Clusters in the three-dimensional Ising model rigorously obey reducibility
and thermal scaling up to the critical temperature. The barriers extracted from
Arrhenius plots depend on the cluster size as $B \propto A^{\sigma}$ where
$\sigma$ is a critical exponent relating the cluster size to the cluster
surface. All the Arrhenius plots collapse into a single Fisher-like scaling
function indicating liquid-vapor-like phase coexistence and the univariant
equilibrium between percolating clusters and finite clusters. The compelling
similarity with nuclear multifragmentation is discussed.Comment: (4 pages, 4 figures

A. Chappars

C. M. Mader

C. S. Kiang

D. De Meo

D. Stauffer

F. Gulminelli

G. J. Wozniak

H. Gould

J. B. Elliott

J. Pan

K. Okunishi

L. G. Moretto

L. Phair

M. E. Fisher

S. K. Samaddar

W. Bauer

English

arXiv

Clusters in the three-dimensional Ising model rigorously obey reducibility and thermal  scaling up to the critical temperature. The barriers extracted from Arrhenius plots depend on the  cluster size as $B \propto A^ sigma$ wqwhere $\ sigma$ is a critical exponent relating the cluster  size to the cluster surface. All the Arrhenius plots collapse into a single Fisher-like scaling  function indicating liquid-vapor-like phase coexistence and the univarian equilibrium between  percolating clusters and finite clusters. The compelling similarity with nuclear  multifragmentation is discussed

Mader, Catherine M.

Chappars, Amber

Elliott, James B.

Moretto, Luciano G.

Phair, Larry

Wozniak, Gordon J.

