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Microcanonical simulation of first-order phase transitions in finite volumes
Volume 155, number 4,5
PHYSICS LETTERS A
13 May 1991
Microcanonical simulation of first-order phase transitions in finite volumes ~ Frank R. Brown and Au Yegulaip Department of Physics, Columbia University, New York, NY 10027, USA Received 26 September 1990; revised manuscript received 8 February 1991; accepted for publication 6 March 1991 Communicated by A.R. Bishop
Microcanonical studies of first-order phase transitions provide essentially the same information as canonical ensemble calculations, and offer no significant advantage over the more common canonical approach. Interpretation of the S-shaped energy— temperature curve that signals a first-order transition as a “van der Waals” curve whose extrema denote “limits of metastability” and bracket an accessible “unstable phase” is unjustified; a simple model offinite-volume effects reproduces such S-shaped curves with good accuracy, as illustrated by the three-dimensional three-state Potts model.
Simulation of the microcanonical ensemble has been promoted as an attractive alternative to canonical ensemble calculations [1—31,especially in the investigation of first-order phase transitions [4—11]. Numerical studies often suffer in practice finite-volume smearing effects that can obscure the structure of a phase transition. In finite volumes microcanonical calculations produce a characteristic S-shaped energy—temperature curve in the vicinity of a firstorder transition. This S-shape is viewed as an especially attractive signal for a first-order phase transition because it can actually sharpen as the volume is made smaller. Furthermore, the interpretation of such a curve in terms of the S-shaped curves derived from the van der Waals equation of state would seem to provide information about the limits of metastability of supercooled and superheated phases, and even about a putative unstable phase inaccessible to canonical ensemble calculations [5—11]. We argue to the contrary ~: microcanonical cal~ Research supported in part by the U.S. Department of Energy. The microcanonical approach has advantages not addressed here. Microcanonical updates are sometimes less expensive [31 and can decorrelate more rapidly [12,131 than standard Canonical updates, and the microcanonical ensemble can be used in large volumes to study mixed-phase configurations that are exponentially suppressed in the canonical ensemble [71.Our analysis also does not directly address the “friction term” method ofref. [8].
252
culations provide essentially the same information as canonical calculations; signals for first-order transitions from the two methods wash out in small volumes in essentially the same way; and the van der Waals interpretation of the S-shaped curve is incorrect. In the van der Waals picture that underlies many discussions of finite-volume microcanonical simulation, an infinite-volume system is approximated as a single homogeneous state whose temperature and other properties vary smoothly with energy. If the energy—temperature curve obtained in this approximation shows an S-shape, the region of negative dT/ dE is interpreted as unphysical; in this region phase separation occurs and the homogeneous phase decays into a mixed-phase state. The van der Waals interpretation of a finite-volume simulation therefore assumes both that the physics of the phase transition can be described by a van der Waals equation of state, and that a finite volume can enforce homogeneity of the system, but still be large enough to display reasonably thermodynamic behavior. These assumptions seem unlikely to be satisfied in general, and we know of no examples where they have been convincingly tested. We show that the van der Waals picture is in fact unnecessary; simple, well-understood finite-volume effects provide a quantitative explanation of the S-shaped curve seen for first-order transitions in finite-volume simulations.
0375-960l/91/$ 03.50 © 1991
—
Elsevier Science Publishers B.V. (North-Holland)
Volume 155, number 4,5
PHYSICS LETTERS A
We discuss the relationship between microcanonical and canonical signals for first-order phase transitions, present a novel finite-volume model and describe its predictions for the shape of the energy— temperature curve, comment on the possible difference in a finite volume between “thermodynamic” and “demon-variable” definitions of temperature, and illustrate our conclusions with numerical results for the three-dimensional three-state Potts model. Microcanonical and canonical calculations sample configurations from the same density of states, w (E), but differ in the choice of weight. The measure ô(E—E0)w(E) dE defines the microcanonical ensemble, while the canonical ensemble has measure P~(E)dE_.,e_PEW(E) dE. Our analysis uses a “thermodynamic” finite-volume definition of inverse temperature, /3(E) =dS(E)/dE, where the entropy, S(E), is defined by S(E) =log w(E). The simple result that /3(E)
=
—
d log P~0(E) dE
(1)
gives the connection between the microcanonical and canonical approaches, as illustrated in fig. 1. The energy—inverse-temperature curve shown in fig. 1 a and the corresponding canonical ensemble probability, P,~(E),in fig. lb can be derived from one another via eq. (1). The S-shape in fig. la signals a firstorder transition in a microcanonical simulation, a) B
-
-l .
