Oct 12, 2007 - Integral equation procedure based on tailored orthogonal functions for the XY spin fluid ...... from the Carnahan-Starling equation of state 23 .
PHYSICAL REVIEW E 76, 041502 共2007兲
Integral equation procedure based on tailored orthogonal functions for the XY spin fluid in an external magnetic field F. Lado Department of Physics, North Carolina State University, Raleigh, North Carolina 27695-8202, USA
E. Lomba Instituto de Química Física Rocasolano, CSIC, Serrano 119, E-28006 Madrid, Spain 共Received 3 September 2007; published 12 October 2007兲 The classical XY model describes particles in three-dimensional space that carry magnetic moments or spins whose motion is restricted to rotations in a plane. Introduction of an external magnetic field lying in the same plane then generates a system that is anisotropic in the azimuthal angle . We use numerical simulations and integral equation techniques to study this system, producing in the latter case a formalism that is identical to that of the simpler isotropic version having no external field. The basis for this simplification is a generalization Em共兲 of the ordinary exponential basis set eim that restores orthogonality in the presence of the external field. We display results of sample calculations obtained with two integral equation closures, reference hypernettedchain and soft mean-spherical approximation, both coupled to the Lovett-Mou-Buff-Wertheim relation, along with results from the numerical simulations for comparison. Construction of the Em共兲 is described in an Appendix. DOI: 10.1103/PhysRevE.76.041502
PACS number共s兲: 61.20.Gy, 02.60.⫺x, 75.10.Hk, 75.50.Mm
I. INTRODUCTION
The response to internal and external magnetic fields of molecular magnetic moments in a classical fluid is variously modeled as occurring along an axis 共Ising model兲, on a plane 共XY model兲, or in space 共Heisenberg model兲. All of these instances have long been studied using the basic techniques of equilibrium statistical mechanics: Mean field theory 关1–7兴, integral equations 关8–16兴, and numerical simulation 关4,7,11,16–19兴. In this work, we focus on one of the less studied cases, the XY spin fluid in an external magnetic field, with the purpose of extending to it the integral equation technique based on generalized orthogonal functions that we have previously used for the Heisenberg model 关10,11兴. We are following on the recent work of Omelyan et al. 关14,15兴, who have presented the first integral equation studies of this model. Academic though this model may appear, it turns out to be useful in the description of superfluid and demixing transitions in 2He- 4He mixtures 关20–22兴. The classical XY spin fluid 共N particles in volume V at temperature T兲 in an external magnetic field B0 is defined by the canonical partition function Z=
1 N!共2⌳3兲N
冋
冕兿
att u0共r兲 = urep 0 共r兲 + u0 共r兲,
共2兲
where urep 0 共r兲 is the repulsive part of the Lennard-Jones potential 共truncated at its minimum and shifted兲, urep 0 共r兲 =
再
4⑀关共/r兲12 − 共/r兲6兴 + ⑀ , r ⬍ 21/6 , r 艌 21/6 ,
0,
冎
共3兲
and uatt 0 共r兲 is an attractive potential of Yukawa form, uatt 0 共r兲 = − J共r兲,
J共r兲 =
⑀ −共r/−1兲 e . r/
共4兲
共5兲
N
共dr jd j兲
j=1
册
⫻exp  兺 j · B0 −  兺 u0共rij兲 −  兺 uss共rij, i, j兲 . j
define the x axis, so that is the azimuthal angle and · B0 = B0 cos , 0 艋 艋 2. The internal energy of the system is written as pairwise sums of a spherically symmetric potential u0共r兲 and a spin-spin potential uss共r , 1 , 2兲. We follow Omelyan et al. 关14,15兴 in putting
i⬍j
i⬍j
Here and are dimensionless strength and range parameters, respectively. Finally, the spin-spin interaction tending to align two spins with one another is taken to be of XY form 关14,15兴,
共1兲 Here  = 1 / kBT is the inverse Kelvin temperature, with kB Boltzmann’s constant, ⌳ the de Broglie thermal wavelength 共which plays only a dimensional role兲, and the orientation of a magnetic dipole moment that is free to rotate in a plane, denoted the xy plane. The uniform external magnetic field B0 also lies in that plane and its direction is taken to 1539-3755/2007/76共4兲/041502共10兲
uss共r, 1, 2兲 = − J共r兲sˆ 1 · sˆ 2 = − J共r兲cos 12 ,
共6兲
where sˆ is the unit spin vector in the direction of and 12 ⬅ 1 − 2. The formal elements of the calculation are neatly displayed by factoring the partition function Z = ZidZex into an ideal part and the excess; we get
041502-1
©2007 The American Physical Society
PHYSICAL REVIEW E 76, 041502 共2007兲
F. LADO AND E. LOMBA
1 Z = N!共2⌳3兲N id
=
冋
冕兿 N
j=1
1 VI0共B0兲 N! ⌳3
册
冋
j
冋 册 冕 冕
册
N
共7兲
,
冕
1 Z = 共2V兲N ex
冋
共dr jd j兲exp  兺 j · B0
d f共1兲 ln = d1 f 0共 1兲 2
兿 j=1 关dr jd j f 0共 j兲兴
册
i⬍j
共8兲
where u共r , 1 , 2兲 ⬅ u0共r兲 + uss共r , 1 , 2兲 is the total pair potential. This defines in Eq. 共8兲 the one-body distribution function f 0共兲 in the noninteracting XY spin system, B0 cos
e , I0共B0兲
冕
2
d f 0共兲 = 1.
共10兲
0
In these expressions, I0共z兲 is the modified Bessel function of order zero.