eScholarship - University of California

Lawrence Berkeley National LaboratoryLawrence Berkeley National LaboratoryTitleThe three-dimensional Ising model: A paradigm of liquid-vapor coexistence in nuclear multifragmentationPermalinkhttps://escholarship.org/uc/item/5gj4n3btAuthorsMader, Catherine M.Chappars, AmberElliott, James B.et al.Publication Date2001-03-05eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaThe three-dimensional Ising model: A paradigm of liquid-vapor coexistence in nuclearmultifragmentationC. M. Mader1, A. Chappars2Hope College, Holland, MI 49423J. B. Elliott, L. G. Moretto, L. Phair and G. J. WozniakNuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720(March 5, 2001)Clusters in the three-dimensional Ising model rigorouslyobey reducibility and thermal scaling up to the critical tem-perature. The barriers extracted from Arrhenius plots dependon the cluster size as B ∝ Aσ where σ is a critical exponent re-lating the cluster size to the cluster surface. All the Arrheniusplots collapse into a single Fisher-like scaling function indi-cating liquid-vapor-like phase coexistence and the univariantequilibrium between percolating clusters and finite clusters.The compelling similarity with nuclear multifragmentation isdiscussed.Nuclear multifragmentation is a process occurring atthe limits of nuclear excitation, and, as such, portraysan appropriate richness and complexity. While the fun-damental problem of dynamics vs. statistics is still de-bated, it appears ever more clearly that many ther-mal/statistical features underlie the empirical body ofdata. In particular, two features associated with the frag-ment multiplicities are found to be quite pervasive in allmultifragmentation reactions. They have been named“reducibility” and “thermal scaling” [1–3].Reducibility is the property of the n-fragment emissionprobability of being expressible in terms of an elementaryone-fragment emission probability. This property can oc-cur only if fragments are created independently from oneanother and it coincides with stochasticity. Both bino-mial, and its limiting form, Poissonian reducibilities havebeen extensively documented experimentally [1–3].Thermal scaling is the linear dependence of the loga-rithm of the one-fragment probability with 1/T (an Ar-rhenius plot). It indicates that the emission probabilityfor fragment type i has a Boltzmann dependencepi = p0e−Bi/T (1)where Bi is a barrier corresponding to the emission pro-cess.The combination of these two empirical features power-fully attests to a statistical mechanism of multifragmen-tation in general, and to liquid-vapor coexistence specif-ically [4].Many statistical models have been proposed as an ex-planation for multifragmentation. It is our intention hereto identify a model which, on one hand, is as simple aspossible, yet on the other captures the essential featuresobserved in the experiments. Percolation in its manyversions has been widely used [5–8]. However, while be-ing simple, it does not lend itself to a non-trivial ther-mal study [9]. The three-dimensional Ising model sat-isfies both the criteria of simplicity in its Hamiltonianand lends itself to a thermal treatment with nontrivialresults. While the Ising model has been widely studiedin terms of its continuous phase transition, the problemof clusterization, of paramount interest to us here, hasreceived relatively little attention [10–19]. We will showthat this model contains both features of reducibility andthermal scaling. In showing the features of thermal scal-ing we will demonstrate that the slopes of the Arrheniusplots associated with the individual masses of the frag-ments, or the barriers in Eq. (1), portray a dependenceon the fragment mass (A) of the form B ∝ Aσ, whereσ is a proper critical exponent which relates the mass tothe cluster surface. In addition, the individual Arrheniusplots for each fragment mass can be absorbed into a sin-gle scaling function identical to that of Fisher’s dropletmodel [21–25], which defines the liquid-vapor coexistenceline up to the critical temperature.The Hamiltonian of the Ising model has two terms:the interaction between nearest neighbor (n.n.) spins ina fixed lattice and the interaction between the fixed spinsand an external applied field Hext:H = −J∑i,j={n.n.}sisj −Hext∑isi (2)where J is the strength of the spin-spin interaction. Inthis model, the two phases of the system are a liquid(clusters of sites with connected up spins) and a gas(clusters of sites with connected down spins). In theabsence of an external field, the distinction between upand down spins vanishes. Thus in the absence of an ex-ternal field, these two phases are in coexistence belowthe critical temperature, Tc. The critical temperaturefor the three-dimensional Ising model has not been de-termined analytically; however, Monte Carlo techniqueshave yielded a value of Tc = 4.513 J/kb [13].In the present study, Monte Carlo techniques are usedto determine the equilibrium cluster distribution as afunction of temperature for a simple cubic lattice withperiodic boundary conditions. The lattice contains 503spins, which is large but may not be large enough to1FIG. 1. Ratio of the variance to the mean for distributionsof clusters with A constituents. The dashed line correspondsto the critical temperature, kbTc/J = 4.513.represent an infinite system. The Swendsen-Wang algo-rithm [16] was used to determine the equilibrium spinconfigurations of the lattice for given temperatures, andclusters were identified using the Coniglio-Klein [12] pre-scription. Details about the model can be found in ref-erence [20].To test if the fragment distributions produced by themodel exhibit binomial or Poissonian characteristics, theratio of the variance to the mean (σ2/ 〈nA〉) of the frag-ment distributions (nA) for various fragment sizes (A)as a function of temperature was determined. For mostclusters, the ratio is very close to one, which is the Poisso-nian limit, over the entire temperature range consideredhere, which extends well beyond the critical tempera-ture. Only for the very lightest clusters (A <∼ 20), doesthe ratio have a discontinuity at the critical tempera-ture. However, the ratio is near one even for these lightclusters at all temperatures away from the critical tem-perature. (See Fig. 1.) Thus, for all but the smallestclusters, the distributions exhibit Poissonian reducibil-ity. This signifies that the clusters are produced nearlyindependently of one another, a feature also observed inpercolation models, Fisher’s droplet model and nuclearfragmentation [4].If the fragment distributions also exhibit thermal re-ducibility, the distributions must be of the form given inEq. (1). Thus in an Arrhenius plot (a semi-log graph ofthe number of clusters of size A (nA) vs. 1/T ), the distri-butions should be linear below the critical temperature.As shown in Fig. 2, this is indeed the case over a widerange of temperatures (0 < T < Tc) and fragment sizes.While we have shown distributions for clusters up to sizeA = 100, the trend continues for larger clusters, howeverstatistics decrease significantly as the size of the clusterincreases. This linearity extends over more than four orfive orders of magnitude. It rigorously confirms the formof Eq. (1) and signifies the independent thermal forma-tion of fragments controlled by a single size-dependentbarrier. This feature has been amply verified in nuclearmultifragmentation. By fitting the fragment distribu-tions below the critical temperature (J/kbT > 0.2216),the barriers can be extracted. The barriers for each clus-ter size are shown in Fig. 3.These barriers should find their origin in the number ofbroken bonds associated with a cluster. Therefore theyshould be well described by a power-law:B(A) = B0Aσ. (3)The fit of the extracted barriers according to Eq. (3) isB = (12.77±0.04)A(0.7230±0.008) and is remarkably goodas shown in Fig. 3. The value for σ (0.7230 ± 0.0004)is close to 2/3, the value one would expect for sphericalclusters of closely packed spherical objects [15], and to0.64 the value expected from scaling relations [11]. Thusthis is a new and effective method to determine the crit-ical exponent σ, never used before to our knowledge.The features of reducibility and thermal scaling dis-cussed above can be found united in Fisher’s formula forthe cluster abundance in a vapor as a function of clustersize and of temperature. The formula isnA(T ) = q0A−τ exp(A∆µT) exp(c0AσTc) exp(−c0AσT) (4)where one can see the thermal scaling up to Tc and thedependence of the “barrier” on the cluster mass throughthe critical exponent σ [21,22]. It is interesting to ex-plore further the applicability of this formula to the Isingmodel.In addition to the linear behavior of the Arrheniusplots below the critical temperature, the Fisher dropletmodel also predicts that the cluster size distribution atthe critical point must follow a power lawnA(Tc) = q0A−τ (5)where q0 is fixed by the normalization relationshipq0 =∑∞A=1 nA(Tc)∑∞A=1A−τ . (6)Away from the critical temperature, the cluster distri-bution should not follow a pure power law. Thus, todetermine τ without a prior knowledge of the criticaltemperature, linear fits to the cluster distributions weredetermined for all temperatures. At the critical tem-perature, the fit should have the lowest χ2/DoF andthus this fit should determine τ . The results for thebest fit are shown in the lower panel of Fig. 3. Fromthis method, the critical temperature was found to be(4.515 ± .010)J/kB with a best fit line of the formnA(Tc) = (30084 ± 7)A(−2.39671±0.00017). This temper-ature is consistent with the value determined for infi-nite systems (4.513J/kB) and the value of τ (2.39671±2FIG. 2. Arrhenius plots of the cluster distributions. Sta-tistical error bars are shown when they exceed the size of thedata point. The lines are fits of the form given in Eq. (1).The critical temperature is indicated by the dashed line.0.00017) is close to the expected value for an infinite sys-tem (τ∞ = 2.21). The fit also determines q0 which canbe compared to the prediction of Eq. (6). The fit pre-dicts qfit0 = 30084 while q∞0 = 27566. The power law fitof the fragment abundances at the critical temperatureis shown in Fig. 3.In the coexistence region, which in the Ising modelshould prevail for T < Tc when the magnetization is notconstrained, the chemical potentials of the liquid and gasphases are equal, thus Eq. (4) can be rewritten as:nA(T )Aτ/q0 = exp(−c0Aσε/T ) (7)with ε = (Tc − T )/Tc. Therefore, a graph of thescaled cluster distributions (nA(T )Aτ/q0) as a function ofεAσ/T should make the distributions of all cluster sizescollapse onto a single curve. This scaling behavior canclearly be seen in Fig. 4. This nearly perfect collapsebelow the critical temperature extends over six orders ofmagnitude for the broadest range of cluster sizes and itis perfectly linear. Therefore the three-dimensional Isingmodel and fluids belong to the same class of universal-ity and can be described by Fisher’s droplet model. TheIsing clusters constructed here can be properly thoughtof as “vapor” in equilibrium with the “liquid” percolat-ing cluster. Coexistence of the two phases is determinedby the observation that the empirical scaling requires∆µ = 0. The fact that both the three-dimensional IsingFIG. 3. The upper panel shows the extracted barriers fromthe fits to the cluster distributions. The line is a fit of theform given in Eq. (3). The lower panel shows the power lawbehavior of the cluster distribution at kbT/J = 4.515. Theline is a fit of the form given in Eq. (5). In both panels, errorbars do not exceed the size of the data point.model and the experimental nuclear multifragmentationdata obey the same scaling predicted by Fisher’s dropletmodel indicates that nuclear multifragmentation can in-deed be identified as the clustering (non-ideality) in anuclear vapor in equilibrium with the nuclear liquid [4].We note that the value of σ determined in this workis larger than 0.64, the usual value of σ from studiesof the thermal properties of three-dimensional Ising sys-tems. There are several possible explanations for the dif-ferences between the value of σ determined in this studyand the values found from other studies. First, while sta-tistical errors are small, systematic errors may not be sosmall. Finite size effects of the 503 lattice could lead toan overabundance of small clusters. This is caused by theFIG. 4. Scaling behavior of cluster distributions.3fact that large clusters near the edge of the lattice are un-able to grow beyond the lattice edges easily, even thoughperiodic boundary conditions are used in this model. Theeffects of lattice size as well as other systematic errors arediscussed in greater detail in [20]. An over-abundance ofsmall clusters, with a corresponding suppression of largeclusters, would increase the slope of the cluster distribu-tion leading to larger than expected values of τ and q0,just as we have observed.In conclusion, we have shown that the clusterizationin the Ising model, like nuclear multifragmentation, por-trays reducibility and thermal scaling. In addition, theArrhenius plots allow for the extraction of “barriers”which are found to have a dependance of B ∝ Aσ,where σ (which is close to 2/3) is a critical exponent.The reducibility and thermal scaling features in the Isingmodel can be incorporated into a Fisher-like scaling with∆µ = 0, which is obeyed rigorously over the exploredtemperature range below the critical temperature. Thusthe observed clusters can be interpreted as the non-idealities of a vapor in equilibrium with a liquid. Finally,nuclear multifragmentation, which is seen to share all thescaling observed here, should be similarly interpreted asthe clusterization of a nuclear vapor in equilibrium withits liquid.AcknowledgementsThis work was supported by the National ScienceFoundation under grants NSF-RUI 9800747 and NSF-REU 9876955 and the Nuclear Physics Division of theUS Department of Energy under contract DE-AC03-76SF00098. One of us (CMM) acknowledges supportfrom LBNL during her sabbatical visit.Present Addresses1 Nuclear Science Division, Lawrence Berkeley NationalLaboratory, Berkeley, CA 947202 Marietta College, Marietta, OH 45750[1] L. G. Moretto et al., Phys. Rep. 287, 249 (1997).[2] L. Beaulieu et al., Phys. Rev. Lett. 81, 770 (1998).[3] L. G. Moretto et al., Phys. Rev. C 60, 031601 (1999).[4] J. B. Elliott et al., Phys. Rev. Lett 85, 1194 (2000).[5] D. Stauffer, Phys. Rep 54, 2 (1979).[6] D. Stauffer and A. Aharony, “Introduction to PercolationTheory”, 2nd ed. (Taylor and Francis, London, 1992).[7] W. Bauer et al., Phys. Lett. B 150, 53 (1985).[8] W. Bauer et al., Prog. Part. Nucl. Phys. 99, 42 (1999).[9] P. D. Gujrati, Phys. Rev. E 51, 957 (1995).[10] D. Stauffer, Phys. Rev. Lett. 35, 394 (1975).[11] C. Domb and E. Stoll, J. Phys. A 10, 1141 (1977).[12] A. Coniglio and W. Klein, J. Phys. A 13, 2775 (1980).[13] J. L. Cambier and M. Nauenberg, Phys. Rev. B 34, 8071(1986).[14] J. Kertész, Physica A 161, 58 (1989).[15] J. S. Wang, Physica A 161, 249 (1989).[16] J. S. Wang and R. H. Swendsen, Physica A 167, 565(1990).[17] D. De Meo, D. W. Heermann and K. Binder, J. Stat.Phys. 60, 585 (1990).[18] A. M. Ferrenberg and D. P. Landau, Phys. Rev. B 44,5081 (1991).[19] J. J. Alonso, A. I. López-Lacomba and J. Marro, Phs.Rev. E 52, 6006 (1995).[20] C. M. Mader et al, in progress.[21] M. E. Fisher, Physics 3, 255 (1967).[22] M. E. Fisher, Rep. Prog. Phys. 30, 615 (1969).[23] C. S. Kiang and D. Stauffer, Z. Physik 235, 130 (1970).[24] D. Stauffer and C. S. Kiang, Advances in Colloid andInterface Science 7, 103 (1977).[25] J. B. Elliott et al., Phys. Rev. C 62, 064603 (2000).4

