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Volumn 127, Issue , 2004, Pages 1-64

Computational strategies for mapping equilibrium phase diagrams

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EID: 78751654267     PISSN: 00652385     EISSN: None     Source Type: Book Series    
DOI: 10.1002/0471466603.ch1     Document Type: Review
Times cited : (47)

References (162)
  • 5
    • 78751658223 scopus 로고    scopus 로고
    • N. B. Wilding (unpublished)
    • N. B. Wilding (unpublished)
  • 6
    • 78751660113 scopus 로고    scopus 로고
    • In the arguments of exponentials we shall generally absorb factors of kT ≡ 1/β into effective energy functions
    • In the arguments of exponentials we shall generally absorb factors of kT ≡ 1/β into effective energy functions.
  • 11
    • 78751652542 scopus 로고    scopus 로고
    • It would be more circumspect to say that the partition function cannot in general be naturally or usefully expressed as a canonical average since that average inevitably falls into the category for which canonical sampling is inadequate [13]
    • It would be more circumspect to say that the partition function cannot in general be naturally or usefully expressed as a canonical average since that average inevitably falls into the category for which canonical sampling is inadequate [13].
  • 13
    • 85086422511 scopus 로고    scopus 로고
    • 0
    • 0
  • 14
    • 78751652684 scopus 로고    scopus 로고
    • Throughout this section we use the notation introduced in Eq. (7) in which the phase label is absorbed into an extended macrostate label ℓ. The path defined by the set of macrostates (ℓ) may (and sometimes will) extend from one phase to another
    • Throughout this section we use the notation introduced in Eq. (7) in which the phase label is absorbed into an extended macrostate label ℓ. The path defined by the set of macrostates (ℓ) may (and sometimes will) extend from one phase to another.
  • 15
    • 78751656306 scopus 로고    scopus 로고
    • We shall refer to a field l and a macrovariable M as conjugate if the l-dependence of the configurational energy is of the form ℰ({q}, c; δ = ℰ({q}, c)- δ M ({q})
    • We shall refer to a field l and a macrovariable M as conjugate if the l-dependence of the configurational energy is of the form ℰ({q}, c; δ = ℰ({q}, c)- δ M ({q})
  • 16
    • 78751677792 scopus 로고    scopus 로고
    • It would be more conventional to refer to "densities," which are intensive. But we prefer to deal with their extensive counterparts, which we shall refer to as macrovariables. The preference reflects our focus on simulation studies which are necessarily conducted on systems of some given (finite) size
    • It would be more conventional to refer to "densities," which are intensive. But we prefer to deal with their extensive counterparts, which we shall refer to as macrovariables. The preference reflects our focus on simulation studies which are necessarily conducted on systems of some given (finite) size.
  • 21
    • 85086423190 scopus 로고    scopus 로고
    • 2 ≡ α̃ c
    • 2 ≡ α̃ c.
  • 40
    • 85086421976 scopus 로고    scopus 로고
    • hcp]/N
    • For hard spheres the Helmholtz free energy is purely entropic. Thus Table I shows the difference between the entropy per particle of the two phases, [Sfcc - Shcp]/N.
  • 42
    • 78751662234 scopus 로고    scopus 로고
    • Equation (10) shows that the logarithm of the probability ratio measures the free-energy difference, which is extensive
    • Equation (10) shows that the logarithm of the probability ratio measures the free-energy difference, which is extensive.
  • 43
    • 78751673828 scopus 로고    scopus 로고
    • In fact, HPT seems to require that all the points chosen for replicas lie suffiently close to coexistence that both phases have detectable signatures
    • In fact, HPT seems to require that all the points chosen for replicas lie suffiently close to coexistence that both phases have detectable signatures.
  • 46
    • 78751660417 scopus 로고    scopus 로고
    • Reference 44 and N. B. Wilding (unpublished
    • Reference 44 and N. B. Wilding (unpublished)
  • 48
    • 78751673960 scopus 로고    scopus 로고
    • Some of the results from Ref. 44 are included in the LJ phase diagram shown in Fig. 4
    • Some of the results from Ref. 44 are included in the LJ phase diagram shown in Fig. 4.
  • 49
    • 78751655296 scopus 로고    scopus 로고
    • Operationally we may regard a phase as "metastable" and assign it a "free energy" (in relation )to some other, also metastable) if the conditions for single-phase sampling [see Eq. (21) and accompanying discussion] are met
    • Operationally we may regard a phase as "metastable" and assign it a "free energy" (in relation to some other, also metastable) if the conditions for single-phase sampling [see Eq. (21) and accompanying discussion] are met.