冓
j=1
冔
f共兲, 共11兲 2
共2兲共r, ,r⬘, ⬘兲 =
冓兺
␦共r − ri兲␦共 − i兲␦共r⬘ − r j兲␦共⬘ − j兲
i⫽j
=
2 f共兲f共⬘兲g共兩r − r⬘兩, , ⬘兲, 共2兲2
d2 f共2兲c共r, 1, 2兲
0
共13兲
冕 冕
2
dr
d2 f共2兲
0
⫻ g共r, 1, 2兲
duss共r, 1, 2兲 . 共14兲 d1
Calculation of f共兲 from these equations requires knowing the pair function c共r , 1 , 2兲 or g共r , 1 , 2兲. In classical liquid state theory, the pair distribution function is obtained from the Ornstein-Zernike 共OZ兲 equation and a closure relation 关23,24兴. The first of these, generalized for anisotropy, reads
2
冕
dr3d3 f共3兲关␥共r13, 1, 3兲
+ c共r13, 1, 3兲兴c共r32, 3, 2兲
Since the molecules are subject to both one-body and two-body forces, a complete statistical description of the system requires knowing both the one-body and two-body density functions, =
2
dr
冋 册
A. One-body and two-body distribution functions
兺 ␦共r − r j兲␦共 − j兲
冕 冕
f共1兲 d ln =− d1 f 0共 1兲 2
II. ANISOTROPIC INTEGRAL EQUATIONS
N
0
冔
共15兲
for the indirect correlation function ␥ = g − 1 − c. The second, or closure, relation expresses the direct correlation function c back in terms of ␥ and the model’s pair interactions, c共r, 1, 2兲 = exp关− u0共r兲 − uss共r, 1, 2兲 + ␥共r, 1, 2兲 + b共r, 1, 2兲兴 − 1 − ␥共r, 1, 2兲.
共16兲
This relation must be supplemented with an approximation for b, the so-called bridge function 关27兴, which is formally defined in terms of a diagram summation 关23兴 that offers little practical benefit. Most approximate closures for c define b implicitly. For a given f共兲, Eqs. 共15兲 and 共16兲 are solved iteratively for ␥共r , 1 , 2兲 and c共r , 1 , 2兲, after which one can find
共12兲
where = N / V is the density and f共兲 the one-body orientational distribution in the interacting fluid. Equation 共12兲 defines the generalized pair distribution function g共r , , ⬘兲 of the anisotropic spin system in an external magnetic field. The angular brackets in these definitions denote a canonical ensemble average exemplified in Eq. 共1兲. The basic equations that determine the distribution functions f共兲 and g共r , 1 , 2兲 are well known 关23,24兴. The onebody distribution function f共兲 can be calculated using the Lovett-Mou-Buff-Wertheim 共LMBW兲 equation 关25,26兴,
df共2兲 d2
where c共r , 1 , 2兲 is the anisotropic direct correlation function 共see below兲. Alternatively, the one-body density, Eq. 共11兲, can be differentiated with respect to to give the first member of a Kirkwood-Born-Green-Yvon 共KBGY兲 hierarchy 关23兴,
␥共r12, 1, 2兲 =
共1兲共r, 兲 =
d2c共r, 1, 2兲
d ln f共2兲 , d2
共9兲
normalized such that 1 2
2 ⫻
N
⫻exp −  兺 u共rij, i, j兲 ,
f 0共 兲 =
=
2
dr
g共r, 1, 2兲 = exp关− u0共r兲 − uss共r, 1, 2兲 + ␥共r, 1, 2兲 + b共r, 1, 2兲兴
共17兲
and evaluate Eq. 共13兲 or 共14兲. 共In the results reported below we have used the LMBW relation.兲 This produces a new f共兲 and so requires a new round for ␥共r , 1 , 2兲 and c共r , 1 , 2兲, which depend on this function through Eq. 共15兲. These iterations continue until both one-body and two-body functions are self-consistently determined. It is obvious that to actually carry out a numerical version of these procedures with Eqs. 共13兲–共16兲, the equations must first be simplified.
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INTEGRAL EQUATION PROCEDURE BASED ON TAILORED … B. Tailored exponential functions
Pair functions such as ␥共r , 1 , 2兲 may be expanded in an angle-dependent basis set so that the orientational information they contain might be effectively conveyed by a small set of expansion coefficients. Thus one could write, for example,
␥共r, 1, 2兲 =
兺 m ,m 1
* ⌫m1m2共r兲Em1共1兲Em 共2兲, 2
PHYSICAL REVIEW E 76, 041502 共2007兲
A specific expansion needed in the sequel is that of the spin-spin interaction, Eq. 共6兲, where we have cos 12 = cos 1 cos 2 + sin 1 sin 2 , while, from Appendix A, 1 cos = 1 + 共1 − 221 + 2兲1/2关E1共兲 + E*1共兲兴, 2
共18兲
E m共 兲 ⬅ e
= Tm共cos 兲 + iVm共cos 兲.
sin = 共19兲
Here Tm共cos 兲 = cos m is the type I Chebyshev polynomial of order m and Vm共cos 兲 = sin m the associated Chebyshev function of order m 关28兴. The familiar exponential functions Em共兲 of course have the orthonormalization 1 2
冕
2
0
d Em共兲Em⬘共兲 = ␦mm⬘ *
共20兲
冕
2
d
0
* f共兲Em共兲Em⬘共兲
= ␦mm⬘ ,
共21兲
again for m , m⬘ positive or negative integers, where Em共兲 = Tm共cos 兲 + iVm共cos 兲, * E−m共兲 = Em 共兲,
uss共r, 1, 2兲 = − J共r兲
兺
␥m1m2共r兲Em1共1兲Em* 2共2兲.