The three-dimensional Ising model: A paradigm of liquid-vapor coexistence in nuclear 
multifragmentation

Crossref

The three-dimensional Ising model and its Fisher analysis: A paradigm of liquid-vapor coexistence in nuclear multifragmentation

Clusters in the three-dimensional Ising model rigorously obey reducibility and thermal scaling up to the critical temperature. The barriers extracted from Arrhenius plots depend on the cluster size as B {proportional_to} A{sup {sigma}} where {sigma} is a critical exponent relating the cluster size to the cluster surface. All the Arrhenius plots collapse into a single Fisher-like scaling function indicating liquid-vapor-like phase coexistence and the univarian equilibrium between percolating clusters and finite clusters. The compelling similarity with nuclear multifragmentation is discussed

Moretto,Luciano G.

UNT (University of North Texas) Digital Library

The three-dimensional Ising model: A paradigm of liquid-vapor coexistence in nuclearmultifragmentationC. M. Mader1, A. Chappars2Hope College, Holland, MI 49423J. B. Elliott, L. G. Moretto, L. Phair and G. J. WozniakNuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720(March 5, 2001)Clusters in the three-dimensional Ising model rigorouslyobey reducibility and thermal scaling up to the critical tem-perature. The barriers extracted from Arrhenius plots dependon the cluster size as B ∝ Aσ where σ is a critical exponent re-lating the cluster size to the cluster surface. All the Arrheniusplots collapse into a single Fisher-like scaling function indi-cating liquid-vapor-like phase coexistence and the univariantequilibrium between percolating clusters and finite clusters.The compelling similarity with nuclear multifragmentation isdiscussed.Nuclear multifragmentation is a process occurring atthe limits of nuclear excitation, and, as such, portraysan appropriate richness and complexity. While the fun-damental problem of dynamics vs. statistics is still de-bated, it appears ever more clearly that many ther-mal/statistical features underlie the empirical body ofdata. In particular, two features associated with the frag-ment multiplicities are found to be quite pervasive in allmultifragmentation reactions. They have been named“reducibility” and “thermal scaling” [1–3].Reducibility is the property of the n-fragment emissionprobability of being expressible in terms of an elementaryone-fragment emission probability. This property can oc-cur only if fragments are created independently from oneanother and it coincides with stochasticity. Both bino-mial, and its limiting form, Poissonian reducibilities havebeen extensively documented experimentally [1–3].Thermal scaling is the linear dependence of the loga-rithm of the one-fragment probability with 1/T (an Ar-rhenius plot). It indicates that the emission probabilityfor fragment type i has a Boltzmann dependencepi = p0e−Bi/T (1)where Bi is a barrier corresponding to the emission pro-cess.The combination of these two empirical features power-fully attests to a statistical mechanism of multifragmen-tation in general, and to liquid-vapor coexistence specif-ically [4].Many statistical models have been proposed as an ex-planation for multifragmentation. It is our intention hereto identify a model which, on one hand, is as simple aspossible, yet on the other captures the essential featuresobserved in the experiments. Percolation in its manyversions has been widely used [5–8]. However, while be-ing simple, it does not lend itself to a non-trivial ther-mal study [9]. The three-dimensional Ising model sat-isfies both the criteria of simplicity in its Hamiltonianand lends itself to a thermal treatment with nontrivialresults. While the Ising model has been widely studiedin terms of its continuous phase transition, the problemof clusterization, of paramount interest to us here, hasreceived relatively little attention [10–19]. We will showthat this model contains both features of reducibility andthermal scaling. In showing the features of thermal scal-ing we will demonstrate that the slopes of the Arrheniusplots associated with the individual masses of the frag-ments, or the barriers in Eq. (1), portray a dependenceon the fragment mass (A) of the form B ∝ Aσ, whereσ is a proper critical exponent which relates the mass tothe cluster surface. In addition, the individual Arrheniusplots for each fragment mass can be absorbed into a sin-gle scaling function identical to that of Fisher’s dropletmodel [21–25], which defines the liquid-vapor coexistenceline up to the critical temperature.The Hamiltonian of the Ising model has two terms:the interaction between nearest neighbor (n.n.) spins ina fixed lattice and the interaction between the fixed spinsand an external applied field Hext:H = −J∑i,j={n.n.}sisj −Hext∑isi (2)where J is the strength of the spin-spin interaction. Inthis model, the two phases of the system are a liquid(clusters of sites with connected up spins) and a gas(clusters of sites with connected down spins). In theabsence of an external field, the distinction between upand down spins vanishes. Thus in the absence of an ex-ternal field, these two phases are in coexistence belowthe critical temperature, Tc. The critical temperaturefor the three-dimensional Ising model has not been de-termined analytically; however, Monte Carlo techniqueshave yielded a value of Tc = 4.513 J/kb [13].In the present study, Monte Carlo techniques are usedto determine the equilibrium cluster distribution as afunction of temperature for a simple cubic lattice withperiodic boundary conditions. The lattice contains 503spins, which is large but may not be large enough to1FIG. 1. Ratio of the variance to the mean for distributionsof clusters with A constituents. The dashed line correspondsto the critical temperature, kbTc/J = 4.513.represent an infinite system. The Swendsen-Wang algo-rithm [16] was used to determine the equilibrium spinconfigurations of the lattice for given temperatures, andclusters were identified using the Coniglio-Klein [12] pre-scription. Details about the model can be found in ref-erence [20].To test if the fragment distributions produced by themodel exhibit binomial or Poissonian characteristics, theratio of the variance to the mean (σ2/ 〈nA〉) of the frag-ment distributions (nA) for various fragment sizes (A)as a function of temperature was determined. For mostclusters, the ratio is very close to one, which is the Poisso-nian limit, over the entire temperature range consideredhere, which extends well beyond the critical tempera-ture. Only for the very lightest clusters (A <∼ 20), doesthe ratio have a discontinuity at the critical tempera-ture. However, the ratio is near one even for these lightclusters at all temperatures away from the critical tem-perature. (See Fig. 1.) Thus, for all but the smallestclusters, the distributions exhibit Poissonian reducibil-ity. This signifies that the clusters are produced nearlyindependently of one another, a feature also observed inpercolation models, Fisher’s droplet model and nuclearfragmentation [4].If the fragment distributions also exhibit thermal re-ducibility, the distributions must be of the form given inEq. (1). Thus in an Arrhenius plot (a semi-log graph ofthe number of clusters of size A (nA) vs. 1/T ), the distri-butions should be linear below the critical temperature.As shown in Fig. 2, this is indeed the case over a widerange of temperatures (0 < T < Tc) and fragment sizes.While we have shown distributions for clusters up to sizeA = 100, the trend continues for larger clusters, howeverstatistics decrease significantly as the size of the clusterincreases. This linearity extends over more than four orfive orders of magnitude. It rigorously confirms the formof Eq. (1) and signifies the independent thermal forma-tion of fragments controlled by a single size-dependentbarrier. This feature has been amply verified in nuclearmultifragmentation. By fitting the fragment distribu-tions below the critical temperature (J/kbT > 0.2216),the barriers can be extracted. The barriers for each clus-ter size are shown in