I 0
-
-
~ -j
L
b)
—1
I 1
0
-~
2
E •
•
while fig. lb shows the bimodal probability distribution that signals a first-order transition in a canonical simulation. Indeed, an S-shape in the sense that the energy—inverse-temperature curve crosses some value /3~three times as shown by the dashed line in fig. 1 a immediately implies bimodality of the energy histogram for the canonical ensemble at inverse temperature flu, and a volume small enough to wash out one consequently washes out the other. It is furthermore now apparent that the equal area construction in the energy—inverse-temperature plane and the requirement that the peaks in the canonical histogram have equal height define the same finitevolume approximation to the critical value of /3, as illustrated by the dashed line and choice of /1~in fig. 1. This shows the equivalence of microcanonical and canonical signals for first-order transitions for the “thermodynamic” definition of temperature that underlies eq. (1). Microcanonical temperatures, however, are normally determined from the energy distribution of demons or other auxiliary variables, and in a finite volume need not coincide with thermodynamic temperatures. Two familiar examples Creutz’s demon variable [3] and the fictitious momenta of Callaway and Rahman [1] illustrate the use of auxiliary variables to define a finite-volume temperature. As a check that the two methods are nonetheless equivalent in practice we have compared demon-derived temperatures, for which the —
—
explicit construction is given below, with eq. (1) in a three-state Potts model calculation. As discussed below, they agree quite well, from which we conelude that our equivalence argument applies to microcanonical calculations as conventionally perOur discussion of finite-volume effects begins with
A
•—~~~[ 0 F —2
13 May 1991
• .
.
Fig. 1. Typical signals for a first-order phase transition in the (a) microcanonical, and (b) canonical approach. The two methods are equivalent because these curves can be derived from one another using eq. (1).
the double-Gaussian approximation advocated by Binder and collaborators for first-order phase tran2/2o-2] sitions [14]. Let G(x;~~, c)~exp[—(x—u) and G(x;~u,a)~(2~a2)~2U(x;j~, a) denote Gaussians of unit height and weight, respectively. We model the Boltzmann distribution at inverse temperature /3~by a sum of two Gaussians with equal height. Each Gaussian approximates the energy dis. tribution of a single pure phase. From -
253
Volume 155, number 4,5
PHYSICS LETTERS A
P~0(E)—.G(E;E1,a1)+G(E;E2,a2)
13 May 1991
M(E)~N$KPG(E;Ep,ap)dp, 0
we obtain w(E) z__e~~0EP~(E), and hence /3(E)=/30
2(E—E
—
aj
2(E—E 1)G1(E)+a~ 2)G2(E) G1 (E) + G2 (E)
The curves in fig. 1 illustrate the double-Gaussian model for E1=—l, E2=1, a1=0.25, and a2=0.45. Although the double-Gaussian model produces an 5shape, it is very different from the van der Waals pieture. For example, in the double-Gaussian model the two pure phases can be taken metastable for all vatues of the temperature (or energy), or in fact absolutely stable if one imagines constructing a model with an infinite barrier between the two phases. The van der Waals interpretation of the points A and B as limits of metastability is incorrect; the locations of these extrema are determined only by how the enorgy distributions of the two pure phases overlap, rather than by any intrinsic properties of the phases taken individually. Similarly, configurations lying along the branch A—B of the energy—inverse-temperature curvewould do not correspond to an interpreunstable phase as they in the van der Waals tation; they are simply sampled separately from the overlapping high-energy tail of the low-temperature phase and low-energy tail of the high-temperature phase. We now present a novel extension of the doubleGaussian model that includes mixed-phase configurations. For simplicity we assume that the domain wall separating the coexisting phases has only two effects; it permits the two phases to coexist, and because it carries surface tension it reduces the probability of such a configuration in the canonical ensemble by a factor that decreases exponentially in the area of the domain wall. Let p be the fraction of the system in the phase characterized by the energy distribution G 1 (E), and I —p be the fraction in the phase corresponding to G2(E). For a given value of p, the system has mean total energy E~=pE1+(l—p)E2 and fluctuations in total energy given by a~= pa~+ (1 —p)a~.Integrating over p yields the mixedphase distribution:
where i~is a weighting factor that accounts for one pure phase being favored over the other and N normalizes M(E) to unit total probability. Adding mixed-phase contributions to the double-Gaussian model we obtain P~O(E)=YM(E)
+ (1 ~Y) [aG1(E)+ (1 —a)G2(E)] (2) For consistency the mixed-phase weighting factor is determined by the relative weight of the two pure phases: K=a/(l —a). Again, the logarithmic denyative in eq. (1) yields the energy—inverse-temperature curve. This model is illustrated in fig. 2. The solid curve is the analog of fig. la; the parameters describing the pure phases are the same, but mixed-phase configurations contribute a fraction of y=O.3 to the distribution. Fig. 2 also shows the results of a simple model for the volume dependence of the S-shaped curve. Letting V be the volume of the system, we the as1”2 and that sume that E1, E2-~ V; that a~, a 2—~ V cost of a domain wall grows as its area, y= exp(—cV213). (Thus energy densities are independent of volume, the fractional widths of the Gaussians decrease as V 1/2, and for simplicity we ap-
—
I
uJ
‘‘I
1
I 0
I
1
I 2
E/V Fig. 2. Volume dependence ofthe microcanonical S-shapedcurve. Dash length increases with volume; the solid curve is the thirdlargest volume. The S-shape is washed out by broader pure-phase distributions and greater mixed-phase contributions in small volumes, and by phase separation in large volumes.
254
Volume 155, number 4,5
PHYSICS LETTERS A
13 May 1991
proximate y by the exponential part of its volume dependence in a cubic volume with periodic boundary conditions.) The solid curve is assigned unit volume, and the dashed curves (in order of increasing
tice, and derives the temperature from the exponential demon-energy distribution that would thus be produced. A single spin flip produces only one of a finite set of changes in energy, MIe{c51}. Lattice con-
dash length) correspond to volumes of 1/10, 1/V/To, ~/iO,and 10. The S-shape washes out in small volumes both because the double-Gaussian distribution loses its bimodality as the fractional widths of the Gaussians increase, and because the valley of the double-Gaussian is filled in by increasingly probable mixed-phase configurations. The 5shape is eliminated by phase separation in the thermodynamic limit [5]; in large volumes it flattens over the energy range for which mixed-phase configurations dominate the tails of the pure-phase Gaussians. This straightforward finite-volume model reproduces the structure and volume dependence of the S-shaped curve without use of homogeneity arguments or a van der Waals equation of state. Note that the double-Gaussian model of Binder and collaborators fails to reproduce the qualitative volume dependence of the S-shaped curve; the mixed-phase model presented above is necessary to obtain the correct large-volume behavior. Lastly we present a calculation performed for the three-dimensional three-state Potts model, a system that undergoes a well-studied first-order transition [15]. Our purpose is three-fold: we illustrate the above arguments in a concrete model, we confirm that thermodynamic and demon-variable measurements yield consistent definitions of temperature, and we show explicitly that eq. (2) gives a good description of finite-volume effects. The model is the standard three-dimensional cubic lattice of three-state spin variables, a1, with nearest-neighbor Hamiltonian H= results ~ inand Boltz1.We present terms of mann weight energy density,e~’ E/V= (satisfied bonds)/sites. We have performed a canonical ensemble Monte Carlo calculation at /3=0.5505, very near the transition, that consists of 100000 sweeps of a multispin Metropolis update [16]. Measurements were taken every sweep. Temperatures as a function ofenergy were computed in two ways, both by numerically differentiating the energy histogram and applying eq. (1), and by measuring the temperature with a demon-like thermometer. In the latter approach one imagines bringing a demon variable in contact with each spin in the lat-
figurations are binned by energy, EL~.E/2
(UI if)
—
I
I
I —1 .8
—1 .6
0
.~‘
‘~
.