m1,m2
冕 冕 2
0
d1
2
1 C11 = C−1−1 = 共1 − 21兲. 2
Numerical evaluation of the Ornstein-Zernike equation is simpler in Fourier transform representation, which deconvolves the direct space integral. In transform space, Eq. 共15兲 becomes ˜␥共k, 1, 2兲 =
¯ ⬅ −m. where m
冕
2
d3 f共3兲关˜␥共k, 1, 3兲
0
共32兲
with one remaining integration. Because the orientations of spins 1 and 2 and that of r12 are decoupled in the XY model, the transforms may be performed holding the former fixed. A transform pair is then ˜␥共k, 1, 2兲 = 4
冕
⬁
0
1 22
冕
⬁
0
dr r2␥共r, 1, 2兲
sin共kr兲 , kr
dk k2˜␥共k, 1, 2兲
sin共kr兲 , kr
共33兲
共34兲
and the transformed functions may themselves be expanded, ˜␥共k, 1, 2兲 =
共26兲
2
˜ 共k, 3, 2兲, + ˜c共k, 1, 3兲兴c
Coefficients of all pair functions satisfy the symmetries
␥m1m2共r兲 = ␥m2m1共r兲 = ␥m¯2m¯1共r兲 = ␥m¯1m¯2共r兲,
共31兲
C. Ornstein-Zernike equation
␥共r, 1, 2兲 = 共25兲
共30兲
1 C10 = C01 = C0−1 = C−10 = 1共1 − 221 + 2兲1/2 , 2
0
1
2
2
1 C1−1 = C−11 = 共2 − 21兲, 2
d2 f共1兲f共2兲
* 共1兲Em2共2兲. ⫻ ␥共r, 1, 2兲Em
* Cm1m2Em1共1兲Em 共2兲,
C00 = 21 ,
In Eq. 共24兲, m1 and m2 range as noted over positive and negative integers. With Eq. 共21兲, its inversion becomes 1 ␥m1m2共r兲 = 共2兲2
共29兲
with nonzero coefficients
共23兲
共24兲
兺 m ,m 1
共22兲
for m 艌 0. Construction of the needed generalized Chebyshev functions Tm共cos 兲 and Vm共cos 兲 is described in Appendix A. Equation 共18兲 is now generalized to read
␥共r, 1, 2兲 =
1 共1 − 2兲1/2关E1共兲 − E*1共兲兴. 2i
Here the k are quasimoments of f共兲, defined in Eq. 共A2兲. Combining these expressions and simplifying, we find
for m , m⬘ positive or negative integers. We write the expansion coefficients in Eq. 共18兲 in upper case ⌫ to distinguish them from the distinct set of coefficients written in lower case ␥ to be introduced next. The external field B0 along the x axis produces a nonuniform distribution f共兲 that enters the integrand of Eq. 共20兲 and spoils the orthogonality of the standard exponential functions Em共兲. To regain the valuable property of orthogonality and work conveniently with the external field turned on, we introduce new basis functions Em共兲 that are orthonormal with weight function f共兲, 1 2
共28兲
2
where the asterisk denotes complex conjugate and im
共27兲
兺 m ,m 1
* ˜␥m m 共k兲Em 共1兲Em 共2兲, 1 2 1 2
共35兲
2
with expansion coefficients ˜␥m1m2共k兲 that are directly the Fourier transforms of the corresponding ␥m1m2共r兲.
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PHYSICAL REVIEW E 76, 041502 共2007兲
F. LADO AND E. LOMBA ⬁
We may now begin to garner the profit from the special orthogonality built into the basis set Em共兲. Expanding the transforms as in Eq. 共35兲, we find the final integral in Eq. 共32兲 can be carried out to yield a relation between expansion coefficients alone, ˜ m m 共k兲, ˜␥m m 共k兲 = 兺 关˜␥m m 共k兲 + ˜cm m 共k兲兴c 1 2 1 3 1 3 3 2
共36兲
m3
兺 Dm m Am m =1 1 2
2
⬁
m2=1 m3=−⬁
兺 Qm m Am m =1 1 2
共37兲
共FT兲−1
OZ
Q m1m2 ⬅
兺 AmTm共cos 兲 m=0
兺
m2=1
A m2
dTm2共cos 1兲 d1
+
兺 兺
where we have used D11 = 21 关共1 − 2兲 / 共1 − 221 + 2兲兴1/2. Then the form of the angular distribution function remains unchanged, ln f共兲 ⬇ const + B cos ,
It follows then that
冕
0
* d f共兲Em 共兲 1
B=
共40兲
where 2
共46兲
共47兲
dTm2共cos 兲 d
.
B0 1 − ˜c11共0兲 + ˜c1−1共0兲
.
共48兲
We will find that Eqs. 共47兲 and 共48兲 are effectively exact in the context of the numerical solutions. We further anticipate the calculations of Sec. IV by noting that the alternative approach based on the KBGY relation, Eq. 共14兲, although not further discussed here, produces essentially the same effective fields.
Em1共1兲˜cm1m3共0兲
⫻iDm3m2Am2 ,
−i D m1m2 ⬅ 2
B0共1 − 221 + 2兲1/2 , 1 − ˜c11共0兲 + ˜c1−1共0兲
but the effective field in the interacting system becomes
⬁
m1,m2=1 m3=−⬁
B0共1 − 2兲1/2 , 2D11关1 − ˜c11共0兲 + ˜c1−1共0兲兴
⬇
共39兲
d cos 1 = B0 d1 ⬁
共45兲
where in matrix notation R ⬅ Q−1. If it happens that A1 is much larger than A2 , A3 , . . ., then we may ignore these higher coefficients in the analysis above to get
and seek to find the Am. 共Actually, A0 is determined by normalization, so we are concerned with A1 , A2 , . . .兲 Expanding c共r , 1 , 2兲, we first find from Eq. 共13兲 that ⬁
共44兲
1 Am = B0共1 − 2兲1/2Rm1 , 2
⬁
ln f共兲 =
兺
m3=−⬁
For m1 , m2 ⬎ 0, the matrix elements Dm1m2 are nonzero only for m2 = m1 , m1 + 1, while Dm¯1m2 = −Dm1m2. We have finally from Eq. 共43兲,
共38兲
Equation 共13兲 also simplifies smartly with expansions in the generalized Chebyshev functions. We put
关␦m1m3 − ˜cm1m3共0兲兴Dm3m2 .
⬁
A1 ⬇
D. LMBW equation for f„…
共43兲
where
˜ m m 共k兲其 兵␥m1m2共r兲其 ——→ 兵cm1m2共r兲其 ——→ 兵c 1 2 → 兵˜␥m1m2共k兲其 ——→ 兵␥m1m2共r兲其.
