Page - 108 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Thisequation is the fundamentalequationof informationgeometry. The“Legendre” transformwas introducedbyAdrien-MarieLegendre in1787 [225] to solvea minimalsurfaceproblemGaspardMonge in1784.Usingaresultof JeanBaptisteMeusnier, astudent ofMonge, it solves theproblembyachangeofvariablecorrespondingto the transformwhichnow entitledwithhisname. Legendrewrote: “I have just arrived by a change of variables that canbeuseful in other occasions.”About this transformation, Darboux [226] in his book gives an interpretation ofChasles: “This comes after a comment byMr. Chasles, to substitute its polar reciprocal on the surface compared toaparaboloïd.”TheequationofClairautwas introduced40yearsearlier in1734byAlexis Clairaut [225]. Solutions“envelopeof theClairautequation”areequivalent to theLegendre transform withunconditional convexity,butonlyunderdifferentiabilityconstraint. Indeed, foranon-convex function,Legendre transformation isnotdefinedwhere theHessianof the function iscanceled, so that the equationofClairaut onlymakes thehypothesis of differentiability. Theportionof the strictly convexfunctiong inClairautequationy=px−g(p) to the function f givingtheenvelopesolutions by the formulay= f(x) is precisely theLegendre transformation. TheapproachofFréchetmaybe reconsidered inamoregeneral contextonthebasisof theworkof Jean-LouisKoszul. AppendixB. BalianGaugeModelofThermodynamicsanditsCompliancewithSouriauModel SupportedbyIndustialgroupTOTAL(previouslyElf-Aquitaine),RogerBalianhas introduceda Gauge theoryof thermodynamics [103]andhasalsodeveloped informationgeometry instatistical physicsandquantumphysics [103,227–235]. Balianhasobservedthat theentropyS (weuseBalian notation, contrary with previous section where we use−S as neg-entropy) can be regarded as an extensive variable q0 = S ( q1,...,qn ) , with qi(i = 1,...,n), n independent quantities, usually extensive and conservative, characterizing the system. The n intensive variables γi are defined as thepartialderivatives: γi= ∂S(q1,...,qn) ∂qi (B1) Balianhas introducedanon-vanishinggaugevariable p0,without physical relevance,which multipliesall the intensivevariables,defininganewsetofvariables: pi=−p0.γi , i=1,...,n (B2) The2n+1-dimensional space is therebyextended into a 2n+2-dimensional thermodynamic spaceT spannedbythevariables pi ,qiwith i=0,1,...,n,where thephysical systemisassociatedwith an+1-dimensionalmanifoldM inT,parameterizedfor instancebythecoordinatesq1,...,qn and p0. Agauge transformationwhichchanges theextravariable p0whilekeeping the ratios pi/p0=−γi invariant isnotobservable, so thatastateof thesystemisrepresentedbyanypointofaone-dimensional ray lying in M, alongwhich thephysical variables q0,...,qn,γ1,...,γn are fixed. Then, the relation betweencontactandcanonical transformationsisadirectoutcomeofthisgaugeinvariance: thecontact structure ω˜ = dq0− n∑ i=1 γi ·dqi inn+1dimensioncanbeembedded intoa symplectic structure in 2n+2dimension,with1-form: ω= n ∑ i=0 pi ·dqi (B3) as symplectization,withgeometric interpretation in the theoryoffiberbundles. Then+1-dimensional thermodynamicmanifoldsMarecharacterizedbythevanishingof this formω=0. The1-forminduces thena symplectic structureonT: dω= n ∑ i=0 dpi∧dqi (B4) 108