0
— ~.,,
—
6 I —2
E/V Fig. 3. Results for the three-state Potts model on a l6~lattice demonstrate the equivalence of demon-variable (squares) and thermodynamic (circles) inverse temperature measurements. The • • solid curve shows how a simple finite-volume model accounts for the S-shaped curve, without recourse to van der Waals homogeneity arguments.
255
Volume 155, number 4,5
PHYSICS LETTERS A
that our demon-like temperature measurements obtamed from energy bins of a canonical calculation differ significantly from temperatures measured with standard microcanonical-update demons. The solid curve in fig. 3 demonstrates the success with which eq. (2) models the structure of the S-shaped curve. The curve corresponds to the parameters E1 / V 1.84, E2/ V= 1.57, a1 / V= 0.0695, a2/ V= 0.0293, a =0.664, y=O.5l8 (obtained from a maximum-likelihood fit to the distribution of energy measurements), and agrees nicely with the Monte Carlo results. It is worthy of note that the relationship analogous to eq. (1) between the constant-volume (NVT) and isobaric (NpT) ensembles has been discussed by Wood [18], whose NpT simulations of the hard-disk system reproduce with some success the S-shaped curve seen in Alder and Wainwright’s molecular dynamics studies [4]. Microcanonical simulation is a valuable approach to the study of statistical systems and first-orderphase transitions, and is expected to be consistent and competitive with canonical Monte Carlo calculations. It is not, however, significantly better, nor does it provide information about first-order transitions complementary to that obtained from the canonical ensemble. The van der Waals picture that underlies many discussions of the microcanonical approach is incorrect; simple, well-understood finite-volume effects (rather than notions that finite volume enforces homogeneity and stabilizes unstable states) accurately account for the S-shaped energy—temperature curve that signals a first-order transition. Microcanonical S-shaped curves and bimodal canonical distributions both provide valid, but essentially equivalent, finite-volume evidence for first-order phase transitions There is no particular benefit in confirming with the microcanonical approach the =
—
256
—
13 May 1991
existence of a first-order transition already seen with similar volume and statistics in a canonical ensemble calculation. It is our pleasure to thank Adrianus Pruisken and Norman Christ for encouragement and discussions, and to thank the referee for drawing our attention to the work of Wood and of Alder and Wainwright.
References [1] D.J.E. Callaway and A. Rahman, Phys. Rev. Lett. 49 (1982) 613. [2] A. Strominger, Ann. Phys. (NY) 146 (1983) 419. [31M. Creutz, Phys. Rev. Lett. 50 (1983) 1411. [4] B.J. Alder and T.E. Wainwright, Phys. Rev. 127 (1962) 359.