1 = B0共1 − 2兲1/2␦m11 , 2
2
2
共FT兲
closure
˜cm1m3共0兲Dm3m2Am2 , 共42兲
or more briefly ⬁
˜ 共k兲 and C ˜ 共k兲 are symmetric matrices with elements where ⌫ ˜␥m m 共k兲 and ˜cm m 共k兲, respectively, and I is the unit matrix. 1 2 1 2 The notable feature of Eq. 共37兲 is that it is identical to that of an isotropic system with B0 = 0. All of the anisotropy is built into the basis functions. The closure relation 共16兲 is also reduced to manipulation of coefficients. For a given r, first the full function g共r , 1 , 2兲 is assembled in Eq. 共17兲 by performing summations like Eq. 共24兲 for the terms in its exponent; this is followed by calculation of its coefficients gm1m2共r兲 with an inversion like that of Eq. 共25兲. Then the cm1m2共r兲 = gm1m2共r兲 − ␦m10␦m20 − ␥m1m2共r兲 follow from Eq. 共16兲 in coefficient form. One cycle of the iterative 共numerical兲 solution that yields a converged set of coefficients 兵␥m1m2共r兲其 will then consist of four steps:
⬁
兺 兺
+
which can be solved for the ˜␥m1m2共k兲; in matrix form, ˜ 共k兲 = C ˜ 共k兲C ˜ 共k兲关I − C ˜ 共k兲兴−1 , ⌫
1 = B0共1 − 2兲1/2␦m11 2
2
共41兲
III. THERMODYNAMICS
Once the one-body and two-body distribution functions have been determined, it is straightforward to calculate the various thermodynamic quantities. For instance, the internal energy E = −共 ln Z / 兲 and the pressure p = kBT共 ln Z / V兲 are found as quadratures,
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INTEGRAL EQUATION PROCEDURE BASED ON TAILORED …
1 E 3 = − B01 + 2 N 2 1 + 2 1 p =1− 6 ⫻
兺
m1,m2
冕
冕
dr
兺 m ,m 1
dr g00共r兲r
冕
dr g00共r兲u0共r兲
共ss兲 gm1m2共r兲um m 共r兲,
共49兲
1 2
2
du0共r兲 1 + 6 dr
冕 冉
dr 1 +
r
共ss兲 gm1m2共r兲um m 共r兲,
冊 共50兲
1 2
in terms of the coefficients of the distribution function g共r , 1 , 2兲 and the pair potentials u0共r兲 and uss共r , 1 , 2兲, while the isothermal compressibility KT is given by
kBTKT = 1 + ˜h00共0兲,
共51兲
where h = g − 1. Finally, an expression for the Helmholtz free energy is presented in Appendix B. One notes again that expansions in the generalized basis set Em共兲 lead to equations that are identical to those of an isotropic system. Further, we can compute the magnetic properties of the XY spin fluid in a uniform magnetic field B0. The net magnetization in the direction ␣ = x , y is M␣ =
1 ln Z =  B 0␣
冓 冔
兺 s j␣ ,
␣ ⬅
1 M ␣  = V B 0 V ⫻
冓 兺 冔册 s j
f 0共 兲 =
s i␣
s j
j
eB0 cos . I0共B0兲
共56兲
For the interacting system, Omelyan et al. 关15兴 find that in setting ⬁
ln f共兲 = const + B0 cos +  兺 amTm共cos 兲, 共57兲 m=1
with standard Chebyshev polynomials Tm共cos 兲, a2 can be as large as 30% of a1 and only terms am with m ⬎ 3 are negligible. In contrast, we find that in setting ⬁
ln f共兲 =
j
兺 AmTm共cos 兲,
共58兲
m=0
冋冓冉兺 冊冉兺 冊冔 冓 兺 冔 i
= sin m. Here we aim to present just a small sample of additional data to illustrate application of the alternative approach using orthogonal basis functions Tm共cos 兲 and Vm共cos 兲 that are specifically tailored to the anisotropic environment created by the external field; aside from the enhanced simplicity of the formal expressions already manifest in the equations displayed above, the most notable difference is found in the results for f共兲. The interaction of an individual spin with the external field involves the familiar dipole potential, − · B0, and leads to the unperturbed distribution function
共52兲
and so the magnetic susceptibility components are 2
PHYSICAL REVIEW E 76, 041502 共2007兲
−
s i␣
i
.
共53兲
j
Now with B0 defining the x axis, we get the longitudinal, L ⬅ xx, and transverse, T ⬅ yy, magnetic susceptibilities as
with tailored Chebyshev polynomials Tm共cos 兲, no more than the linear term in the expansion is significant; the coefficients of higher terms 共A2 , A3 , . . . 兲 are smaller than A1 by two or more orders of magnitude. Correspondingly, the approximate A1 of Eq. 共46兲 differs from the exact A1 of Eq. 共45兲 by no more than 0.1%; within the overall precision of a numerical calculation, Eq. 共46兲 may be treated as exact. Thus f共兲 retains the same functional form as the unperturbed system,
L/2 = 21关1 + ˜h00共0兲兴 + 21共1 − 221 + 2兲1/2˜h10共0兲
f共兲 =
1 + 共1 − 221 + 2兲关1 + ˜h11共0兲 + ˜h1−1共0兲兴, 2 共54兲 1 T/2 = 共1 − 2兲关1 + ˜h11共0兲 − ˜h1−1共0兲兴. 2
共55兲
The off-diagonal elements of vanish, while the magnetization is M = N1. IV. SAMPLE RESULTS
A very thorough integral equation study of the XY spin fluid in an external magnetic field has recently been published by Omelyan et al. 关15兴 using a soft mean-spherical approximation 共SMSA兲 and we need not duplicate that scope of data. That work was based on expansions in the ordinary Chebyshev functions Tm共cos 兲 = cos m and Vm共cos 兲
eB cos , I0共B兲
共59兲
but with an effective field B given by Eq. 共48兲. For the sample calculations here, we use both the SMSA and the reference hypernetted-chain 共RHNC兲 closures, the latter optimized to achieve a minimum in the free energy. The SMSA closure 关23兴 is a generalization of the meanspherical approximation that is widely used for potentials with a hard core. Splitting the total potential u共r , 1 , 2兲 = uSR共r , 1 , 2兲 + uLR共r , 1 , 2兲 into short-range 共SR兲 and long-range 共LR兲 parts, one replaces the formally exact but uncomputable 关because of the unknown b共r , 1 , 2兲兴 Eq. 共17兲 with the approximation gSMSA共r, 1, 2兲 = e−u
SR共r, , 兲 1 2
关1 + ␥共r, 1, 2兲
− uLR共r, 1, 2兲兴.
