Reconsidering the variational procedure for uniaxial systems modeled by
continuous free energy functionals, we derive new general conditions for
thermodynamic extrema. The utility of these conditions is briefly illustrated
on the models for the classes I and II of incommensurate-commensurate systems.Comment: 5 pages, to be published in Phys. Rev. Let

A. Bjeliš

A. Michelson

H. Z. Cummins

J. C. Tolédano

L. N. Bulaevski

R. M. Hornreich

S. Aubry

V. Dananić

V. I. Arnold

W. L. McMillan

English

arXiv

Reconsidering the variational procedure for uniaxial systems modeled by continuous free energy functionals, we derive new general conditions for thermodynamic extrema. The utility of these conditions is briefly illustrated on the models for the classes I and II of incommensurate-commensurate systems

Dananić, V.

Bjeliš, Aleksa

University of Zagreb Repository

VOLUME 80, NUMBER 1 P HY S I CA L REV I EW LE T T ER S 5 JANUARY 199810General Criteria for the Stability of Uniaxially Ordered Statesof Incommensurate-Commensurate SystemsV. Dananic´Department of Physics, Faculty of Chemical Engineering and Technology, University of Zagreb,Marulic´ev trg 19, 10000 Zagreb, CroatiaA. BjelisˇDepartment of Theoretical Physics, Faculty of Science, University of Zagreb, P.O. Box 162, 10001 Zagreb, Croatia(Received 5 February 1997; revised manuscript received 23 July 1997)Reconsidering the variational procedure for uniaxial systems modeled by continuous free energy func-tionals, we derive new general conditions for thermodynamic extrema. The utility of these conditionsis briefly illustrated on the models for the classes I and II of incommensurate-commensurate systems.[S0031-9007(97)04969-7]PACS numbers: 05.70.Ce, 64.70.Rh, 82.60.–sNumerous materials which are under intense investi-gations in the contemporary condensed matter physicsare thermodynamically one dimensional. The well-knownexamples are various uniaxial materials with incommen-surate and commensurate orderings [1] and quasi-one-dimensional conductors with charge or spin density waveinstabilities [2]. Order parameters for such systems aregenerally multicomponent, u ­ su1, u2, . . . , uN d, and de-pend on a single spatial variable x. The principal taskis then to find thermodynamically stable configurationsucsxd, those which minimize the free energy functionalF . Since the latter is the one-dimensional integral, it istempting to treat this variational problem as an equivalentto the standard classical mechanical one [3], with the rolesof time variable, vectors in the N-dimensional mechanicalconfiguration space, action functional, and Lagrangian at-tributed to x, u, F , and f, respectively, the latter beingthe free energy density.In the present Letter we do not follow this widely ac-cepted attitude, but start from two obvious, yet substantial,differences between these two variational schemes. Thefirst one is present in the very extremalization procedure.In contrast to the classical mechanical trajectories, the re-alizable solutions of the Euler-Lagrange (EL) equationsfor thermodynamic problems follow after an additionalextremalization with respect to the initial (or boundary)conditions. The second difference concerns the content ofthe free energy densities. In the most interesting modelsfor incommensurate-commensurate (IC) systems, includ-ing the basic ones, they contain either terms linear inthe first derivatives u0 ­ su01, u02, . . . , u0N d, or terms withhigher derivatives us jd ; ›j u›xj s j . 1d (or both), in con-trast to the standard mechanical Lagrangians which do notcontain analogs of such terms.Starting from the first observation, we reformulate theprocedure of thermodynamic extremalization, and derive,under assumptions specified below, the following neces-sary conditions for any thermodynamic extremum uc.0031-9007y98y80(1)y10(4)$15.00Condition A:1LZ L0"NXa­1nXj­1j us jdc,a›f›us jdc,a2 x›f›x#dx ­ 0 , (1)where n is the order of highest derivative of u presentin the free energy functional, and L is the length ofthe system taken in the thermodynamic limit L ! ‘. Inparticular, for free energy densities which do not dependexplicitly on x the condition (1) reduces to the simpleequalityFc 