  • 50
    • 78751662391 scopus 로고    scopus 로고
    • The configurations unearthed by the ES algorithm used in Lattice Switch studies of hard spheres (Section IV.D.2) have some distinctive features [48]
    • The configurations unearthed by the ES algorithm used in Lattice Switch studies of hard spheres (Section IV.D.2) have some distinctive features [48].)
  • 53
    • 78751675638 scopus 로고    scopus 로고
    • 1/d the edge of the ddimensional simulation cube
    • When the interphase traverse proceeds by way of such two-phase configurations the free-energy barrier to be surmounted must scale with the area of the interface supporting two phases in equal measure. The probability of the ravine macrostates (and thus the probability of tunneling between the two phases at coexistence) must vanish as e-FI where the free-energy cost of the interface is of order FI ∼ δsLd-1, with δs the interface tension and L ∼ N1/d the edge of the ddimensional simulation cube.
  • 54
    • 78751668640 scopus 로고    scopus 로고
    • See, for example, Ref. 41
    • See, for example, Ref. 41.
  • 55
    • 78751674541 scopus 로고    scopus 로고
    • The method to be discussed in the next section belongs in this family; so does HPT, but less compellingly because of the limitations we have already touched on in Section IV.B.2
    • The method to be discussed in the next section belongs in this family; so does HPT, but less compellingly because of the limitations we have already touched on in Section IV.B.2.
  • 56
    • 78751651934 scopus 로고    scopus 로고
    • See Ref. 29, p. 176
    • See Ref. 29, p. 176.
  • 57
    • 78751658523 scopus 로고    scopus 로고
    • The traverse might work acceptably well if the dynamics of the interface between the two phases is favorable: systems with martensitic phase transitions may fall into this category: Z. Nishiyama, Martensitic Transformations, Academic Press, New York, 1978. Note also that the special case in which the structural phase transition involves no change of symmetry can be handled within the standard multicanonical framework [55]
    • The traverse might work acceptably well if the dynamics of the interface between the two phases is favorable: systems with martensitic phase transitions may fall into this category: Z. Nishiyama, Martensitic Transformations, Academic Press, New York, 1978. Note also that the special case in which the structural phase transition involves no change of symmetry can be handled within the standard multicanonical framework [55].
  • 62
    • 78751658367 scopus 로고    scopus 로고
    • In general the partition function Eq. (2) should already incorporate a Jacobean reflecting the transformation from cartesian coordinates to the generalized coordinates {q}. It may always be absorbed into the configurational energy and we have adopted that convention
    • In general the partition function Eq. (2) should already incorporate a Jacobean reflecting the )transformation from cartesian coordinates to the generalized coordinates {q}. It may always be absorbed into the configurational energy and we have adopted that convention.
  • 63
    • 78751672469 scopus 로고    scopus 로고
    • The constraint is rarely made explict in MC studies: it is tacitly assumed that the conditions for "single-phase sampling" [Eq. (21) and accompanying discussion] are met
    • The constraint is rarely made explict in MC studies: it is tacitly assumed that the conditions for "single-phase sampling" [Eq. (21) and accompanying discussion] are met.
  • 67
    • 78751679034 scopus 로고    scopus 로고
    • One sees this more clearly when one views this relationship as the result of integrating Eq. (55)
    • One sees this more clearly when one views this relationship as the result of integrating Eq. (55)
  • 68
    • 78751670236 scopus 로고    scopus 로고
    • But a judicious choice of the representations chosen for the two phases, reflected in the transformation τ [Eq. (44)], can bring one closer to this attractive limit. See Refs. 70 and 71 and further remarks below
    • But a judicious choice of the representations chosen for the two phases, reflected in the transformation τ [Eq. (44)], can bring one closer to this attractive limit. See Refs. 70 and 71 and further remarks below.
  • 69
    • 78751653233 scopus 로고    scopus 로고
    • We use the term "lattice" rather loosely. The set of vectors {R→ }α identifies the orthodox lattice convoluted with the orthodox basis
    • We use the term "lattice" rather loosely. The set of vectors {R→ }α identifies the orthodox lattice convoluted with the orthodox basis.
  • 70
    • 78751678436 scopus 로고    scopus 로고
    • This will frequently be true for solid-solid phase behavior too; in the systems studied with Lattice Switch methods to date, the effects have proved rather small
    • This will frequently be true for solid-solid phase behavior too; in the systems studied with Lattice Switch methods to date, the effects have proved rather small.
  • 74
    • 78751657023 scopus 로고    scopus 로고
    • A. R. Acharya, A. D. Bruce, and G. J. Ackland, in preparation
    • A. R. Acharya, A. D. Bruce, and G. J. Ackland, in preparation.