共60兲
A particularly convenient choice for the potential splitting which we shall adopt here is
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PHYSICAL REVIEW E 76, 041502 共2007兲
F. LADO AND E. LOMBA
For convenience, in the present calculation we have chosen as reference a hard sphere fluid at the same density,
as described in Ref. 关35兴, with the orientational degrees of freedom restricted to lie in the xy plane. We have used a sample of 864 particles and performed averages over 50 000 configurations. Each configuration implies 864 displacement attempts, with an acceptance ratio of 50%, and either 864 single-particle moves or one cluster move. The initial 20 000 configurations in each run were discarded to attain thermal equilibrium. Thermodynamic and magnetic properties of the XY spin fluid computed with an RHNC-LMBW combination are presented in Table I. Additionally, in Table II the results obtained using the SMSA-LMBW combination are laid out for just the case of nonzero field. A global assessment of the quality of the closure approximations can be made from inspection of Fig. 1, in which these results are compared with Monte Carlo simulation data. From this figure, it is clear that the RHNC-LMBW approach is practically exact in the zero field case. At B0 / = 5, thermodynamic properties from RHNC-LMBW compare well with simulation data and the approach is clearly superior to the SMSA-LMBW. Small discrepancies show up as density increases, especially for the net magnetization M / N = 1. Interestingly, the simulation data for this quantity are bracketed by the SMSA-LMBW and RHNC-LMBW results. This points to a possible route to improve the quality of the theoretical approach. In the spirit of the Zerah-Hansen HMSA approximation 关36兴, one might use a continuous interpolation scheme between the SMSA and HNC 共or RHNC兲 closures to implement a thermodynamically self-consistent approach based on the net magnetization and the longitudinal susceptibility as the key quantities to check for consistency. To illustrate the structure, we define the functions
bref共r, 1, 2兲 ⬇ bHS共r; 0兲,
g00共r兲 ⬅ 具g共r, 1, 2兲典12 = g00共r兲,
rep
gSMSA共r, 1, 2兲 = e−u0
共r兲
关1 + ␥共r, 1, 2兲 − uatt 0 共r兲
− uss共r, 1, 2兲兴,
共61兲
since this linearizes the angular dependence of the pair distribution function. Its coefficients can then be immediately obtained by inspection, rep
SMSA 共r兲 = e−u0 g00
共r兲
共ss兲 关1 + ␥00共r兲 − uatt 0 共r兲 − u00 共r兲兴,
共62兲 rep
SMSA −u0 gm m 共r兲 = e 1 2
共r兲
共ss兲 关␥m1m2共r兲 − um m 共r兲兴, 1 2
for m1m2 ⫽ 00, 共63兲
which saves some computation time in the iteration step closure
兵␥m1m2共r兲其 → 兵cm1m2共r兲其 compared to the RHNC procedure. 共It should be noted that the full RHNC calculation is already extremely fast on any current personal computer—on the order of seconds per thermodynamic state.兲 If urep 0 共r兲 were the hard sphere potential, these equations would describe the standard mean-spherical approximation 关23兴. The RHNC closure 关29,30兴 consists in replacing the unknown function b共r , 1 , 2兲 in Eq. 共17兲 with the bridge function bref共r , 1 , 2兲 obtained from some calculable reference model, gRHNC共r, 1, 2兲 = exp关− u0共r兲 − uss共r, 1, 2兲 + ␥共r, 1, 2兲 + bref共r, 1, 2兲兴.
共64兲
共65兲
where 0 is the reference sphere diameter that is varied to attain a minimized free energy 关31兴. For the modeled hard sphere functions we use the parametrizations of Verlet-Weis 关32兴 and Henderson-Grundke 关33兴. In the RHNC calculation, expansions in m such as Eqs. 共24兲 and 共39兲 have been extended through 兩m兩 = 4; reduction to a maximum of 兩m兩 = 3 yields unchanged results. For SMSA, the linearization of the closure in Eq. 共61兲 limits the nonvanishing terms in all expansions to just those in the potential; i.e., through 兩m兩 = 1. We note that given the symmetries 共26兲, the number of distinct coefficients in the expansion of a pair function is 共mmax + 1兲2 for a 共maximum兩m兩 ⬅ mmax兲 = 0 , 1 , 2 , 3 , 4 , . . .. We have set = = 1 in the pair potentials for all calculations reported 关see Eqs. 共4兲 and 共5兲兴. Transform functions of r and k are connected through the fast Fourier transform 关34兴 and are evaluated on uniform grids r j = j⌬r, k j = j⌬k, j = 0 , 1 , 2 , . . . , Nr, with ⌬r⌬k = / Nr and Nr = 1024, ⌬r = 0.02. Integrals over r and k for thermodynamic quantities are calculated with the trapezoidal rule using the same grid points, while Gauss-Chebyshev quadratures over angle, Eq. 共A10兲, are carried out with n 艌 10 Gaussian root points. With regard to the simulation, we have used a standard Monte Carlo method whose only peculiarity is the implementation of orientational single-particle and cluster moves
g11共r兲 ⬅ 具g共r, 1, 2兲cos 12典12 =
兺
m1,m2
共66兲
Cm1m2gm1m2共r兲, 共67兲
where the averages are calculated as 具H共r, 1, 2兲典12 ⬅
1 共2兲2
冕 冕 2
d1
0
2
d2
0
⫻ f共1兲f共2兲H共r, 1, 2兲.
共68兲
Plots of g00共r兲 and g11共r兲 for various representative states are shown in Figs. 2 and 3. The radial coefficient g00共r兲 is very well reproduced by both approximations; departures in the long-range behavior of g11共r兲 become apparent at the larger density and nonzero field. This is an immediate consequence of the deviations observed in the calculated magnetization, since lim g11共r兲 = 共M/N兲2 .
r→⬁
Again, we see that the angular coefficient from the simulation lies within the boundaries formed by the SMSA-LMBW and RHNC-LMBW results.
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PHYSICAL REVIEW E 76, 041502 共2007兲
TABLE I. Thermodynamic and magnetic properties of the XY spin fluid in an external magnetic field B0 at reduced temperature kBT / ⑀ = 10, calculated using the optimized RHNC closure with the LMBW equation; 0 is the optimized sphere diameter of the reference hard sphere system. The excess free energy is defined here as Aex / N ⬅ A / N − ln共⌳3兲 + 1.