Fig. 3.These barriers should find their origin in the number ofbroken bonds associated with a cluster. Therefore theyshould be well described by a power-law:B(A) = B0Aσ. (3)The fit of the extracted barriers according to Eq. (3) isB = (12.77±0.04)A(0.7230±0.008) and is remarkably goodas shown in Fig. 3. The value for σ (0.7230 ± 0.0004)is close to 2/3, the value one would expect for sphericalclusters of closely packed spherical objects [15], and to0.64 the value expected from scaling relations [11]. Thusthis is a new and effective method to determine the crit-ical exponent σ, never used before to our knowledge.The features of reducibility and thermal scaling dis-cussed above can be found united in Fisher’s formula forthe cluster abundance in a vapor as a function of clustersize and of temperature. The formula isnA(T ) = q0A−τ exp(A∆µT) exp(c0AσTc) exp(−c0AσT) (4)where one can see the thermal scaling up to Tc and thedependence of the “barrier” on the cluster mass throughthe critical exponent σ [21,22]. It is interesting to ex-plore further the applicability of this formula to the Isingmodel.In addition to the linear behavior of the Arrheniusplots below the critical temperature, the Fisher dropletmodel also predicts that the cluster size distribution atthe critical point must follow a power lawnA(Tc) = q0A−τ (5)where q0 is fixed by the normalization relationshipq0 =∑∞A=1 nA(Tc)∑∞A=1A−τ . (6)Away from the critical temperature, the cluster distri-bution should not follow a pure power law. Thus, todetermine τ without a prior knowledge of the criticaltemperature, linear fits to the cluster distributions weredetermined for all temperatures. At the critical tem-perature, the fit should have the lowest χ2/DoF andthus this fit should determine τ . The results for thebest fit are shown in the lower panel of Fig. 3. Fromthis method, the critical temperature was found to be(4.515 ± .010)J/kB with a best fit line of the formnA(Tc) = (30084 ± 7)A(−2.39671±0.00017). This temper-ature is consistent with the value determined for infi-nite systems (4.513J/kB) and the value of τ (2.39671±2FIG. 2. Arrhenius plots of the cluster distributions. Sta-tistical error bars are shown when they exceed the size of thedata point. The lines are fits of the form given in Eq. (1).The critical temperature is indicated by the dashed line.0.00017) is close to the expected value for an infinite sys-tem (τ∞ = 2.21). The fit also determines q0 which canbe compared to the prediction of Eq. (6). The fit pre-dicts qfit0 = 30084 while q∞0 = 27566. The power law fitof the fragment abundances at the critical temperatureis shown in Fig. 3.In the coexistence region, which in the Ising modelshould prevail for T < Tc when the magnetization is notconstrained, the chemical potentials of the liquid and gasphases are equal, thus Eq. (4) can be rewritten as:nA(T )Aτ/q0 = exp(−c0Aσε/T ) (7)with ε = (Tc − T )/Tc. Therefore, a graph of thescaled cluster distributions (nA(T )Aτ/q0) as a function ofεAσ/T should make the distributions of all cluster sizescollapse onto a single curve. This scaling behavior canclearly be seen in Fig. 4. This nearly perfect collapsebelow the critical temperature extends over six orders ofmagnitude for the broadest range of cluster sizes and itis perfectly linear. Therefore the three-dimensional Isingmodel and fluids belong to the same class of universal-ity and can be described by Fisher’s droplet model. TheIsing clusters constructed here can be properly thoughtof as “vapor” in equilibrium with the “liquid” percolat-ing cluster. Coexistence of the two phases is determinedby the observation that the empirical scaling requires∆µ = 0. The fact that both the three-dimensional IsingFIG. 3. The upper panel shows the extracted barriers fromthe fits to the cluster distributions. The line is a fit of theform given in Eq. (3). The lower panel shows the power lawbehavior of the cluster distribution at kbT/J = 4.515. Theline is a fit of the form given in Eq. (5). In both panels, errorbars do not exceed the size of the data point.model and the experimental nuclear multifragmentationdata obey the same scaling predicted by Fisher’s dropletmodel indicates that nuclear multifragmentation can in-deed be identified as the clustering (non-ideality) in anuclear vapor in equilibrium with the nuclear liquid [4].We note that the value of σ determined in this workis larger than 0.64, the usual value of σ from studiesof the thermal properties of three-dimensional Ising sys-tems. There are several possible explanations for the dif-ferences between the value of σ determined in this studyand the values found from other studies. First, while sta-tistical errors are small, systematic errors may not be sosmall. Finite size effects of the 503 lattice could lead toan overabundance of small clusters. This is caused by theFIG. 4. Scaling behavior of cluster distributions.3fact that large clusters near the edge of the lattice are un-able to grow beyond the lattice edges easily, even thoughperiodic boundary conditions are used in this model. Theeffects of lattice size as well as other systematic errors arediscussed in greater detail in [20]. An over-abundance ofsmall clusters, with a corresponding suppression of largeclusters, would increase the slope of the cluster distribu-tion leading to larger than expected values of τ and q0,just as we have observed.In conclusion, we have shown that the clusterizationin the Ising model, like nuclear multifragmentation, por-trays reducibility and thermal scaling. In addition, theArrhenius plots allow for the extraction of “barriers”which are found to have a dependance of B ∝ Aσ,where σ (which is close to 2/3) is a critical exponent.The reducibility and thermal scaling features in the Isingmodel can be incorporated into a Fisher-like scaling with∆µ = 0, which is obeyed rigorously over the exploredtemperature range below the critical temperature. Thusthe observed clusters can be interpreted as the non-idealities of a vapor in equilibrium with a liquid. Finally,nuclear multifragmentation, which is seen to share all thescaling observed here, should be similarly interpreted asthe clusterization of a nuclear vapor in equilibrium withits liquid.AcknowledgementsThis work was supported by the National ScienceFoundation under grants NSF-RUI 9800747 and NSF-REU 9876955 and the Nuclear Physics Division of theUS Department of Energy under contract DE-AC03-76SF00098. One of us (CMM) acknowledges supportfrom LBNL during her sabbatical visit.Present Addresses1 Nuclear Science Division, Lawrence Berkeley NationalLaboratory, Berkeley, CA 947202 Marietta College, Marietta, OH 45750[1] L. G. Moretto et al., Phys. Rep. 287, 249 (1997).[2] L. Beaulieu et al., Phys. Rev. Lett. 81, 770 (1998).[3] L. G. Moretto et al., Phys. Rev. C 60, 031601 (1999).[4] J. B. Elliott et al., Phys. Rev. Lett 85, 1194 (2000).[5] D. Stauffer, Phys. Rep 54, 2 (1979).[6] D. Stauffer and A. Aharony, “Introduction to PercolationTheory”, 2nd ed. (Taylor and Francis, London, 1992).[7] W. Bauer et al., Phys. Lett. B 150, 53 (1985).[8] W. Bauer et al., Prog. Part. Nucl. Phys. 99, 42 (1999).[9] P. D. Gujrati, Phys. Rev. E 51, 957 (1995).[10] D. Stauffer, Phys. Rev. Lett. 35, 394 (1975).[11] C. Domb and E. Stoll, J. Phys. A 10, 1141 (1977).[12] A. Coniglio and W. Klein, J. Phys. A 13, 2775 (1980).[13] J. L. Cambier and M. Nauenberg, Phys. Rev. B 34, 8071(1986).[14] J. Kertész, Physica A 161, 58 (1989).[15] J. S. Wang, Physica A 161, 249 (1989).[16] J. S. Wang and R. H. Swendsen, Physica A 167, 565(1990).[17] D. De Meo, D. W. Heermann and K. Binder, J. Stat.Phys. 60, 585 (1990).[18] A. M. Ferrenberg and D. P. Landau, Phys. Rev. B 44,5081 (1991).[19] J. J. Alonso, A. I. López-Lacomba and J. Marro, Phs.Rev. E 52, 6006 (1995).[20] C. M. Mader et al, in progress.[21] M. E. Fisher, Physics 3, 255 (1967).[22] M. E. Fisher, Rep. Prog. Phys. 30, 615 (1969).[23] C. S. Kiang and D. Stauffer, Z. Physik 235, 130 (1970).[24] D. Stauffer and C. S. Kiang, Advances in Colloid andInterface Science 7, 103 (1977).[25] J. B. Elliott et al., Phys. Rev. C 62, 064603 (2000).4

