Page - 109 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 AnythermodynamicmanifoldMbelongs to thesetof theso-calledLagrangianmanifolds inT, whichare the integral submanifoldsofdωwithmaximumdimension(n+1).Moreover,Misgauge invariant,which is impliedbyω=0. Theextensivityof theentropyfunctionS ( q1,...,qn ) is expressed bytheGibbs-DuhemrelationS= n ∑ i=1 qi ∂S ∂qi , rewrittenwithpreviousrelation n ∑ i=0 piqi=0,defininga2n+ 1-dimensionalextensivitysheet inT,where the thermodynamicmanifoldsM should lie. Considering aninfinitesimalcanonical transformation,generatedbytheHamiltonianh(q0,q1,...,qn,p0,p1,...,pn), . qi= ∂h ∂pi and . pi= ∂h ∂qi , theHamilton’sequationsaregivenbyPoissonbracket: . g={g,h}= n ∑ i=0 ∂g ∂qi ∂h ∂pi − ∂h ∂qi ∂g ∂pi (B5) Theconcavityof theentropyS ( q1,...,qn ) , as functionof theextensivevariables, expresses the stabilityofequilibriumstates. ThispropertyproducesconstraintsonthephysicalmanifoldsM in the 2n+2-dimensionalspace. Itentails theexistenceofametricstructure in then-dimensionalspaceqi relyingonthequadratic form: ds2=−d2S=− n ∑ i,j=1 ∂2S ∂qi∂qj dqidqj (B6) whichdefinesadistancebetweentwoneighboringthermodynamicstates. Asdγi= n ∑ j=1 ∂2S ∂qi∂qj dqj, then:ds2=− n ∑ i=1 dγidqi= 1 p0 n ∑ i=0 dpidqi (B7) The factor1/p0 ensuresgauge invariance. Inacontinuous transformationgeneratedbyh, themetric evolvesaccordingto: d dτ (ds2)= 1 p0 ∂h ∂q0 ds2+ 1 p0 n ∑ i,j=0 ( ∂2h ∂qi∂pj dpidpj− ∂ 2h ∂qi∂qj dqidqj ) (B8) We can observe that this gauge theory of thermodynamics is compatible with Souriau Lie groupTthermodynamics, wherewe have to consider the Souriau vector β = ⎡⎢⎣ γ1... γn ⎤⎥⎦, transformed inanewvector: pi=−p0.γi, p= ⎡⎢⎣ −p0γ1... −p0γn ⎤⎥⎦ =−p0 ·β (B9) AppendixC. Casalis-LetacAffineGroupInvarianceforNaturalExponentialFamilies Thecharacterizationof thenatural exponential familiesofRdwhicharepreservedbyagroup ofaffine transformationshasbeenexaminedbyMurielCasalis inherPh.D. [173]andherdifferent papers [172,174–178].Hermethodhasconsistedof translatingthe invariancepropertyof the family intoapropertyconcerningthemeasureswhichgenerate it, andtocharacterizesuchmeasures. LetEavectorspaceoffinitesize,E∗ itsdual. 〈θ,x〉dualitybracketwith(θ,x)∈E∗×E.μpositive RadonmeasureonE,Laplace transformis: Lμ :E∗→ [0,∞]withθ →Lμ(θ)= E e〈θ,x〉μ(dx) (C1) 109