3
0 /
B0 / ⑀
B / ⑀
E / N − 3 / 2
p / − 1
共kBTKT兲−1
M / N
L / 2
T / 2
Aex / N
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
0.8889 0.8917 0.8935 0.8943 0.8942 0.8930 0.8914 0.8873 0.8900 0.8917 0.8924 0.8923 0.8915 0.8900 0.8877 0.8851 0.8819
0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 5
0 0 0 0 0 0 0 5.749 6.738 8.069 9.858 12.174 14.956 18.032 21.238 24.483 27.730
−0.105 −0.207 −0.304 −0.394 −0.475 −0.545 −0.604 −0.254 −0.394 −0.547 −0.719 −0.914 −1.127 −1.342 −1.542 −1.714 −1.848
0.041 0.115 0.231 0.401 0.638 0.960 1.383 0.031 0.087 0.173 0.292 0.448 0.656 0.944 1.349 1.906 2.656
1.098 1.302 1.654 2.201 3.006 4.143 5.642 1.074 1.229 1.479 1.835 2.337 3.096 4.282 6.064 8.613 12.116
0 0 0 0 0 0 0 0.276 0.319 0.374 0.441 0.518 0.595 0.663 0.717 0.759 0.792
0.578 0.687 0.848 1.110 1.610 2.928 14.07 0.596 0.693 0.799 0.883 0.886 0.772 0.592 0.424 0.300 0.214
0.578 0.687 0.848 1.110 1.610 2.928 14.07 0.552 0.638 0.748 0.883 1.036 1.189 1.324 1.432 1.516 1.581
0.034 0.083 0.150 0.238 0.350 0.492 0.666 −0.037 0.000 0.051 0.116 0.196 0.293 0.411 0.556 0.738 0.967
V. CONCLUSIONS
ACKNOWLEDGMENTS
In summary, we have shown that a general integral equation procedure based on tailored orthogonal basis functions, successfully applied in previous works 关10,11兴 to describe the Heisenberg spin fluid in an external field, can be readily extended to the classical XY model in an external field. The specific RHNC-LMBW approximation is found to work reasonably well, although some deviations are noticed at high densities and nonzero fields. We speculate that the quality of the theoretical approximation might be improved by implementing self-consistency at the level of the magnetic quantities 共i.e., magnetic susceptibility vs. magnetization兲.
E.L. gratefully acknowledges financial support from the Dirección General de Investigación Científica y Técnica under Grant No. FIS2004-02954-C03-01 and the Dirección General de Universidades e Investigación de la Comunidad de Madrid under Grant No. S0505/ESP/0299, Program MOSSNOHO-CM. APPENDIX A: GENERALIZED CHEBYSHEV FUNCTIONS
In principle, the needed generalization of the type I Chebyshev polynomials Tm共cos 兲 = cos m can be produced
TABLE II. Thermodynamic and magnetic properties of the XY spin fluid in an external magnetic field B0 at reduced temperature kBT / ⑀ = 10, calculated using the SMSA closure with the LMBW equation.
3
B0 / ⑀
B / ⑀
E / N − 3 / 2
p / − 1
共kBTKT兲−1
M / N
L / 2
T / 2
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
5 5 5 5 5 5 5 5 5 5
5.718 6.640 7.837 9.383 11.319 13.608 16.138 18.786 21.468 24.142
−0.253 −0.392 −0.541 −0.706 −0.890 −1.090 −1.297 −1.500 −1.685 −1.846
0.028 0.079 0.156 0.265 0.407 0.593 0.842 1.179 1.633 2.233
1.083 1.257 1.546 1.977 2.606 3.555 5.024 7.268 10.596 15.419
0.275 0.315 0.365 0.424 0.491 0.560 0.623 0.677 0.720 0.755
0.592 0.680 0.771 0.840 0.847 0.763 0.616 0.464 0.339 0.248
0.550 0.630 0.729 0.848 0.982 1.120 1.246 1.354 1.441 1.511
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PHYSICAL REVIEW E 76, 041502 共2007兲
F. LADO AND E. LOMBA 0
0.75
-1
0.5
MC µB0/ε=0 MC µB0/ε=5
11
g (r)
βE/N-3/2
µB0/ε = 5
βp/ p/ρ ρ-1
-2 RHNC-LMBW µB0/ε=0 RHNC-LMBW µB0/ε=5 SMSA-LMBW µB0/ε=5
2
0
1
µB0/ε = 0 0.1
g (r)
0
3
ρσ =0.3
11
0.6
M/N M/Nµ µ
RHNC-LMBW SMSA-LMBW
0.25
3
ρσ =0.7
0.05
0.4 0.2 0
0
0
0.2
0.4
ρσ
3
1
0.8
0.6
FIG. 1. 共Color online兲 Density dependence of the excess internal energy, excess pressure, and magnetization for the XY spin fluid at kBT / = 10, with and without external field. Symbols denote data from Monte Carlo simulation and lines correspond to the integral equation results using the RHNC-LMBW and SMSA-LMBW combinations.
0
2
cos − 1 共1 − 221 + 2兲1/2
k ⬅
1
冕
d f共兲cos共k兲.
共A2兲
0
eB cos , I0共B兲
共A3兲
they are readily found to be
µB0/ε=5
k = 1
00
g (r)
共A1兲
Here and for higher-order polynomials we need the quasimoments
f共兲 =
RHNC-LMBW SMSA-LMBW 3 MC ρσ =0.3
0.5
3
MC ρσ =0.7
0 0
.
For the distribution
T0共cos 兲 = 1,
1.5
4
FIG. 3. 共Color online兲 Angular coefficient g11共r兲 calculated in the RHNC-LMBW 共solid lines兲 and SMSA-LMBW 共dotted lines兲 combinations along with Monte Carlo simulation data 共symbols兲; with external field 共top figure兲 and without external field 共bottom figure兲.
