Microcanonical phase transitions in small systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Microcanonical phase transitions in small systems
Michele Campisi ∗ Department of Physics,University of North Texas Denton, TX 76203-1427, U.S.A. (Dated: November 22, 2018)When studying the thermodynamic properties of mesoscopic systems the most appropriate mi-crocanonical entropy is the volume entropy, i.e. the logarithm of the volume of phase space enclosedby the hypersurface of constant energy. For systems with broken ergodicity, the volume entropy hasdiscontinuous jumps at values of energy that correspond to separatrix trajectories. Simultaneouslythere is a convex intruder in the entropy function and a region of negative specific heat below suchcritical energies. We illustrate this with a simple model composed of a chain of 3 particles whichinteract via a Lennard-Jones potential.
PACS numbers: 05.20.-y; 05.70.Fh; 05.70.Ce
I. INTRODUCTION
The work of D.H.E. Gross [1] has recently pointed outthe attention on the fact that a microcanonical descrip-tion of systems which may display phase transitions isin general more adequate than the traditional canoni-cal one. This is because the canonical description may“smear out” important information contained in the mi-crocanonical description which is richer [1]. For exam-ple negative specific heats, which have recently been ob-served experimentally in mesoscopic systems [2, 3, 4],can be accounted for in the microcanonical ensemble butnot in the canonical one [5]. Indeed it is well knownthat canonical ensemble and microcanonical ensemble arenot in general equivalent, even when the thermodynamiclimit is considered [1, 6, 7, 8].The statistical mechanical analysis of physical sys-tems based on canonical ensemble is quite well estab-lished and universally agreed upon. Roughly speakingone has to compute the partition function Z ( β ) and de-rive the thermodynamics of the system from the free en-ergy F = − β − ln Z ( β ). Things are not quite so broadlyagreed upon in the case of the microcanonical ensem-ble. In fact since the pioneering works of Boltzmann andGibbs two possibilities were given for the microcanon-ical analysis of physical systems, which correspond totwo different definitions of entropy (see for example thetextbook of Gibbs [9] or the more recent textbook ofHuang [10]). Following [11] and [12] we shall refer tothese two entropies as “surface entropy” and “volumeentropy”. The surface entropy is defined as:[34] S Ω ( E ) = ln Ω( E ) (1)where Ω( E ) = Z d z δ ( E − H ( z )) (2)with δ ( x ) denoting Dirac delta function. Sometimes this ∗ Electronic address: [email protected] is also referred to as Boltzmann entropy or Boltzmann-Planck entropy. The volume entropy is defined as: S Φ ( E ) = ln Φ( E ) (3)where Φ( E ) = Z d z θ ( E − H ( z )) (4)with θ ( x ) = R x −∞ dyδ ( y ) denoting Heaviside step func-tion. The quantities Ω and Φ are related through thedifferential equation [13]:Φ ′ ( E ) = Ω( E ) (5)where the prime symbol denotes derivation with respectto E . These entropies are named surface entropy andvolume entropy because they are calculated as the loga-rithm of the area of the hyper-surface of constant energyin phase space and the volume of phase space that itencloses respectively.With reference to the literature about microcanonicalphase transitions the surface entropy is certainly the mostpopular. For example Barr´e et. al. [7] method basedon large deviation techniques uses the surface entropy.The surface entropy is used also in Rugh’s microcanoni-cal formalism adopted in Ref. [14]. The strongest advo-cate of surface entropy is perhaps Gross [15]. Nonethe-less pioneers of microcanonical phase transitions, such asThirring [16] and Lynden-Bell [5], used the volume en-tropy.The two entropies coincide in the thermodynamic limitbut when the number of degrees of freedom of the sys-tem under study is small relevant differences may appear,therefore it is necessary to choose properly. Some au-thors [12, 17] have already pointed out that the surfaceentropy is not adequate when dealing with small systemsbecause it does not account properly for finite-size ef-fects. On the other hand there is a number of theoreticalreasons to prefer the volume entropy when the number ofdegrees of freedom becomes small. Here we shall reviewthese reasons and we will illustrate the employment ofthe volume entropy in the study of phase transition witha small Lennard-Jones chain which displays a region ofnegative heat capacity. II. WHY VOLUME ENTROPY