The three-dimensional Ising model: A paradigm of liquid-vaporcoexistence in nuclear multifragmentation

Mader, C M

Chappars, A

Elliott, J B

Moretto, L G

Phair, L

Wozniak, G J

Lawrence Berkeley National LaboratoryLBL PublicationsTitleThe Three-Dimensional Ising Model: A Paradigm of Liquid-Vapor Coexistence in Nuclear MultifragmentationPermalinkhttps://escholarship.org/uc/item/4gn139fkAuthorsMader, C MChappars, AElliott, J Bet al.Publication Date2001-03-01Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, available at https://creativecommons.org/licenses/by/4.0/eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaLBNL-47575 Preprint ERNEST DRLAND_O LAWRENCE ... ,.t•/'t . BERKELEY NATIONAL LABD .. RATDRY The Three-Dimensional Ising Model: A Paradigm of Liquid-Vapor Coexistence in Nuclear Multifragmentation C.M. Mader, A. Chappars, J.B. Elliott, L.G. Moretto, L. Phair, and G.J. Wozniak Nuclear Science Division March 2001 Submitted for publication DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. LBNL-47575 The Three-Dimensional Ising Model: A Paradigm of Liquid-Vapor Coexistence in Nuclear Multifragmentation C.M. Mader and A. Chappars Hope College Holland, Michigan 49423 J.B. Elliott, L.G. Moretto, L. Phair, and G.J. Wozniak Nuclear Science Division Ernest Orlando Lawrence Berkeley National Laboratory University of California Berkeley, California 94720 March 2001 This work was supported by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of Nuclear Physics, ofthe U.S. Department of Energy under Contract No. DE-AC03-76SF00098, and by the National Science Foundation Grant Nos. NSF-RUI 9800747 and NSF-REU 9876955. The three-dimensional Ising model: A paradigm of liquid"'vapor coexistence in nuclear multifragmentation C. M. Mader1 , A. Chappars2 Hope College, Holland, MI 49423 J. B. Elliott, L. G. Moretto, L. Phair and G. J. Wozniak Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94 720 (March 5, 2001) Clu~ters in the three-dimensional Ising model rigorously obey reducibility and thermal scaling up to the critical tem-perature. The barriers extracted from Arrhenius plots depend on the duster size as B ex: Au where u is a critical exponent re-lating the cluster size to the cluster surface. All the Arrhenius plots collapse into a single Fisher-like scaling function indi-cating liquid~ vapor-like phase coexistence and the univariant equilibrium between percolating clusters and finite clusters. The compelling similarity with nuclear multifragmentation is discussed. Nuclear multifragmentation is a process occurring at the limits of nuclear excitation, and, as such, portrays an appropriate richness and complexity. While the fun-damental problem of dynamics vs. statistics is still de-bated, it appears ever more clearly that many ther-mal/statistical features underlie the empirical body of data. In particular, two features associated with the frag-ment multiplicities are found to be quite pervasive in all multifragmentation reactions. They have been named "reducibility" and "thermal scaling" [1-3]. Reducibility is the property of then-fragment emission probability of being expressible in terms of an elementary one-fragment emission probability. This property can oc-cur only if fragments are created independently from one another and it coincides with stochasticity. Both bino-mial, and its limiting form, Poissonian reducibilities have been extensively documented experimentally [1-3]. Thermal scaling is the linear dependence of the loga-rithm of the one-fragment probability with 1/T (an Ar-rhenius plot). It indicates that the emission probability for fragment type i has a Boltzmann dependence (1) where Bi is a barrier corresponding to the emission pro-cess. The combination of these two empirical features power-fully attests to a statistical mechanism of multifragmen-tation in general, and to liquid-vapor coexistence specif-ically [4]. Many statistical models have been proposed as an ex-planation for multifragmentation. It is our intention here to identify a model which, on one hand, is as simple as possible, yet on the other captures the essential features observed in the experiments. Percolation in its many 1 versions has been widely used [5-8]. However, while be-ing simple, it does not lend itself to a non-trivial ther-mal study [9]. The three-dimensional Ising model sat-isfies both the criteria of simplicity in its Hamiltonian and lends itself to a thermal trea:tment with nontrivial results. While the Ising model has been widely studied in terms of its continuous phase transition, the problem of clusterization, of paramount interest to us here, has received relatively little attention [10-19]. We will show that this model contains both features of reducibility and thermal scaling. In showing the features of thermal scal-ing we will demonstrate that the slopes of the Arrhenius plots associated with the individual masses of the frag-ments, or the barriers in Eq. (1), portray a dependence on the fragment mass (A) of the form B ex: Au, where a is a proper critical exponent which relates the mass to the cluster surface. In addition, the individual Arrhenius plots for each fragment mass can be absorbed into a sin-gle scaling function identical to that of Fisher's droplet model [21-25], which defines the liquid-vapor coexistence line up to the critical temperature. The Hamiltonian of the Ising model has two terms: the interaction between nearest neighbor ( n. n.) spins in a fixed lattice and the interaction between the fixed spins and an external applied field Hext: H = -J L SiSj- Hext LSi (2) i,j={n.n.