1 H ­ 0 , (2)where Fc is the averaged value of free energy and H is theintegral constant which has the meaning of Hamiltonianin the equivalent classical mechanical problem (but doesnot have a direct physical meaning in the thermodynamiccounterpart).Conditions B:1LZ L0nXj­0us jdc,a›f›us jdc,adx ­ 0 , (3)where a ­ 1, . . . , N .The ensuing discussion will show that in the case ofthermodynamic functionals of the standard “mechanical”form the conditions (1) and (3) are of almost trivialmeaning. They, however, have far-reaching implicationsjust in IC models, for which, as was already pointed out,free energy densities depend in more complex ways onderivatives usnd. These conditions also appear to be apowerful tool in the numerical determination of phasediagrams, particularly for systems with nonintegrable freeenergy functionals.In order to derive the conditions A and B we start fromthe general expression for the free energy functionalF ­ 1LZ L0fhusxd, u0sxd, u00sxd, . . . , usndsxd; xidx ,(4)© 1997 The American Physical SocietyVOLUME 80, NUMBER 1 P HY S I CA L REV I EW LE T T ER S 5 JANUARY 1998where f is an analytical function of its arguments,bounded from below. Each thermodynamic extremumucsxd of this functional has to obey the variational con-dition dF shucjd ­ 0, equivalent to the Hamilton varia-tional principle in classical mechanics. This necessarycondition leads to the EL equationsnXj­0s21djdjdxj›f›usjda­ 0 sa ­ 1, . . . , Nd , (5)equivalent to the Lagrange equations in classicalmechanics. The solutions of the EL Eqs. (5) forma set husx; Adj which generally depends on 2nNcontinuous parameters sa1, . . . , a2nN d ; A. Thereis a freedom in the definition of the parametersA, the most usual choices being initial conditionsfusx0d, u0sx0d, . . . , us2n21dsx0dg where x0 is an arbitraryinitial spatial position, and boundary conditions fusx1d,u0sx1d, . . . usn21dsx1d; usx2d, u0sx2d, . . . usn21dsx2dg wherex1 and x2 are arbitrary end points. In classical mechanicsthese two choices correspond to the Newton and theHamilton (variational) axiomatizations, respectively.Thermodynamic extrema, including thermodynamicallystable configurations for which d2F $ 0, are thosemembers of the set husx; Adj which extremalize the freeenergy F shusx; Adjd as a function of the parametersA. This additional property completes, together with theEL Eqs. (5), the sufficient condition for thermodynamicextrema. In particular, a configuration which fulfills theconditions dF ­ 0 and d2F $ 0 is thermodynamicallystable only if it is also a minimum in the set husx; Adj.The dependence of F shusx; Adjd on the parametersA is generally intricate. It may be at least partlynonanalytic, as is usually the case for the functionals (4)with nonintegrable EL equations [4], and in particular forthose with free energy densities f which are explicitlyx dependent. Thus, there is no efficient general wayto extract local extrema of F from the set husx; Adj.However, we can now conveniently reformulate the aboveproposition that the thermodynamic extrema follow fromthe succession of the first order variation (5) and theextremalization with respect to the parameters A, intoan equivalent, and again sufficient, requirement thatthe solutions of the EL equations are thermodynamicextrema if they are local extrema in the set husxdj ofall configurations allowed by the functional (4). By thisenlargement of the set within which we are looking forthe local extrema ucsxd, we get a freedom to choosearbitrarily (and suitably) the parameters with respectto which the set husxdj is analytic and correspondingextremalizations reduce to simple differentiations. Thisfreedom will be partly exploited here, by making twochoices of continuous parameters which will lead to theconditions A and B.The first continuous parameter is introduced in the fol-lowing way. Let us take one thermodynamic extremum,ucsxd, and define a set of functions husx; qdj byusx; qd ; ucsqxd . (6)The free energy functional (4) for this set becomes afunction of q given byF shusx; qdjd ; Fsqd ­ 1qLZ qL0fhucszd, qu0cszd, q2u00c szd, . . . , qnusndc szd; q21zidz , (7)with z ; qx and us jdc szd ; ›ujcszdy›zj . The requirementthat ucsxd is an