[51U.M. Heller and N. Seiberg, Phys. Rev. D 27 (1983) 2980. [6] J. Kogut et al., NucI. Phys. B 251 [FSI3] (1985) 311. [7] R. Harris, Phys. Lett. A ill (1985) 299; Comput. Phys. 4 (1990) 314. [8] Y. Morikawa and A. Iwazaki, Phys. Lett. B 165 (1985) 361. [91DR. Stump and J.H. Hetherington, Phys. Lett. B 188 (1987) 359. [10] W.G. Wilson and C.A. Vause, Phys. Rev. B 36 (1987) 587. [11] J.B. Kogut and D.K. Sinclair, Nuci. Phys. B 295 [FS21] (1988) 465. [12] 5. Duane, NucI. Phys. B 257 [FS1 41(1985) 652. [131FR. Brown and T.J. Woch, Phys. Rev. Lett. 58 (1987) 2394. [14] K. Binder and D.P. Landau, Phys. Rev. B 30 (1984) 1477; M.S.S. Challa, D.P. Landau and K. Binder, Phys. Rev. B 34 (1986) 1841. [15] R.V. Gavai, F. Karsch and B. Petersson, Nucl. Phys. B 322 (1989) 738; M. Fukugita and M. Okawa, Phys. Rev. Lett. 63 (1989) 13, and references therein. [16] F.R. Brown, Phys. Lett. B 224 (1989) 412. [17] B. Efron, The Jackknife, the bootstrap, and other resampling 1982). 18] plans W.W. (SIAM, Wood, J.Philadelphia, Chem. Phys. 48 ( 1968) 415; 52 ( 1970) 729; in: Physics of simple liquids, eds. H.N.V. Temperley, J.S. Rowlinson and G.S. Rushbrooke (North-Holland, Amsterdam, 1968) p. 115.
PHYSICS LETTERS A
13 May 1991
Microcanonical simulation of first-order phase transitions in finite volumes ~ Frank R. Brown and Au Yegulaip Department of Physics, Columbia University, New York, NY 10027, USA Received 26 September 1990; revised manuscript received 8 February 1991; accepted for publication 6 March 1991 Communicated by A.R. Bishop
Microcanonical studies of first-order phase transitions provide essentially the same information as canonical ensemble calculations, and offer no significant advantage over the more common canonical approach. Interpretation of the S-shaped energy— temperature curve that signals a first-order transition as a “van der Waals” curve whose extrema denote “limits of metastability” and bracket an accessible “unstable phase” is unjustified; a simple model offinite-volume effects reproduces such S-shaped curves with good accuracy, as illustrated by the three-dimensional three-state Potts model.
Simulation of the microcanonical ensemble has been promoted as an attractive alternative to canonical ensemble calculations [1—31,especially in the investigation of first-order phase transitions [4—11]. Numerical studies often suffer in practice finite-volume smearing effects that can obscure the structure of a phase transition. In finite volumes microcanonical calculations produce a characteristic S-shaped energy—temperature curve in the vicinity of a firstorder transition. This S-shape is viewed as an especially attractive signal for a first-order phase transition because it can actually sharpen as the volume is made smaller. Furthermore, the interpretation of such a curve in terms of the S-shaped curves derived from the van der Waals equation of state would seem to provide information about the limits of metastability of supercooled and superheated phases, and even about a putative unstable phase inaccessible to canonical ensemble calculations [5—11]. We argue to the contrary ~: microcanonical cal~ Research supported in part by the U.S. Department of Energy. The microcanonical approach has advantages not addressed here. Microcanonical updates are sometimes less expensive [31 and can decorrelate more rapidly [12,131 than standard Canonical updates, and the microcanonical ensemble can be used in large volumes to study mixed-phase configurations that are exponentially suppressed in the canonical ensemble [71.Our analysis also does not directly address the “friction term” method ofref. [8].
252
culations provide essentially the same information as canonical calculations; signals for first-order transitions from the two methods wash out in small volumes in essentially the same way; and the van der Waals interpretation of the S-shaped curve is incorrect. In the van der Waals picture that underlies many discussions of finite-volume microcanonical simulation, an infinite-volume system is approximated as a single homogeneous state whose temperature and other properties vary smoothly with energy. If the energy—temperature curve obtained in this approximation shows an S-shape, the region of negative dT/ dE is interpreted as unphysical; in this region phase separation occurs and the homogeneous phase decays into a mixed-phase state. The van der Waals interpretation of a finite-volume simulation therefore assumes both that the physics of the phase transition can be described by a van der Waals equation of state, and that a finite volume can enforce homogeneity of the system, but still be large enough to display reasonably thermodynamic behavior. These assumptions seem unlikely to be satisfied in general, and we know of no examples where they have been convincingly tested. We show that the van der Waals picture is in fact unnecessary; simple, well-understood finite-volume effects provide a quantitative explanation of the S-shaped curve seen for first-order transitions in finite-volume simulations.