In this section we will summarize some old and recentresults concerning the volume entropy. These results in-dicate that, no matter the number of degrees of freedomof the system under study, the volume entropy alwaysprovides a good mechanical analogue of thermodynamicentropy. For example Helmholtz [18] proved that thelogarithm of the area enclosed in phase space by the tra-jectory of a 1-dimensional system (i.e., the 1D volumeentropy S Φ ) provides a mechanical analogue of physicalentropy in the sense that if one considers the quantities P . = h ∂H∂V i t and T . = h K i t where K is the kinetic energy, V is an external parameter on which the Hamiltonian de-pends and h·i t denotes time average, then, dE + P dVT = exact differential = dS Φ (6)This result, known as Helmholtz Theorem [6, 11], saysthat the volume entropy is a good mechanical analogueof thermodynamic entropy in the sense that it repro-duces exactly the fundamental law of thermodynamics(i.e. the heat theorem (6)). The Helmholtz Theoremhas been recently generalized to multi-dimensional er-godic (i.e. metrically indecomposable) systems (see Ref.[11] or Ref. [19] for a different but equivalent approach).The resulting Generalized Helmholtz Theorem essentiallystates that the volume entropy reproduces the heat theo-rem no matter the number of degrees of freedom N . Thesame cannot be said about surface entropy which hasbeen proved to reproduce it only up to corrections of theorder O (1 /N ) [6]. As a matter of fact Gibbs presents thevolume entropy in his celebrated Principles of Statisti-cal Mechanics [9] as the entropy that naturally satisfiesthe fundamental principle of thermodynamics (that is theheat theorem) [20].Hertz [21] pointed out that the volume entropy isan adiabatic invariant already in 1910, and based hisapproach to statistical mechanics on it. Among thetextbooks that adopt the same approach, those ofM¨unster [22], Becker [23], and the more recent book ofBerdichevsky [24] are worth mentioning. Adiabatic in-variance is another good property of volume entropy be-cause it reproduces quite well Clausius’ requirement that“For every quasi static process in a thermally isolatedsystem which begins and ends in an equilibrium state,the entropy of the final state is equal to that of the ini-tial state” [25]. Of course the surface entropy is not anadiabatic invariant, although it becomes approximatelysuch as the number of degrees of freedom increases [11].Very recently it has been also proved that non-adiabatic transformations occurring in isolated systemswhich are initially in a state of thermal equilibrium al-ways result in an increase of the expectation value of thevolume entropy [26, 27]. This result too does hold nomatter the number of degrees of freedom N and cannotbe proved in general for surface entropy. Thus the volumeentropy explains quite satisfactorily also Clausius’ law of entropy increase “For every non quasi static process ina thermally isolated system which begins and ends in anequilibrium state, the entropy of the final state is greaterthan that of the initial state” [25].Recently more and more authors are becoming awareof the theoretical value of volume entropy. For example,on the basis of a Laplace transform technique for the mi-crocanonical ensemble, Pearson et. al. [28] reached theconclusion that the volume entropy “is the most correctdefinition for the entropy, even though it is unimportantfor any explicit numerical calculation”, meaning that inthe thermodynamic limit the difference with surface en-tropy becomes negligible. On the other hand for smallsystem, such intrinsic correctness of the volume entropybecomes very important. Adib [12] argues that the fi-nite size corrections to surface entropy found in Ref. [29]would be unneeded if the volume entropy were used in-stead.It is worth mentioning that the volume entropy hasanother property that is particularly important for smallsystems which have negative heat capacity, namely