Comment on Is Tsallis Thermodynamics Nonextensive by E. Vives and A. Planes [cond-mat010642

Comment on Is Tsallis Thermodynamics Nonextensive by E. Vives and A. Planes [cond-mat0106428]

Commenton“IsTsallisThermodynamicsNonextensive?”byE.VivesandA.Planes

[cond-mat/0106428]

VladimirGarc´ a-Morales andJavierCervera

DepartmentofThermodynamics,UniversityofValencia,E-46100Burjassot,Spain

Oct 2003

Wecommentonletter“IsTsallisThermodynamicsNonextensive?”byE.VivesandA.Planes

[Phys.Rev.Lett.88,020601(2002)cond-mat/0106428].ItispointedoutthattheEulerandGibbs-DuhemequationsderivedinthelettercanservetojustifyanappropriateformfortheLa-grangeparameterscontrollingthermalequilibrium,withoutneedofanychangeofvariables.ThisleadstoaframeworkforTsallisThermodynamicswhichisfreefromrecentcriticismsraisedbyNauenberg[Phys.Rev.E67,036114(2003)cond-mat/0210561]andGross[Physica(Amsterdam)305,99(2002)cond-mat/0106496].ThisisaccomplishedthroughadirectconnectionwithHill’s82Nanothermodynamics.

]PACSnumbers:05.70.Ln,05.20.Gg,05.40.-a

chemInarecentletter[1]VivesandPlanes(VP)discussTsallisThermodynamics(TT)derivingageneralization-toftheGibbs-Duhem(GD)equationwhichreducestotheattraditionalonewhentheentropicparameterqtendstosunity.Theauthorssuggestachangeofvariablesthat.taallowstorecoverstandardThermodynamics,proposingmexpressionsfortheLagrangeparameters(LPs)whichare-supposedtobethosecontrollingmutualequilibriumbe-dtweenthermodynamicsystems.AlthoughtheapproachnleadingtoEq.(9)in[1]iscorrectandinsightful,theochangeofvariablesandsubsequentanalysisisunnec-c[essaryandleadstocontroversialanalyticalexpressions fortheLPsthathavebeenthekernelofthemaincrit-1vicismsraisedagainstTT[2].Wepointoutherethat2Eqs.(4),(8)and(9)in[1]canserveasabasistojus-4tifythattheLPsyαin[1]arethephysicallymeaningful6onescontrarilytowhatissuggestedbytheauthorsand0otherpractitionersinthe eld(seeRefs.[12]and[13]1in[1]).Thisleadsustoestablishequilibriumproperties30forTTovercomingpreviousdi culties[2].Theyαcan/beviewedastheLPscontrolling(nano)thermodynamictaequilibriumand,contrarilytowhatisclaimedbythemauthors,thesearethenintensivevariables.Toproceed-furtherletusintroduceHill’sformalismofNanothermo-ddynamics(NT)[3,4].Hill’sNTisarigorousextensionnofstandardthermodynamicstosystemsthatexperiencecoequilibrium uctuationsofarbitrarystrenght(“smallsys-:vtems”)inwhichallquantitiesinvolvedhaveaclearphys-iicalmeaning.WeprovidenextaconnectionbetweenTTXandNT.Itisinterestingtonotethatthe(entropic)EulerrandGDequationsinNT(seeEqs.(1-72)and(1-75)inaRef.[4])are,respectively

(rewrittenhereinVPnotation)

yα,H Xα H=S J(1)

α

Xα Hdyα,H=dJ

(2)

α

SubindexHmeans“Hill’svariables”whicharethephys-ical(averaged)extensive( Xα )andintensive(yα)ones.

SisthephysicalentropyforonesystemandJisthe

subdivision(entropic)potential.Jisanintensivevari-ableandthenumberofsystemsλisitsconjugateexten-siveone(inourcase,asin[1],wearedealingwithonly

onesystemandλ=1isimplicitlyconsideredintheseequations).Bydi erentiatingEq.(1)andusingEq.(2)itcanbeseenthattheentropyfor onesystemSsatis esthedi erentialequationdS=αyα,Hd Xα formallyidenticaltoEq.(4)in[1].ItisclearHwhichisthatthestructureofVPequationsandthoseofHillisthesameandtheformercanbeobtainedfromthelatterifS≡S ,yα,H≡yα, Xα H≡ Xα qandthefollowingcorrespon-denceismade

J≡S

[1+(1 q)S

]

ln[1+(1 q)S ]

1 q[1+(1 qA

A)SA ]ln[1+(1 qA)SA

]

=SB

1

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