  • 80
  • 84
    • 84957235540 scopus 로고
    • See, for example, H. Falk, Am. J. Phys. 38, 858 (1970)
    • (1970) Am. J. Phys. , vol.38 , pp. 858
    • Falk, H.1
  • 89
    • 78751657636 scopus 로고    scopus 로고
    • A continuous space transformation like this can also be implemented in the lattice switch framework described in Section IV.D. By the same token, one can generally devise a continuous version of any LS transformation. Thus, for example, the obvious continuous counterpart of the fcc-hcp LS transformation depicted in Fig. 6 entails a gradual shear of the close-packed planes. Since the LS implementation remains always in the space of one or other of the crystalline structures, one might expect it to prove the safer choice. Where an explicit comparison of the two strategies has been made [87], this was indeed the conclusion
    • A continuous space transformation like this can also be implemented in the lattice switch framework described in Section IV.D. By the same token, one can generally devise a continuous version of any LS transformation. Thus, for example, the obvious continuous counterpart of the fcc-hcp LS transformation depicted in Fig. 6 entails a gradual shear of the close-packed planes. Since the LS implementation remains always in the space of one or other of the crystalline structures, one might expect it to prove the safer choice. Where an explicit comparison of the two strategies has been made [87], this was indeed the conclusion.
  • 91
    • 78751663182 scopus 로고    scopus 로고
    • We use {δ} to denote a set of physically different fields δ 2δ2 . . . . In contrast, [δ] (appearing in our discussion of paths) is shorthand for a set of different values δ(1) δ (1) . . . of some (particular) field δ. We use {M} and [M] in a similar way
    • We use {δ} to denote a set of physically different fields δ 2δ2 . . . . In contrast, [δ] (appearing in our discussion of paths) is shorthand for a set of different values δ(1) δ (1) . . . of some (particular) field δ. We use {M} and [M] in a similar way.
  • 92
    • 78751663736 scopus 로고    scopus 로고
    • It reduces to the familiar form with the macrovariables M1 and M2 chosen as respectively the enthalpy and the volume whose conjugate fields are 1=T and P=T
    • It reduces to the familiar form with the macrovariables M1 and M2 chosen as respectively the enthalpy and the volume whose conjugate fields are 1=T and P=T.
  • 99
    • 78751671877 scopus 로고    scopus 로고
    • Recall that knowing the weights that define a multicanonical distribution of some quantity M over some range is equivalent to knowing the true canonical M-distribution throughout the chosen range
    • Recall that knowing the weights that define a multicanonical distribution of some quantity M over some range is equivalent to knowing the true canonical M-distribution throughout the chosen range.
  • 100
    • 78751672016 scopus 로고    scopus 로고
    • For a review of FSS see Finite Size Scaling and Numerical Simulation of Statistical Systems, V. Privman, ed., World Scientific Publishing, Singapore, 1990
    • For a review of FSS see Finite Size Scaling and Numerical Simulation of Statistical Systems, V. Privman, ed., World Scientific Publishing, Singapore, 1990.
  • 104
    • 78751669844 scopus 로고    scopus 로고
    • As an immediate corollary, the field values that satisfy the equal-areas criterion will identify the coexistence curve of two such ensembles to within exponentially small corrections [95]
    • As an immediate corollary, the field values that satisfy the equal-areas criterion will identify the coexistence curve of two such ensembles to within exponentially small corrections [95].
  • 110
    • 12944333198 scopus 로고    scopus 로고
    • The treatment of scaling-field mixing in Ref. 100 has recently been corrected: M. E. Fisher and G Orkoulas, Phys. Rev. Lett. 85, 696 (2000)
    • (2000) Phys. Rev. Lett. , vol.85 , pp. 696
    • Fisher, M.E.1    Orkoulas, G.2
  • 112
    • 0037030028 scopus 로고    scopus 로고
    • In a number of recent papers the ideas outlined here have been developed to yield a framework in which the liquid-vapor coexistence curve can be tracked with remarkable precision deep into the critical region: E. Luijten, M. E. Fisher and A. Z. Panagiotopoulos, Phys. Rev. Lett. 88, 185701 (2002)
    • (2002) Phys. Rev. Lett. , vol.88 , pp. 185701
    • Luijten, E.1    Fisher, M.E.2    Panagiotopoulos, A.Z.3
  • 113
    • 78751669249 scopus 로고    scopus 로고
    • Y. C. Kim, M. E. Fisher and E Luitjen, cond-mat/0304032
    • Y. C. Kim, M. E. Fisher and E Luitjen, cond-mat/0304032.
  • 123
    • 78751676495 scopus 로고    scopus 로고
    • Ecclesiastes 1:9. [The Holy Bible, King James Version, American Bible Society, New York, 1999).]
    • Ecclesiastes 1:9. [The Holy Bible, King James Version, American Bible Society, New York, 1999).]