it isa naturally nonextensive entropy. According to Lynden-Bell [5], in fact, systems with negative heat capacity arenecessarily nonextensive . The property of nonextensivityof volume entropy follows directly from the compositionrule of enclosed volumes, Φ i i = 1 , E = E + E [13]:Φ( E ) = Z dE Φ ′ ( E )Φ ( E − E ) (7)which is not a simple multiplication but a form of convo-lution which accounts for all possible partitions of ener-gies between the two systems. It has to be stressed that,despite of what is often stated in literature, the surfaceentropy is nonextensive too, as the composition rule forsurface integrals Ω i is the convolution, not the multipli-cation [13]:Ω( E ) = Z dE Ω ( E )Ω ( E − E ) (8)In sum, the volume entropy accounts for certain basicprinciples of thermodynamics, like the heat theorem andClausius formulation of the second law equally well forlarge and small systems, whereas the surface entropy ac-counts for them only in the case of large system. For thisreason it is the most appropriate mechanical analogueof thermodynamic entropy when dealing with small sys-tems. III. LENNARD-JONES CHAIN
According to the Helmholtz Theorem [11] the mechan-ical analogue of thermodynamic entropy of a one dimen-sional system is S Φ ( E, V ) = log 2 Z x + ( E,V ) x − ( E,V ) dx p m ( E − ϕ ( x, V )) (9) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1−0.8−0.6−0.4−0.200.20.40.60.81 x p FIG. 1: Phase space structure for a particle of mass m = 1in a Lennard-Jones box of size V = 4 > V c . The separatrixcorresponds to the critical energy E c = − . where x ± ( E, V ) denote the turning points of the tra-jectory. If the potential is such that there is only onetrajectory per energy level (ergodicity), then S Φ satis-fies Eq. (6) [6, 11]. Nonetheless, if the system has morethan one trajectory per energy level for a certain rangeof energies and the system is on one of them, still wecan use the above formula and say that the heat theoremis satisfied as long as the energies considered lye withinthat energy range. In this case P and T would be calcu-lated as time averages over the actual trajectory and S Φ would be given by the area enclosed by that trajectoryonly. Let us illustrate this with a practical example. Letus consider a 1D chain composed of three particles whichinteract via a Lennard-Jones potential. Let us fix theposition of two of them and let us place the third one inbetween, so that the first two particles act as walls of a1D box. Let us now study the behavior of the particleinside the box. Let the interaction potential be: u ( x ) = 1 x − x (10)and let us place the “walls” at x = ± V /
2. Then theparticle in the box is subject to the following potential: ϕ ( x, V ) = u ( x + V /
2) + u ( x − V /
2) (11)For values of V larger than a certain critical value V c ≃ .
5, this system has a critical energy E c ( V ) = φ (0 , V )such that for energy below E c ( V ) ergodicity is brokenand there are two trajectories per energy level. Above E c the dynamics is ergodic and there is only one trajectoryper energy level. Figure 1 shows a contour plot of variousenergy levels in phase space for a particle of mass m = 1in the Lennard-Jones box of size V = 4 > V c . Forenergy E = E c we have a separatrix. Below E c the curveof constant energy splits into two disconnected curves,whereas for values of E larger than E c we have only onecurve. Below the critical energy the volume integral Φis given by the area enclosed by one of the two possibletrajectories. As the energy crosses the critical value the −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−3.5−3−2.5−2−1.5−1−0.500.5 energy en t r op y FIG. 2: Entropy versus Energy for a particle of mass m = 1in a Lennard-Jones box of size V = 4 > V c . The discontinuityof Entropy at the critical energy signals a discontinuous phasetransition. −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.1500.050.10.150.20.250.30.350.40.450.5 energy T e m pe r a t u r e FIG. 3: Temperature versus Energy for a particle in of mass m = 1 in a Lennard-Jones box of size V = 4 > V c . The curvedisplays a region of negative specific heat. At the criticalenergy the temperature goes to zero. integral Φ jumps discontinuously. In formulae we have:Φ( E ) = (cid:20) θ ( E c − E ) + θ ( E − E c ) (cid:21) × Z [2 m ( E − ϕ ( x, V ))] / dx (12)The symbol [ y ] / denotes a function that is equal to √ y for y ≥ S as a function of E , forthe values m = 1 and V = 4 > V c . The critical energy is E c = − . T . = h K i t = Φ( E )Ω( E ) = (cid:18) ∂S Φ ∂E (cid:19) − (13)There is a region of negative slope in the graph whichcorrespond to a negative heat capacity . −0.1 −0.09 −0.08 −0.07 −0.06 −0.05 −0.04−2.2−2.1−2−1.9−1.8−1.7−1.6−1.5−1.4 energy en t r op y FIG. 4: Entropy versus Energy for
