Page - 32 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Themotionof theparticlecanbemathematicallydescribedbymeansof theEuler–Lagrangeequations, with theLagrangian L=−mc2 √ 1− v 2 c2 . Thecomponentsof the linearmomentum−→p of theparticle, inanorthonormal frameat rest in the consideredGalileanreference frame,are pi= ∂L ∂vi = mvi√ 1− v 2 c2 , therefore −→p = m −→v√ 1− v 2 c2 . Denotingby p themodulusof−→p , theHamiltonianof theparticle is H=−→p ·−→v −L= mc 2√ 1− v 2 c2 = c √ p2+m2c2 . Letusconsiderarelativisticgas,madeofNpointparticles indexedby i∈{1,. . . ,N},mibeing therestmassof the i-thparticle.With thesameassumptionsas thosemade inSection6.3.1,wecan take forHamiltonianof thegas H= c N ∑ i=1 √ pi2+m2c2 . With the samenotationsas thoseof Section6.3.1, thepartition functionPof thegas takes the value, foreachb>0, P(b)= ∫ D exp ( −bc N ∑ i=1 √ (pi)2+m2c2 ) N ∏ i=1 (d−→xid−→pi) . This integralcanbeexpressedintermsof theBessel functionK2,whoseexpression is, foreach x>0, K2(x)= x ∫ +∞ 0 exp(−xchχ)sh2χchχdχ . Wehave P(b)= ( 4πVc b )N N ∏ i=1 ( mi2K2(mibc2) ) , ρb= 1 P(b) exp ( −bc N ∑ i=1 √ pi2+mi2c2 ) . This probability density of the Gibbs state shows that the 2N stochastic vectors−→xi and−→pi are independent, thateach−→xi isuniformlydistributed in thevessel containingthegasandthat the probabilitydensityofeach−→pi isexactlytheprobabilitydistributionofthelinearmomentumofparticles inarelativisticgascalledtheMaxwell–Jüttnerdistribution, obtainedbyFerenczJüttner (1878–1958) in 1911,discussed in thebookbythe IrishmathematicianandphysicistSynge[54]. Ofcourse, the formulaegiven inProposition11allowthecalculationof the internalenergyE(b), theentropyS(b)andthepressureΠ(b)of therelativisticgas. 6.3.4. Relativistic IDealGasofMasslessParticles We have seen in the previous Chapter that in an inertial reference frame, the Hamiltonian of a relativistic point particle of rest mass m is c √ p2+m2c2, where p is the modulus of the linear momentum vector−→p of the particle in the considered reference frame. This expression 32