extremum in the set husx; qdj is ex-pressed byf›Fsqdy›qgq­1 ­ 0 , (8)provided Fsqd is a smooth function of q for q > 1. Letus also take the thermodynamic limit L ! ‘ and assumethat Fsqd then does not depend on L [up to the correctionsof the order O s1yLd]. Under these assumptions, whichwill be critically examined later on, Fsqd may dependon q through only the density f in Eq. (7). The latteris an analytic function of q since it is analytic withrespect to u0, . . . , usnd by assumption. The derivative›Fsqdy›q is then well defined and the requirement (8),applied onto the function (7), gives the condition A,Eq. (1).The further simplification takes place for the function-als (4) in which the free energy density does not dependexplicitly on x. Then, as in classical mechanics, there ex-ists an integral constant (Hamiltonian),H ­ 2f 1NXa­1"nXi­1usida›f›usida2nXi­2i22Xj­0s21djusi2j21dadj11dxj11›f›usida#,(9)for each solution of the EL Eqs. (5). Using the obviousidentity H ­ 1LRL0 H dx, and the identity1LZ L0ˆuskddldxlg!dx ­1L"l21Xm­0s21dmusk1mddl2m21dxl2m21g#L01 s21dl1LZ L0‡usk1ldg·dx , (10)which follows after l successive partial integrations of the left-hand side, one reduces the expression (9) toH ­ 2Fc 21LNXa­1"nXi­2i22Xj­0jXk­0s21dj1kusi2j1k21dadj2kdxj2k›f›usida#L011LZ L0NXa­1nXj­1j us jda›f›us jdadx . (11)11VOLUME 80, NUMBER 1 P HY S I CA L REV I EW LE T T ER S 5 JANUARY 1998Here g in Eq. (10) is identified with ›fy›usida fromEq. (9), and Fc ; F shucjd. The second term on theright-hand side in Eq. (11) is negligible in the limitL ! ‘, provided ucsxd and its derivatives are finite.All thermodynamically stable extrema have this propertysince f is bounded from below. The third term vanishesfor each thermodynamic extremum due to the condition(1). The expression (11) thus reduces to the condition A,Eq. (2).The equality (2) is the consequence of the invarianceof the functional (4) with respect to translations in x,and of its noninvariance with respect to the changes ofx scale. As in classical mechanics, the former invarianceensures the existence of the integral constant H and thedegeneracy of the solutions of EL equations with respectto the choice of “initial position” x0. The number ofparameters on which the set husx; Adj explicitly dependsis then 2nN 2 1. Note that for all nontrivial functionals(4) one has N $ 1 and n $ 1, so that 2nN 2 1 $ 1. Inthe simplest nontrivial case N ­ n ­ 1 the set A hasone parameter, i.e., just H.For functionals (4) with an explicit x dependence off, the insertion of the EL Eqs. (5) into the expression(1) leads to the relation Fc ­ 2HsLd, where HsLd isgiven by the, now x dependent, expression (9) at x ­ L.Since the right-hand side in this relation depends on L,it is inconsistent with at least one of two assumptions onthe analyticity of Fsqd specified below Eq. (8). We cometo the conclusion that whenever the free energy densitydepends explicitly on x, all thermodynamic extremaare isolated nonanalytical points of the correspondingfunctional (4) with respect to changes of x scale. Thisfundamental property is the reason why the condition Adoes not hold for such functionals.Our second choice of continuous parameters from theset husxdj is defined by the scaling ua ! saua for any1 # a # N. The steps equivalent to those specified byEqs. (6)–(8) can be repeated now for each a for whichFssad is a smooth function. The corresponding conditionsf›Fssady›sagsa­1 ­ 0 (12)then reduce to the conditions B. Performing partialintegrations and inserting EL Eqs. (5) into Eq. (3), onefinally gets the conditions1L"n21Xl­0usldan2l21Xj­0s21djdjdxj›f›us j111lda#L0­ 0 , (13)which are constraints on the boundary values of the ther-modynamic extrema. Note that the boundary (“surface”)terms are the leading ones here, in contrast to the condi-tion A in which the analogous terms are only negligibleO s1yLd corrections to the finite volume terms of the orderO sL0d. Obviously, any periodic solution of EL equationssatisfies the conditions (13). To this end it suffices to takeinto account corrections of the order O s1yLd coming fromthe boundary terms in conditions A and B, in particular a12correction which adjusts the period to be a divisor of