RELATIVISTIC EQUIPARTITION VIA A MASSIVE DAMPED SLIDING PARTITION

A cylinder partitioned by a massive sliding slab undergoing nonrelativistic damped one-dimensional (1D) motion under bombardment from the left (i=1) and right (i=2) by particles having rest mass m(i), speed v(i), relativistic momentum (magnitude) p(i), and (let c=1) total energy E(i)=(p(i)2+m(i)2)1/2 is considered herein. The damped slab of mass M transforms the system from its initial p(i) distributions (i=1,2) to a state, first, of pressure (P) equilibrium with P1=P2, but temperature T1 not-equal T2, then, to P-T equilibrium with P1=P2 and T1=T2, given by the (1D) ''first moment'' equipartition relation (kappa is Boltzmann's constant), [q1]=[q2]=kappaT [Eq. (A1)], where q(i)=p(i) v(i)=E(i)v(i)2=p(i)2/E(i). In achieving first-moment equilibrium at a given kappaT the slab M can be taken sufficiently large, hence slab oscillation period tau sufficiently long (tau much greater than t(max) where t(max)=2L(i)/v(min) is the round trip period of the slowest particle) to give ''mechanical adiabatic invariance'' (MAI), hence conservation of mechanical ''action'' p(i)L(i) of each particle. This first-moment equilibrium is not yet ''thermal'' equilibrium, since the MAI process leaves the higher moments [q(i)2], [q(i)3], etc., with their original values, relative to [q(i)]. To achieve thermal equilibrium the slab damping is turned off and slab mass M is reduced, hence tau decreases, until tau much less than t(max), whereupon MAI becomes ''broken'' and we achieve complete thermal equilibrium, given by Eq. (A1) plus the appropriate higher moments. Using straightforward extension of the relativistic technique used by Menon and Agrawal to find the first-moment relation (A1) we find that all of the moments of q1 satisfy the recursion relation [q(i)n]=nkappaT[q(i)n-1]+(n-1)m(i)2kappaT[q(i)n-1/E(i)2], i=1 or 2, n=1, 2, 3, 4,... [Eq. (A2)].