0375-960l/91/$ 03.50 © 1991
—
Elsevier Science Publishers B.V. (North-Holland)
Volume 155, number 4,5
PHYSICS LETTERS A
We discuss the relationship between microcanonical and canonical signals for first-order phase transitions, present a novel finite-volume model and describe its predictions for the shape of the energy— temperature curve, comment on the possible difference in a finite volume between “thermodynamic” and “demon-variable” definitions of temperature, and illustrate our conclusions with numerical results for the three-dimensional three-state Potts model. Microcanonical and canonical calculations sample configurations from the same density of states, w (E), but differ in the choice of weight. The measure ô(E—E0)w(E) dE defines the microcanonical ensemble, while the canonical ensemble has measure P~(E)dE_.,e_PEW(E) dE. Our analysis uses a “thermodynamic” finite-volume definition of inverse temperature, /3(E) =dS(E)/dE, where the entropy, S(E), is defined by S(E) =log w(E). The simple result that /3(E)
=
—
d log P~0(E) dE
(1)
gives the connection between the microcanonical and canonical approaches, as illustrated in fig. 1. The energy—inverse-temperature curve shown in fig. 1 a and the corresponding canonical ensemble probability, P,~(E),in fig. lb can be derived from one another via eq. (1). The S-shape in fig. la signals a firstorder transition in a microcanonical simulation, a) B
-
-l .
I 0
-
-
~ -j
L
b)
—1
I 1
0
-~
2
E •
•
while fig. lb shows the bimodal probability distribution that signals a first-order transition in a canonical simulation. Indeed, an S-shape in the sense that the energy—inverse-temperature curve crosses some value /3~three times as shown by the dashed line in fig. 1 a immediately implies bimodality of the energy histogram for the canonical ensemble at inverse temperature flu, and a volume small enough to wash out one consequently washes out the other. It is furthermore now apparent that the equal area construction in the energy—inverse-temperature plane and the requirement that the peaks in the canonical histogram have equal height define the same finitevolume approximation to the critical value of /3, as illustrated by the dashed line and choice of /1~in fig. 1. This shows the equivalence of microcanonical and canonical signals for first-order transitions for the “thermodynamic” definition of temperature that underlies eq. (1). Microcanonical temperatures, however, are normally determined from the energy distribution of demons or other auxiliary variables, and in a finite volume need not coincide with thermodynamic temperatures. Two familiar examples Creutz’s demon variable [3] and the fictitious momenta of Callaway and Rahman [1] illustrate the use of auxiliary variables to define a finite-volume temperature. As a check that the two methods are nonetheless equivalent in practice we have compared demon-derived temperatures, for which the —
—
explicit construction is given below, with eq. (1) in a three-state Potts model calculation. As discussed below, they agree quite well, from which we conelude that our equivalence argument applies to microcanonical calculations as conventionally perOur discussion of finite-volume effects begins with
A
•—~~~[ 0 F —2
13 May 1991
• .
.
Fig. 1. Typical signals for a first-order phase transition in the (a) microcanonical, and (b) canonical approach. The two methods are equivalent because these curves can be derived from one another using eq. (1).