  • 126
    • 0035506610 scopus 로고    scopus 로고
    • Phys. Rev. E, 64 056101 (2001)
    • (2001) Phys. Rev. e , vol.64 , pp. 056101
  • 134
    • 78751665109 scopus 로고    scopus 로고
    • M. Kolesik, M. A. Novotny, and P. A. Rikvold, cond-mat 0207405
    • M. Kolesik, M. A. Novotny, and P. A. Rikvold, cond-mat 0207405.
  • 135
    • 78751664017 scopus 로고    scopus 로고
    • In this section we refer to E rather than its dimensionless counterpart ℰ ≡ βE since it seems advisable to make the measure of temperature β = 1/kT explicit
    • In this section we refer to E rather than its dimensionless counterpart ℰ ≡ βE since it seems advisable to make the measure of temperature β = 1/kT explicit.
  • 138
    • 78751658366 scopus 로고    scopus 로고
    • Different authors have made different choices of notation and adopted different sign conventions at this point. We adopt those of Ref. 122
    • Different authors have made different choices of notation and adopted different sign conventions at this point. We adopt those of Ref. 122.
  • 140
    • 78751656757 scopus 로고    scopus 로고
    • The constant k merely serves to assign a convenient lower (or upper) bound to the weights; the value chosen is absorbed in the normalization of the sampling distribution
    • The constant k merely serves to assign a convenient lower (or upper) bound to the weights; the value chosen is absorbed in the normalization of the sampling distribution.
  • 143
    • 78751656305 scopus 로고    scopus 로고
    • The WL scheme is terminated when the histogram is " sufficiently" flat and the weightmodification factor f is "sufficiently" close to unity
    • The WL scheme is terminated when the histogram is " sufficiently" flat and the weightmodification factor f is "sufficiently" close to unity.
  • 146
    • 78751670653 scopus 로고    scopus 로고
    • One may count the macrostate-transitions that are actually implemented, as in Ref. 55 or (retaining somewhat more information) store the acceptance probabilities of all attempted transitions, as in Ref. 126b
    • One may count the macrostate-transitions that are actually implemented, as in Ref. 55 or (retaining somewhat more information) store the acceptance probabilities of all attempted transitions, as in Ref. 126b.
  • 147
    • 0034341458 scopus 로고    scopus 로고
    • J. Stat. Phys. 98, 321 (2000)
    • (2000) J. Stat. Phys. , vol.98 , pp. 321
  • 148
    • 78751666159 scopus 로고    scopus 로고
    • There is a caveat here: The traverse through the interesting region needs to be slow enough to allow a rough local equilibrium in each macrostate to be established; this condition is satisfied relatively poorly at the outset (where the MC trajectory heads rapidly for the equilibrium states); but it gets better the closer the sampling distribution comes to the multicanonical limit; one requires no more than this
    • There is a caveat here: The traverse through the interesting region needs to be slow enough to allow a rough local equilibrium in each macrostate to be established; this condition is satisfied relatively poorly at the outset (where the MC trajectory heads rapidly for the equilibrium states); but it )gets better the closer the sampling distribution comes to the multicanonical limit; one requires no more than this.
  • 154
    • 78751652226 scopus 로고    scopus 로고
    • J. S. Wang and R. H. Swendsen, cond-mat/0104418
    • J. S. Wang and R. H. Swendsen, cond-mat/0104418.
  • 155
    • 78751667288 scopus 로고    scopus 로고
    • We subsume into the definitions of the macrovariables any parameters that do not have the credentials ("linearity") of the fields flg.)
    • We subsume into the definitions of the macrovariables any parameters that do not have the credentials ("linearity") of the fields flg.
  • 156
    • 78751673372 scopus 로고    scopus 로고
    • The unspecified constant in Eq. (B3) is prescribed by normalization
    • The unspecified constant in Eq. (B3) is prescribed by normalization.)
  • 160
    • 78751653086 scopus 로고    scopus 로고
    • The name seems to reflect no more than the fact that the resulting estimates of PDFs are as always based on (normalized) histograms accumulated in the sampling process
    • The name seems to reflect no more than the fact that the resulting estimates of PDFs are as always based on (normalized) histograms accumulated in the sampling process.)
  • 161
    • 78751674695 scopus 로고    scopus 로고
    • The observable Q is then a delta function that picks out prescribed values for the macrovariables in the setfMg
    • The observable Q is then a delta function that picks out prescribed values for the macrovariables in the setfMg.
  • 162
    • 78751665393 scopus 로고    scopus 로고
    • To do so, one must determine a set of weights that moderate the relative contributions of each histogram. There is more than a passing similarity here to the expanded ensemble framework discussed in Section III.C
    • To do so, one must determine a set of weights that moderate the relative contributions of each histogram. There is more than a passing similarity here to the expanded ensemble framework discussed in Section III.C.


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