Page - 27 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 6.2. ThermodynamicEquilibria andThermodynamicFunctions 6.2.1.AssumptionsMadein thisSection. Any Hamiltonian H defined on a symplectic manifold (M,ω) considered in this section will be assumed to be smooth, bounded by below and such that for any real b > 0, each one of the three functions, defined on M, z → exp(−bH(z)), z → ∣∣H(z)∣∣exp(−bH(z)) and z →( H(z) )2exp(−bH(z)) iseverywheresmaller thansomefunctiondefinedonM integrablewithrespect to theLiouvillemeasureλω. The integralswhichdefine P(b)= ∫ M exp(−bH)dλω and Eρb(H)= ∫ M Hexp(−bH)dλω thereforeconverge. Proposition11. LetHbeaHamiltoniandefinedonasymplecticmanifold (M,ω) satisfying theassumptions indicated inSection6.2.1. Foranyreal b>0 let P(b)= ∫ M exp(−bH)dλω and ρb= 1P(b) exp(−bH) be thevalueat bof thepartition functionPandtheprobabilitydensityof theGibbs statistical stateassociated tob, and E(b)=Eρb(H)= 1 P(b) ∫ M Hexp(−bH)dλω be themeanvalueofHwithrespect to theprobabilitydensityρb. Thefirst andsecondderivativeswith respect to b of thepartition functionPexist, are continuous functionsof bgivenby dP(b) db =−P(b)E(b) , d 2P(b) db2 = ∫ M H2exp(−bH)dλω=P(b)Eρb(H2) . Thederivativewith respect to b of the functionEexists and is a continuous functionof bgivenby dE(b) db =− 1 P(b) ∫ M ( H−Eρb(H) )2dλω=−Eρb((H−Eρb(H))2) . LetS(b)be the entropy s(ρb)of theGibbs statistical state associated to b. The functionScanbe expressed in termsofPandEas S(b)= log ( P(b) ) +bE(b) . Itsderivativewith respect to b exists and is a continuous functionof bgivenby dS(b) db = b dE(b) db . Proof. UsingtheassumptionsSection6.2.1,wesee that the functionsb →P(b)andb →Eρb(H)=E(b), definedbyintegralsonM,haveaderivativewithrespect tobwhich iscontinuousandwhichcanbe calculatedbyderivationunder the sign ∫ M . The indicated results easily follow, ifweobserve that for any function f on M such that Eρb(f) and Eρb(f2) exist, we have the formula,well known in Probability theory, Eρb(f2)− (Eρb(f))2=Eρb((f−Eρb(f))2) . 6.2.2. PhysicalMeaningof the IntroducedFunctions Let us consider aphysical system, for example agas contained in avessel boundedby rigid, thermally insulatedwalls, at rest in aGalilean reference frame. Weassume that its evolution can 27