the double-Gaussian approximation advocated by Binder and collaborators for first-order phase tran2/2o-2] sitions [14]. Let G(x;~~, c)~exp[—(x—u) and G(x;~u,a)~(2~a2)~2U(x;j~, a) denote Gaussians of unit height and weight, respectively. We model the Boltzmann distribution at inverse temperature /3~by a sum of two Gaussians with equal height. Each Gaussian approximates the energy dis. tribution of a single pure phase. From -
253
Volume 155, number 4,5
PHYSICS LETTERS A
P~0(E)—.G(E;E1,a1)+G(E;E2,a2)
13 May 1991
M(E)~N$KPG(E;Ep,ap)dp, 0
we obtain w(E) z__e~~0EP~(E), and hence /3(E)=/30
2(E—E
—
aj
2(E—E 1)G1(E)+a~ 2)G2(E) G1 (E) + G2 (E)
The curves in fig. 1 illustrate the double-Gaussian model for E1=—l, E2=1, a1=0.25, and a2=0.45. Although the double-Gaussian model produces an 5shape, it is very different from the van der Waals pieture. For example, in the double-Gaussian model the two pure phases can be taken metastable for all vatues of the temperature (or energy), or in fact absolutely stable if one imagines constructing a model with an infinite barrier between the two phases. The van der Waals interpretation of the points A and B as limits of metastability is incorrect; the locations of these extrema are determined only by how the enorgy distributions of the two pure phases overlap, rather than by any intrinsic properties of the phases taken individually. Similarly, configurations lying along the branch A—B of the energy—inverse-temperature curvewould do not correspond to an interpreunstable phase as they in the van der Waals tation; they are simply sampled separately from the overlapping high-energy tail of the low-temperature phase and low-energy tail of the high-temperature phase. We now present a novel extension of the doubleGaussian model that includes mixed-phase configurations. For simplicity we assume that the domain wall separating the coexisting phases has only two effects; it permits the two phases to coexist, and because it carries surface tension it reduces the probability of such a configuration in the canonical ensemble by a factor that decreases exponentially in the area of the domain wall. Let p be the fraction of the system in the phase characterized by the energy distribution G 1 (E), and I —p be the fraction in the phase corresponding to G2(E). For a given value of p, the system has mean total energy E~=pE1+(l—p)E2 and fluctuations in total energy given by a~= pa~+ (1 —p)a~.Integrating over p yields the mixedphase distribution:
where i~is a weighting factor that accounts for one pure phase being favored over the other and N normalizes M(E) to unit total probability. Adding mixed-phase contributions to the double-Gaussian model we obtain P~O(E)=YM(E)
+ (1 ~Y) [aG1(E)+ (1 —a)G2(E)] (2) For consistency the mixed-phase weighting factor is determined by the relative weight of the two pure phases: K=a/(l —a). Again, the logarithmic denyative in eq. (1) yields the energy—inverse-temperature curve. This model is illustrated in fig. 2. The solid curve is the analog of fig. la; the parameters describing the pure phases are the same, but mixed-phase configurations contribute a fraction of y=O.3 to the distribution. Fig. 2 also shows the results of a simple model for the volume dependence of the S-shaped curve. Letting V be the volume of the system, we the as1”2 and that sume that E1, E2-~ V; that a~, a 2—~ V cost of a domain wall grows as its area, y= exp(—cV213). (Thus energy densities are independent of volume, the fractional widths of the Gaussians decrease as V 1/2, and for simplicity we ap-
—
I
uJ
‘‘I
1
I 0
I
1
I 2
E/V Fig. 2. Volume dependence ofthe microcanonical S-shapedcurve. Dash length increases with volume; the solid curve is the thirdlargest volume. The S-shape is washed out by broader pure-phase distributions and greater mixed-phase contributions in small volumes, and by phase separation in large volumes.