T1共cos 兲 = by a straightforward application of the Gram-Schmidt method 关28兴. Thus we can directly find for the first two generalized Chebyshev polynomials, used earlier in the expansion of the spin-spin potential uss共r , 1 , 2兲,
r/σ r/ σ
1
r/σ r/ σ
2
3
FIG. 2. 共Color online兲 Center-to-center pair distribution function g00共r兲 calculated in the RHNC-LMBW 共solid line兲 and SMSALMBW 共dotted line兲 combinations along with Monte Carlo simulation data 共symbols兲 for B0 / = 5.
Ik共B兲 . I0共B兲
共A4兲
In these expressions, I j共z兲 is the modified Bessel function of order j. The simple Gram-Schmidt process, however, employs successive subtractions and so risks a progressive loss of numerical precision as higher-order polynomials are generated. This is of particular concern for accurate GaussChebyshev quadratures based on the roots of a polynomial Tn共x兲 with n = 10 or larger. 共In this appendix we put x ⬅ cos .兲 Thus we have instead followed a robust alternative approach described by Press and Teukolsky 关37兴. The monic version m共x兲 ⬅ Tm共x兲 / 2m−1 for m ⬎ 0 of the Chebyshev polynomials Tm共x兲 have by construction a leading coefficient of unity and satisfy a standard recursion relation,
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INTEGRAL EQUATION PROCEDURE BASED ON TAILORED …
PHYSICAL REVIEW E 76, 041502 共2007兲
−1共x兲 ⬅ 0,
V1共cos 兲 =
0共x兲 ⬅ 1,
共A11兲
By construction, then, these generalized Chebyshev functions Tm共cos 兲 and Vm共cos 兲 satisfy
m+1共x兲 = 共x − ␣m兲m共x兲 − mm−1 , 共A5兲
m = 0,1,2, . . . ,
sin . 共1 − 2兲1/2
with ␣m = 0 and
1
冕
0
冦
1,
m = m⬘ = 0,
d f共兲Tm共cos 兲Tm⬘共cos 兲 = 1/2, m = m⬘ ⫽ 0, 0, m ⫽ m⬘ ,
共A12兲
1 = 1/2, m = 1/4,
m ⬎ 1.
共A6兲
The Press-Teukolsky algorithm then uses these monic coefficients along with the k to recursively generate a new set of coefficients am and bm 共here am = 0兲 such that the monic polynomials pm共x兲 generated with these,
1
p−1共x兲 ⬅ 0,
pm+1共x兲 = 共x − am兲pm共x兲 − bm pm−1 , 共A7兲
m = 0,1,2, . . . , are orthogonal with weight function f共兲, 1
冕
0
if m ⫽ m⬘ .
共A8兲
T0共x兲 = 1, m ⬎ 0.
共A9兲
The algorithm further generates the roots x1 , x2 , . . . , xn of Tn共x兲 = 0 and the weights w j for n-point Gauss-Chebyshev quadrature, 1
冕
0
共A10兲
j=1
The quadrature is exact if H共x兲 is a polynomial of degree less than or equal to 2n − 1. A similar procedure could now be followed for the generalization Um共cos 兲 of type II Chebyshev polynomials Um共cos 兲 关28兴, where then Vm共cos 兲 = sin Um−1共cos 兲. Here however we need the generalizations of Vm共cos 兲 = sin m only through m = 4 共in fact, through m = 3 is adequate兲 and for that limited purpose the simpler GramSchmidt orthogonalization suffices. Explicitly, the first two functions, already used above for the pair potential uss共r , 1 , 2兲, are V0共cos 兲 = 0,
d f共兲Vm共cos 兲Vm⬘共cos 兲 = 1/2, m = m⬘ ⫽ 0, 0, m ⫽ m⬘ ,
冕
共A13兲
d f共兲Tm共cos 兲Vm⬘共cos 兲 = 0.
0
冧
共A14兲
To calculate the total free energy we use the familiar “charging” process, turning on the interactions with a parameter , 0 艋 艋 1. However, as emphasized by Sullivan 关38兴, the one-body distribution f共兲 should remain unchanged as the interaction is turned on. Thus we will also adopt an effective external field B0共兲 designed to maintain fixed the f共兲 found by calculation, Eq. 共59兲. Define then the partition function 1 Z共兲 = N!共2⌳3兲N
冕兿 N
j=1
冋
共dr jd j兲exp B0共兲 兺 cos j
册
j
−  兺 u共rij, i, j兲 ,
n
d f共兲H共cos 兲 ⬇ 兺 w jH共x j兲.
m = m⬘ = 0,
APPENDIX B: HELMHOLTZ FREE ENERGY
A final normalization then yields the desired generalized Chebyshev polynomials,
pm共x兲 , 关2具pm兩pm典兴1/2
冦
0,
Collectively, Eqs. 共A12兲–共A14兲 give rise to the orthonormalization of Em共兲 expressed in Eq. 共21兲. For B0 = 0 and f共兲 = 1 so that k = 0 for all k, the Tm共cos 兲 and Vm共cos 兲 generated in the fashion described above are of course just the standard Chebyshev functions Tm共cos 兲 = cos m and Vm共cos 兲 = sin m.
d f共兲pm共cos 兲pm⬘共cos 兲 = 0,
Tm共x兲 =
0
1
p0共x兲 ⬅ 1,
具pm兩pm⬘典 ⬅
冕
冧
i⬍j
共B1兲
where the external field is such that B0共 = 0兲 = B and B0共 = 1兲 = B0. Then following the analysis of Ref. 关11兴, we find first
A = ln共⌳3兲 − 1 − ln关I0共B兲兴 + 共B − B0兲1 N + ⫻
1 2 共2兲2
冕
冕
dr d1d2 f共1兲f共2兲
1
d g共r, 1, 2兩兲u共r, 1, 2兲.