} where J is the strength of the spin-spin interaction. In this model, the two phases of the system are a liquid (clusters of sites with connected up spins) and a gas (clusters of sites with connected down spins). In the absence of an external field, the distinction between up and down spins vanishes. Thus in the absence ofan ex-ternal field, these two phases are in coexistence below the critical temperature, Tc. The critical temperature for the three-dimensional Ising model has not been de-termined analytically; however, Monte Carlo techniques have yielded a value of Tc = 4.513 Jjkb [13]. In the present study, Monte Carlo techniques are used to determine the equilibrium cluster distribution as a function of temperature for a simple cubic lattice with periodic boundary conditions. The lattice contains 503 spins, which is large but may not be ·large ·enough to 1.25 1.00 0.75 0.50 0.25 . 1' i i o Oo· o------4------t--it ··!'· t a A=S • A=20 .._ A=40 'Y A=60 + A=80 • A=lOO • i • 0 represent an infinite system. The Swendsen-Wang algo-rithm [16] was used to determine the equilibrium spin configurations of the lattice for given temperatures, and clusters were identified using the Coniglio-Klein [12] pre-scription. Details about the model can be found in ref-erence [20]. To test if the fragment distributions produced by the model exhibit binomial or Poissonian characteristics, the ratio of the variance to the mean (o-2 I (nA)) of the frag-ment distributions (nA) for various fragment sizes (A) as a function of temperature was determined. For most clusters, the ratio is very close to one, which is the Poisso-nian limit, over the entire temperature range considered here, which extends well beyond the critical tempera-ture. Only for the very lightest clusters (A ;S 20), does the ratio have a discontinuity at the critical tempera-ture. However, the ratio. is near one even for these light clusters at all temperatures away from the critical tem-perature. (See Fig. 1.) Thus, for all but the smallest clusters, the distributions exhibit Poissonian reduCibil-ity. This signifies that the clusters are produced nearly independently of one another, a feature also observed in percolation models, Fisher's droplet model and nuclear fragmentation [4]. 1 If the fragment distributions also exhibit thermal re-ducibility, the distributions must be of.the form given in Eq. (1). Thus in an Arrhenius plot (a semi-log graph of the number of clusters of size A (nA) vs. 1IT), the distri-butions should be linear below the critical temperature. As shown in Fig, 2, this is indeed the case over a wide range of temperatures (0 < T < Tc) and fragment sizes. While we have shown distributions for clusters up to size A = 100, the trend continues for larger clusters, however statistics decrease significantly as the size of the cluster increases. This linearity extends over more than four or five orders of magnitude. It rigorously confirms the form of Eq. (1) and signifies the independent thermal forma-2 tion of fragments controlled by a single size-dependent barrier. This feature has been amply verified in nuclear multifragmentation. By fitting the fragment distribu-tions below the critical temperature ( J I kbT > 0.2216), the barriers can be extracted. The barriers for each clus-ter size are shown in Fig. 3. These barriers should find their origin in the number of broken bonds associated with a cluster. Therefore they should be well described by a power-law: B(A) =BoA". (3) The fit of the extracted barriers according to Eq. (3) is B = (12.77±0.04)A<0·7230±o.oos) and is remarkably good as shown in Fig. 3. The value for u (0.7230 ± 0.0004) is close to 213, the value one would expect for spherical clusters of closely packed spherical objects [15], and to 0.64 the value expected from scaling relations [11]. Thus this is a new and effective method to determine the crit-ical exponent u, never used before to our knowledge. The features of reducibility and thermal scaling dis-cussed above can be found united in Fisher's formula for the cluster abundance in a vapor as a function of cluster size and of temperature. The formula is A~JL eo A" eo A"· nA(T) = qoA--r exp( 'f') exp( r) exp(-'f') (4) where one can see the thermal scaling up to Tc and the dependence of the "barrier" on the cluster mass through the critical exponent u [21,22]. It is interesting to ex-plore further the applicability of this formula to the Ising model. In addition to the linear behavior of the Arrhenius plots below the critical temperature, the Fisher droplet model also predicts that the cluster size distribution at the critical point must follow a power law (5) where q0 is fixed by the normalization relationship E~-l nA(Tc) qo = "'oo A--r . L,.,A=l (6) Away from the critical temperature, the cluster distri-bution should not follow a pure power law. Thus, ·to determine r without a prior knowledge of the critical temperature, linear fits to the cluster distributions were determined for all temperatures. At the critical tem-perature, the fit should have the lowest x2 I DoF and thus this fit should determine T. The results for the best fit are shown in the lower panel of Fig. 3. From this method, the critical temperature was found to be (4.515 ± .010)JikB with a best fit line of the form nA(Tc) = (30084 ± 7)A<-2·39671±0-00017). This temper-ature is consistent with the value determined for infi-nite systems ( 4.5131 I kB) and the value ofT (2.39671 ± 0.20 0.25 0.30 0.20 0.25 0.30 J~T J~T FIG. 2. Arrhenius plots of the cluster distributions. Sta-tistical error bars are shown when they exceed the size of the data point. The lines are fits of the form given in Eq. (1). The critical temperature is indicated by the dashed line. 0.00017) is close to the expected value for an infinite sys-tem (700 = 2.21). The fit also determines q0 which can be compared to the prediction of Eq. (6). The fit pre-dicts q6it ~ 30084 while q(f = 27566. The power law fit of the fragment abundances at the critical temperature is shown in Fig. 3. In the coexistence region, which in the Ising model should prevail for T < Tc when the magnetization is not constrained, the chemical potentials of the liquid and gas phases are equal, thus Eq. (4) can be rewritten as: nA(T)Ar Jqo = exp( -caAu efT) (7) with e = (Tc - T) fTc. Therefore, a graph of the scaled cluster distributions ( n A (T)A r J qo) as a function of eAu JT should make the distributions of all cluster sizes collapse onto a single curve. This scaling behavior can clearly be seen in Fig. 4. This nearly perfect collapse below the critical temperature extends over six orders of magnitude for the broadest range of cluster sizes and it is perfectly linear. Therefore the three-dimensional Ising -model and fluids belong to the same class of universal-ity and can be described by Fisher's droplet model. The Ising clusters constructed here can be properly thought of as "vapor" in equilibrium with the "liquid" percolat-ing cluster. Coexistence of the two phases is determined by the observation that the empirical scaling requires· D..p, = 0. The fact that both the three-dimensional Ising 3 1 o-1 .