254
Volume 155, number 4,5
PHYSICS LETTERS A
13 May 1991
proximate y by the exponential part of its volume dependence in a cubic volume with periodic boundary conditions.) The solid curve is assigned unit volume, and the dashed curves (in order of increasing
tice, and derives the temperature from the exponential demon-energy distribution that would thus be produced. A single spin flip produces only one of a finite set of changes in energy, MIe{c51}. Lattice con-
dash length) correspond to volumes of 1/10, 1/V/To, ~/iO,and 10. The S-shape washes out in small volumes both because the double-Gaussian distribution loses its bimodality as the fractional widths of the Gaussians increase, and because the valley of the double-Gaussian is filled in by increasingly probable mixed-phase configurations. The 5shape is eliminated by phase separation in the thermodynamic limit [5]; in large volumes it flattens over the energy range for which mixed-phase configurations dominate the tails of the pure-phase Gaussians. This straightforward finite-volume model reproduces the structure and volume dependence of the S-shaped curve without use of homogeneity arguments or a van der Waals equation of state. Note that the double-Gaussian model of Binder and collaborators fails to reproduce the qualitative volume dependence of the S-shaped curve; the mixed-phase model presented above is necessary to obtain the correct large-volume behavior. Lastly we present a calculation performed for the three-dimensional three-state Potts model, a system that undergoes a well-studied first-order transition [15]. Our purpose is three-fold: we illustrate the above arguments in a concrete model, we confirm that thermodynamic and demon-variable measurements yield consistent definitions of temperature, and we show explicitly that eq. (2) gives a good description of finite-volume effects. The model is the standard three-dimensional cubic lattice of three-state spin variables, a1, with nearest-neighbor Hamiltonian H= results ~ inand Boltz1.We present terms of mann weight energy density,e~’ E/V= (satisfied bonds)/sites. We have performed a canonical ensemble Monte Carlo calculation at /3=0.5505, very near the transition, that consists of 100000 sweeps of a multispin Metropolis update [16]. Measurements were taken every sweep. Temperatures as a function ofenergy were computed in two ways, both by numerically differentiating the energy histogram and applying eq. (1), and by measuring the temperature with a demon-like thermometer. In the latter approach one imagines bringing a demon variable in contact with each spin in the lat-
figurations are binned by energy, EL~.E/2
(UI if)
—
I
I
I —1 .8
—1 .6
0
.~‘
‘~
.
0
— ~.,,
—
6 I —2
E/V Fig. 3. Results for the three-state Potts model on a l6~lattice demonstrate the equivalence of demon-variable (squares) and thermodynamic (circles) inverse temperature measurements. The • • solid curve shows how a simple finite-volume model accounts for the S-shaped curve, without recourse to van der Waals homogeneity arguments.
255
Volume 155, number 4,5
PHYSICS LETTERS A
that our demon-like temperature measurements obtamed from energy bins of a canonical calculation differ significantly from temperatures measured with standard microcanonical-update demons. The solid curve in fig. 3 demonstrates the success with which eq. (2) models the structure of the S-shaped curve. The curve corresponds to the parameters E1 / V 1.84, E2/ V= 1.57, a1 / V= 0.0695, a2/ V= 0.0293, a =0.664, y=O.5l8 (obtained from a maximum-likelihood fit to the distribution of energy measurements), and agrees nicely with the Monte Carlo results. It is worthy of note that the relationship analogous to eq. (1) between the constant-volume (NVT) and isobaric (NpT) ensembles has been discussed by Wood [18], whose NpT simulations of the hard-disk system reproduce with some success the S-shaped curve seen in Alder and Wainwright’s molecular dynamics studies [4]. Microcanonical simulation is a valuable approach to the study of statistical systems and first-orderphase transitions, and is expected to be consistent and competitive with canonical Monte Carlo calculations. It is not, however, significantly better, nor does it provide information about first-order transitions complementary to that obtained from the canonical ensemble. The van der Waals picture that underlies many discussions of the microcanonical approach is incorrect; simple, well-understood finite-volume effects (rather than notions that finite volume enforces homogeneity and stabilizes unstable states) accurately account for the S-shaped energy—temperature curve that signals a first-order transition. Microcanonical S-shaped curves and bimodal canonical distributions both provide valid, but essentially equivalent, finite-volume evidence for first-order phase transitions There is no particular benefit in confirming with the microcanonical approach the =
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existence of a first-order transition already seen with similar volume and statistics in a canonical ensemble calculation. It is our pleasure to thank Adrianus Pruisken and Norman Christ for encouragement and discussions, and to thank the referee for drawing our attention to the work of Wood and of Alder and Wainwright.
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