and SHO potential energy
degrees of freedom, good ole (1/2)*K*x^2 contribute to the average energy.
Different power laws follow for these as well as per David's comments.

Joel