______..~ .................... ..._._....__.__....__.--'-"~...__. 1 10 100 A FIG. 3. The upper panel shows the extracted barriers from the fits to the cluster distributions. The line is a fit of the form given in Eq. (3). The lower panel shows the power law behavior of the cluster distribution at kbT/ J = 4.515. The line is a fit of the form given in Eq. (5). In both panels, error bars do not exceed the size of the data point. model and the experimental nuclear multifragmentation data obey the same scaling predicted by Fisher's droplet model indicates that nuclear multifragmentation can in-deed be identified as the clustering (non-ideality) in a nuclear vapor in equilibrium with the nuclear liquid [4]. We note that the value of a determined in this work is larger than 0.64, the usual value of a from studies of the thermal properties of three-dimensional Ising sys-tems. There are several possible explanations for the dif-ferences between the value of a determined in this study and the values found from other studies. First, while sta-tistical errors are small, systematic errors may not be so small.· Finite size effects of the 503 lattice could lead to an overabundance of small clusters. This is caused by the ~.,. 10"2 • A=lO \ • A=20 \ --o ... A=30 ~ "' A=40 • A=SO - ... A=60 .. ;:- ~~ ~ .. A=70 • A=80 ··11~1 s:l 10"" * A=90 • A=lOO • A=llO l !I + A=120 .,. X A=l30 'r * A=140 Ill A=150 l 10-6 -1.0 -0.5 0.0 0.5 1.0 eAa/T FIG. 4. Scaling behavior of cluster distributions. fact that large clusters near the edge of the lattice are un-able to grow beyond the lattice edges easily, even though periodiC boundary conditions are used in this model. The effects of lattice size as well as other systematic errors are discussed in greater detail in [20]. An over-abundance of small clusters, with a corresponding suppression of large clusters, would increase the slope of the cluster distribu-tion leading to larger than expected values of T and qo, just as we have observed. In conclusion, we have shown that the clusterization in the Ising model, like nuclear multifragmentation, por-trays reducibility and thermal scaling. In addition, the Arrhenius plots allow for the extraction of "barriers" which are found to have a dependance of B ex: Au, where a (which is close to 2/3) is a critical exponent. The reducibility and thermal scaling features in the Ising model can be incorporated into a Fisher-like scaling with D.J.L = 0, which is obeyed rigorously over the explored temperature range below the critical temperature. Thus the observed clusters can be interpreted as the non-idealities of a vapor in equilibrium with a liquid. Finally, nuclear multifragmentation, which is seen to share all the scaling observed here, should be similarly interpreted as the clusterization of a nuclear vapor in equilibrium with its liquid. Acknowledgements This work was supported by the National Science Foundation under grants NSF-RUI 9800747 and NSF-RED 9876955 and the Nuclear Physics Division of the US Department of Energy under contract DE-AC03-76SF00098. One of us (CMM) acknowledges support from LBNL during her sabbatical visit. Present Addresses 1 Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 2 Marietta College, Marietta, OH 45750 [1] L. G. Moretto et al., Phys. Rep. 287, 249 (1997). [2] L. Beaulieu et al., Phys. Rev. Lett. 81, 770 (1998). [3] L. G. Moretto et al., Phys. Rev. C 60, 031601 (1999). [4] J. B. Elliott et al., Phys. Rev. Lett 85, 1194 (2000). [5] D. Stauffer, Phys. Rep 54, 2 (1979). [6] D. Stauffer and A. Aharony, "Introduction to Percolation Theory", 2nd ed. (Taylor and Francis, London, 1992). [7] W. Bauer et al., Phys. Lett. B 150, 53 (1985). [8] W. Bauer et al., Prog. Part. Nucl. Phys. 99, 42 (1999). [9] P. D .. Gujrati, Phys. Rev. E 51, 957 (1995). [10] D. Stauffer, Phys. Rev. Lett. 35, 394 (1975). [11] C. Domb and E. Stoll, J. Phys. A 10, 1141 (1977). [12] A. Coniglio and W. Klein, J. Phys. A 13, 2775 (1980). [13] J. L. Cambier and M. Nauenberg, Phys. Rev. B 34, 8071 (1986). 4 [14] J. Kertesz, Physica A 161, 58 (1989). [15] J. S. Wang, Physica A 161, 249 (1989). [16] J. S. Wang and R. H. Swendsen, Physica A 167, 565 (1990). [17] D. De Meo, D. W. Heermann and K. Binder, J. Stat. Phys. 60, 585 (1990). [18] A. M. Ferrenberg and D. P. Landau, Phys. Rev. B 44, 5081 (1991). [19] J. J. Alonso, A. I. L6pez-Lacomba and J. Marro, Phs. Rev. E 52, 6006 (1995). [20] C. M. Mader et al, in progress. [21] M. E. Fisher, Physics 3, 255 (1967). [22] M. E. Fisher, Rep. Prog. Phys. 30, 615 (1969). [23] C. S. Kiang and D. Stauffer, Z. Physik 235, 130 (1970). [24] D. Stauffer and C. S. Kiang, Advances in Colloid and Interface Science 7, 103 (1977). [25] J. B. Elliott et al., Phys. Rev. C 62, 064603 (2000). ~~~aiil,~liil'ii ~~~ ~~~~~.~~'7/~ 

The Three-Dimensional Ising Model: A Paradigm of Liquid-Vapor Coexistence in Nuclear Multifragmentation

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The three-dimensional Ising model: A paradigm of liquid-vapor
  coexistence in nuclear multifragmentation

The three-dimensional Ising model: A paradigm of liquid-vapor coexistence in nuclear multifragmentation

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