Lwith an integer ratio. No analogous adjustment for thequasiperiodic and nonperiodic solutions is apparent. Theconditions (13) are therefore expected to represent restric-tive constraints on these solutions as possible candidatesfor thermodynamic configurations.The extremalization of the thermodynamic functional(4) with respect to the parameter set A, and its non-invariance with respect to the transformations x ! qxand ua ! saua , in particular, become short of physicaljustification when transposed to its mechanical counter-part. For free energy densities which have the form ofconservative Lagrangians one has n ­ 1, and the deriva-tives u01, u02, . . . , u0N enter only through a positive definitequadratic form (“kinetic energy”). The criterion (2) thensingles out only equilibrium points (homogeneous con-figurations) u ­ cte as possible extrema. For such so-lutions the condition (8) is trivially fulfilled, since ucsqxdand the corresponding free energy Fsqd does not dependon q. The same is true for the conditions B which reduceto fuau0agL0 ­ 0.As was announced in the introduction, the utility of theconditions A and B becomes apparent for the functionals(4) which have richer dependences on the derivatives of uand allow for the x-dependent stable configurations. Forillustrations we take the basic models for the classes I andII of IC systems [1], defined by [5]f ­12sf0 2 dd2 2 V sfd (14)and [6]f ­ su00d2 2 su0d2 1 lu2 112u4 , (15)respectively.The decisive term in the model (14) is the Lifshitzinvariant df0. f is the phase variable, so that V sf 12pd ­ V sfd, the simplest choice being the sine-Gordonmodel with a single umklapp term, V sfd ~ cospf, wherep is an integer. The problem (4), (14) is entirely solvable[5,7], since the corresponding EL equation is integrableand the set A has one parameter, e.g., H. Here we showhow the condition A enables an elegant derivation and anoriginal interpretation of the solution. The condition A forthe functional (4), (14) reduces to2pd ­ Ic ;Z 2p0f0csfddf­Z 2p0q2f2Fc 1 d2y2 2 V sfdgdf . (16)The determination of the thermodynamic phase diagram,i.e., of the dependence of Fc on the control parameterspresent in the model (14), is thus reduced to the calcula-tion of the integral Ic. The relation (16) also states that fora thermodynamic extremum the corresponding mechanicalaction variable is just equal to the Lifshitz parameter d!VOLUME 80, NUMBER 1 P HY S I CA L REV I EW LE T T ER S 5 JANUARY 1998The dependence of the period P of the stable configura-tion on control parameters follows from the known rela-tion for mechanical systems with one degree of freedom,P ­ ›Icy›H [8]. Finally, the corresponding configura-tion fcsxd follows from the quadrature of the EL equationwith an already determined value of H. Thus, using theequality (2) we avoid a more tedious procedure used in theanalyses of the models (14) [5,7], namely, the entire inte-gration of the EL equation (with free H) followed by theminimization of the free energy FsHd as a function of H.Since the transformation (6) already exhausts the free-dom in the choice of variational parameters A for themodel (14), the condition B which is now given byffsf0 2 ddgL0 ­ 0, cannot be an additional constraint,but may only reproduce some already derived propertyof the extremum fc. This condition states that the con-figuration fc has a slope f0 ­ d at the points x ­0, P, 2P, . . . , NP, where P is a period and N is a large(macroscopic) integer. It indeed follows independentlyfrom the EL equation and the condition (16).Various criteria suggest [9] that the model (15) isnonintegrable due to the presence of the second derivativeof the real order parameter u. Very probably H ­su00d2 2 su0d2 2 2u0u000 2 lu2 2 12 u4 is the only integralconstant among three parameters in the set A. Thecondition A now readsZ L0f2su00c d2 2 su0cd2gdx ­ 0 . (17)Without using this condition, we have minimized numeri-cally the functional (4), (15) in the Fourier basis andshowed that the phase diagram contains an enumerableset of metastable periodic solutions with homogeneousdomains connected by sinusoidal segments [9]. The sub-sequent check [10] verifies that all these solutions satisfythe condition (17). Furthermore, by using it, one signifi-cantly facilitates the numerical calculation of (meta)stableconfigurations for the model (15). Namely, the search forlocal minima in the Fourier basis gives, as a rule, continu-ous families of periodic configurations. In order to findthe proper thermodynamic configurations within one fam-ily it suffices to determine zeros of the diagonal quadraticform of Fourier components to which the left-hand sideof Eq. (17) reduces. By this we directly confirm that theobtained configuration satisfies the EL equation and deter-mines its period. The more detailed presentation of thisprocedure for the model (15) and its various extensions isgiven elsewhere [10].Applying the condition B to the model (15) we obtainan additional constraint on the boundary points,fu0u00 2 usu0 1 u000dgL0 ­ 0 , (18)which, together with the arguments given below Eq. (13),reinforces the expectation based on the independent nu-merical analysis [9] that all thermodynamic extrema ofthe problem (15) are very probably periodic. Note thatby conditions (17) and (18) we have fixed two out ofthree parameters from the set A for the problem (15).Very probably F is not analytic for any choice of the re-maining third variational parameter, in close connectionwith the nonintegrability of the EL equation and the cor-responding chaotic structure of the portrait in the phasespace su, u0, u00, u000d.Having these and other [10] examples in mind, weconnect the limitations of the present method with thedegree of the nonintegrability of a given functional bythe following conjecture: larger is the number of missingintegral constants (in the classical mechanical sense),smaller is the number of analytic conditions for thethermodynamic extrema (like those given by conditionsA and B).In conclusion, necessary conditions for uniaxial ther-modynamic extrema are obtained from the extremalizationwith respect to space and order parameter scales. Thisprocedure proves to be feasible for the free energy densi-ties which are not explicitly dependent on the space vari-able. In particular, we show that in this case the sum ofthe averaged free energy and the integral constant (Hamil-tonian) vanishes for each thermodynamic extremum. Be-sides their general significance, the present results willbe certainly of practical use in analytical and numericalanalyses of particular models for uniaxial systems.We acknowledge discussions with M. Latkovic´. Thework is supported by the Ministry of Science and Tech-nology of the Republic of Croatia through the ProjectNo. 119201.[1] Incommensurate Phases in Dielectrics, edited by R. Blincand A. P. Levanyk (North-Holland, Amsterdam, 1986);H. Z. Cummins, Phys. Rep. 185, 211 (1990).[2] Charge Density Waves in Solids, edited by L. P. Gor’kovand G. Grüner (Elsevier, New York, 1989).[3] For example, S. Aubry, in Structures & Instabilités, editedby C. Godrèche (Les éditions de physique, Paris, 1986),p. 73; D. Escande, ibid. p. 195.[4] A. Bjelisˇ and S. Barisˇic´, Phys. Rev. Lett. 48, 684 (1982).[5] W. L. McMillan, Phys. Rev. B 14, 1496 (1976); L. N.Bulaevski and D. I. Khomski, Zh. Eksp. Teor. Fiz. 74,1863 (1978) [Sov. Phys. JETP 74, 971 (1978)].[6] R.M. Hornreich, M. Luban, and S. Shtrikman, Phys. Rev.Lett. 35, 1678 (1975); A. Michelson, Phys. Rev. B 16, 577(1977).[7] J. C. Tolédano and P. Tolédano, The Landau Theory ofPhase Transitions (World Scientific, Singapore, 1987).[8] V. I. Arnold, Mathematical Methods of Classical Mechan-ics (Springer-Verlag, Heidelberg, 1982).[9] V. Dananic´, A. Bjelisˇ, M. Rogina, and E. Coffou, Phys.Rev. A 46, 3551 (1992); V. Dananic´ and A. Bjelisˇ, Phys.Rev. E 50, 3900 (1994).[10] V. Dananic´, Ph.D. thesis, University of Zagreb, 1996;V. Dananic´; and A. Bjelisˇ (unpublished).13

General Criteria for the Stability of Uniaxially Ordered States of Incommensurate-Commensurate Systems

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General criteria for the stability of uniaxially ordered states of
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General criteria for the stability of uniaxially ordered states of Incommensurate-Commensurate Systems

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