共B2兲
0
Finally, evaluation of three of the integrals in Eq. 共B2兲 leads to 关11兴
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F. LADO AND E. LOMBA
A A1 = ln共⌳3兲 − 1 − ln关I0共B兲兴 + 共B − B0兲1 + N N +
A2 A3 + , N N
A3 1 = 2 N
共B3兲
with
1 A1 =− 2 N
冕
1 A2 =− N 2
再
dr c00共r兲 +
冕
冎
1 兺 关c2 共r兲 − ␥m2 1m2共r兲兴 , 2 m1,m2 m1m2 共B4兲
1
dr
0
d
兺 m ,m 1
gm1m2共r兩兲 2
bm1m2共r兩兲
. 共B6兲
The final integral over in Eq. 共B6兲 cannot be evaluated in closed form and this term must be approximated. In the hypernetted-chain closure 关39兴, it is simply neglected. The effect of the RHNC approximation is to replace Eq. 共B6兲 with the corresponding integral of the reference system, ref ref ref which is then evaluated as Aref 3 = A − A1 − A2 . In practice, only the hard sphere system is currently known well enough for this last step; in this case, we have finally
ARHNC AHS ⌬A1 = − ln关I0共B兲兴 + 共B − B0兲1 + N N N +
dk ˜ 共k兲兴 − tr关H ˜ 共k兲兴其, 兵ln det关I + H 共2兲3
冕 冕
⌬A2 , N
共B7兲
共B5兲
where ⌬Ak = Ak − AHS k . For the excess hard sphere free energy ex AHS / N ⬅ AHS / N − ln共⌳3兲 + 1 we use the form derived from the Carnahan-Starling equation of state 关23兴.
关1兴 N. E. Frankel and C. J. Thompson, J. Phys. C 8, 3194 共1975兲. 关2兴 P. C. Hemmer and D. Imbro, Phys. Rev. A 16, 380 共1977兲. 关3兴 J. M. Tavares, M. M. Telo da Gama, P. I. C. Teixeira, J.-J. Weis, and M. J. P. Nijmeijer, Phys. Rev. E 52, 1915 共1995兲. 关4兴 J.-J. Weis, M. J. P. Nijmeijer, J. M. Tavares, and M. M. Telo da Gama, Phys. Rev. E 55, 436 共1997兲. 关5兴 A. Oukouiss and M. Baus, Phys. Rev. E 55, 7242 共1997兲. 关6兴 F. Schinagl, H. Iro, and R. Folk, Eur. Phys. J. B 8, 113 共1999兲. 关7兴 W. Fenz, R. Folk, I. M. Mryglod, and I. P. Omelyan, Phys. Rev. E 68, 061510 共2003兲. 关8兴 J. S. Høye and G. Stell, Phys. Rev. Lett. 36, 1569 共1976兲. 关9兴 T. G. Sokolovska, Physica A 253, 459 共1998兲. 关10兴 F. Lado and E. Lomba, Phys. Rev. Lett. 80, 3535 共1998兲. 关11兴 F. Lado, E. Lomba, and J.-J. Weis, Phys. Rev. E 58, 3478 共1998兲; 60, 2429 共1999兲. 关12兴 T. G. Sokolovska and R. O. Sokolovskii, Phys. Rev. E 59, R3819 共1999兲. 关13兴 I. P. Omelyan, I. M. Mryglod, R. Folk, and W. Fenz, Phys. Rev. E 69, 061506 共2004兲. 关14兴 I. P. Omelyan, W. Fenz, I. M. Mryglod, and R. Folk, Phys. Rev. Lett. 94, 045701 共2005兲. 关15兴 I. P. Omelyan, W. Fenz, I. M. Mryglod, and R. Folk, Phys. Rev. E 72, 031506 共2005兲. 关16兴 E. Lomba, J.-J. Weis, N. G. Almarza, F. Bresme, and G. Stell, Phys. Rev. E 49, 5169 共1994兲. 关17兴 M. J. P. Nijmeijer and J.-J. Weis, Phys. Rev. Lett. 75, 2887 共1995兲; Phys. Rev. E 53, 591 共1996兲. 关18兴 M. J. P. Nijmeijer and J.-J. Weis, in Annual Review of Computational Physics IV, edited by D. Stauffer 共World Scientific, Singapore, 1996兲. 关19兴 A. L. Ferreira and W. Korneta, Phys. Rev. E 57, 3107 共1998兲. 关20兴 M. Blume, V. J. Emery, and R. B. Griffiths, Phys. Rev. A 4, 1071 共1971兲.
关21兴 A. Maciołek, M. Krech, and S. Dietrich, Phys. Rev. E 69, 036117 共2004兲. 关22兴 D. J. Tulimieri, J. Yoon, and M. H. W. Chan, Phys. Rev. Lett. 82, 121 共1999兲. 关23兴 J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids 共Academic, London, 1986兲. 关24兴 D. Henderson, in Fundamentals of Inhomogeneous Fluids, edited by D. Henderson 共Dekker, New York, 1992兲, Chap. 4. 关25兴 R. Lovett, C. Y. Mou, and F. P. Buff, J. Chem. Phys. 65, 570 共1976兲. 关26兴 M. S. Wertheim, J. Chem. Phys. 65, 2377 共1976兲. 关27兴 So named by M. S. Green 关J. Chem. Phys. 33, 1403 共1960兲; 39, 1367 共1963兲兴 because of the resemblance of the first diagram in its density expansion to a Wheatstone bridge. 关28兴 See, for example, G. Arfken, Mathematical Methods for Physicists 共Academic, Orlando, 1985兲. 关29兴 F. Lado, Phys. Rev. A 8, 2548 共1973兲. 关30兴 Y. Rosenfeld and N. W. Ashcroft, Phys. Rev. A 20, 1208 共1979兲. 关31兴 F. Lado, Phys. Lett. 89A, 196 共1982兲. 关32兴 L. Verlet and J.-J. Weis, Phys. Rev. A 5, 939 共1972兲. 关33兴 D. Henderson and E. W. Grundke, J. Chem. Phys. 63, 601 共1975兲. 关34兴 See, for example, E. O. Brigham, The Fast Fourier Transform 共Prentice-Hall, Englewood Cliffs, NJ, 1974兲, Chap. 10. 关35兴 E. Lomba, C. Martín, and N. G. Almarza, Mol. Phys. 101, 1667 共2003兲. 关36兴 G. Zerah and J.-P. Hansen, J. Chem. Phys. 84, 2336 共1986兲. 关37兴 W. H. Press and S. A. Teukolsky, Comput. Phys. 4, 423 共1990兲. 关38兴 D. E. Sullivan, Phys. Rev. A 25, 1669 共1982兲. 关39兴 T. Morita and K. Hiroike, Prog. Theor. Phys. 23, 1003 共1960兲.
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