E < E c . Right below thecritical energy the entropy function (thick line) is convex (thethin straight line is only a guide for the eye). IV. DISCUSSION
The example provided in the previous section is per-haps too simple to be of interest to any specific physi-cal problem. Nonetheless it illustrates qualitatively themechanism of microcanonical phase transition as cap-tured by the volume entropy. Such phase transitionsare associated with the crossing of separatrix trajecto-ries, for which the dynamics of the system has no finitetime scale. The figures show neatly that at the separatrixenergy the entropy has a discontinuous jump, the tem-perature goes to zero, and for energies below the criticalenergy we have a region of negative heat capacity. Theseare not specific features of the system studied[36]. When-ever a separatrix is crossed there is a sudden open-up ofa larger portion of phase space for the trajectory to en-close which leads to a discontinuity in the entropy[37].Further, at the separatrix, the period of motion, whichfor a well known theorem of classical mechanics is given by Φ ′ = Ω [30], becomes infinite. Therefore the temper-ature, i.e. , T = Φ / Ω goes to zero. Since the temperature
T . = 2 < K > t = < p /m > t is definite positive, belowthe critical energy there necessarily is a region of nega-tive slope, that is negative heat capacity. The appearanceof a negative heat capacity is associated with a convexintruder in the entropy (see Fig. 4) which signals theapproach to the separatrix from below. It is importantto notice that using the surface entropy would lead toa drastically different result. In this case the tempera-ture would be calculated as T Ω = Ω / Ω ′ , which might nottend to zero at the critical energy! Note also that T Ω isnot proportional to the average kinetic energy and canbe negative. Therefore, in agreement with Ref. [17] webelieve that surface entropy is not suited for low dimen-sional systems with broken ergodicity.The volume entropy could be used to address micro-canonical phase transitions in small dimensional systemswith either long or short range interactions, like the φ model, chains of particles interacting via Lennard-Jonespotential [31] or the Hamiltonian Mean Field model [32].All these models are expected to undergo a breaking ofergodicity [31, 33], thus there are separatrix trajectoriesand possible phase transitions that the volume entropycan detect.The advantage of using the volume entropy is that itprovides a good mechanical analogue of thermodynamicentropy even for small system, thus accounting properlyfor the finite-size effects. As the development of technol-ogy is allowing experimentalists to probe the thermody-namic behavior of smaller and smallers systems, this isbecoming an increasingly important task. The main lim-itation of the present approach is that it is restricted toclassical statistical mechanics, thus it does not accountfor quantum-mechanical phenomena. [1] D. H. E. Gross, Physics Reports , 119 (1997).[2] M. Schmidt, R. Kusche, T. Hippler, J. Donges,W. Kronm¨uller, B. von Issendorff, and H. Haberland,Phys. Rev. Lett. , 1191 (2001).[3] F. Gobet, B. Farizon, M. Farizon, M. J. Gaillard, J. P.Buchet, M. Carr´e, and T. D. M¨ark, Phys. Rev. Lett. ,203401 (2001).[4] F. Gobet, B. Farizon, M. Farizon, M. J. Gaillard, J. P.Buchet, M. Carr´e, P. Scheier, and T. D. M¨ark, Phys.Rev. Lett. , 183403 (2002).[5] D. Lynden-Bell, Physica A , 293 (1999).[6] G. Gallavotti, Statistical mechanics. A short treatise (Springer Verlag, Berlin, 1995).[7] J. Barre, Journal of Statistical Physics , 677 (May2005).[8] H. Touchette, R. S. Ellis, and B. Turkington, Physica A , 138 (2004).[9] J. Gibbs,
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