Relativistic equipartition via a massive damped sliding partition

Abstract

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 mi, speed vi, relativistic momentum (magnitude) pi, and (let c≡1) total energy Ei=(pi2+mi2)1/2 is considered herein. The damped slab of mass M transforms the system from its initial pi distributions (i=1,2) to a state, first, of pressure (P) equilibrium with P1=P2, but temperature T1≠T2, then, to P-T equilibrium with P1=P2 and T1=T2, given by the (1D) ‘‘first moment’’ equipartition relation (κ is Boltzmann’s constant), <q1≳=<q2≳≡κT [Eq. (A1)], where qi≡pi vi=Eivi2=pi2/Ei. In achieving first-moment equilibrium at a given κT the slab M can be taken sufficiently large, hence slab oscillation period τ sufficiently long (τ≫tmax where tmax=2Li/vmin is the round trip period of the slowest particle) to give ‘‘mechanical adiabatic invariance’’ (MAI), hence conservation of mechanical ‘‘action’’ piLi of each particle. This first-moment equilibrium is not yet ‘‘thermal’’ equilibrium, since the MAI process leaves the higher moments <qi2≳, <qi3≳, etc., with their original values, relative to <qi≳. To achieve thermal equilibrium the slab damping is turned off and slab mass M is reduced, hence τ decreases, until τ≪tmax, 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 <qin≳= nκT<qin−1≳ +(n−1)mi2κT<qin−1/Ei2≳, i=1 or 2, n=1, 2, 3, 4,... [Eq. (A2)].