Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model
The critical behavior of a 3D Ising-like system is studied at the microscopic level of consideration. The free energy of ordering is calculated analytically as an explicit function of temperature, an external field and the initial parameters of the model. Within a unified approach, both Gibbs and He...
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Інститут фізики конденсованих систем НАН України
2011
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nasplib_isofts_kiev_ua-123456789-1200332025-02-09T22:40:30Z Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model Вiльна енергiя Гiббса та вiльна енергiя Гельмгольца для тривимiрної iзiнгоподiбної моделi Kozlovskii, M.P. Romanik, R.V. The critical behavior of a 3D Ising-like system is studied at the microscopic level of consideration. The free energy of ordering is calculated analytically as an explicit function of temperature, an external field and the initial parameters of the model. Within a unified approach, both Gibbs and Helmholtz free energies are obtained and the dependencies of them on the external field and the order parameter, respectively, are presented graphically. The regions of stability, metastability, and unstability are established on the order parameter--temperature plane. The way of implementation of the well-known Maxwell construction is proposed at microscopic level. В данiй роботi на мiкроскопiчному рiвнi розгляду вивчається критична поведiнка тривимiрної iзiнго-подiбної системи. Аналiтично обчислюється вiльна енергiя впорядкування як функцiя температури, зовнiшнього поля i початкових параметрiв моделi. В межах запропонованого пiдходу отримано ви-рази для вiльної енергiї Гiббса i Гельмгольца, їх залежностi вiд поля i параметра порядку приведенi також графiчно. На площинi параметр порядку–температура знайденi областi стабiльностi, метаста-бiльностi та нестабiльностi. Запропоновано спосiб реалiзацiї правила Максвелла на мiкроскопiчно-му рiвнi. 2011 Article Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model / M.P. Kozlovskii, R.V. Romanik // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 43002:1-10. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 05.50.+q, 05.70.Ce, 64.60.F-, 75.10.Hk DOI:10.5488/CMP.14.43002 arXiv:1202.4569 https://nasplib.isofts.kiev.ua/handle/123456789/120033 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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The critical behavior of a 3D Ising-like system is studied at the microscopic level of consideration. The free energy of ordering is calculated analytically as an explicit function of temperature, an external field and the initial parameters of the model. Within a unified approach, both Gibbs and Helmholtz free energies are obtained and the dependencies of them on the external field and the order parameter, respectively, are presented graphically. The regions of stability, metastability, and unstability are established on the order parameter--temperature plane. The way of implementation of the well-known Maxwell construction is proposed at microscopic level. |
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Article |
| author |
Kozlovskii, M.P. Romanik, R.V. |
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Kozlovskii, M.P. Romanik, R.V. Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model Condensed Matter Physics |
| author_facet |
Kozlovskii, M.P. Romanik, R.V. |
| author_sort |
Kozlovskii, M.P. |
| title |
Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model |
| title_short |
Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model |
| title_full |
Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model |
| title_fullStr |
Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model |
| title_full_unstemmed |
Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model |
| title_sort |
gibbs free energy and helmholtz free energy for a three-dimensional ising-like model |
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Інститут фізики конденсованих систем НАН України |
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2011 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120033 |
| citation_txt |
Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model / M.P. Kozlovskii, R.V. Romanik // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 43002:1-10. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
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2025-12-01T11:26:59Z |
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2025-12-01T11:26:59Z |
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| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 4, 43002: 1–10
DOI: 10.5488/CMP.14.43002
http://www.icmp.lviv.ua/journal
Gibbs free energy and Helmholtz free energy
for a three-dimensional Ising-like model
M.P. Kozlovskii, R.V. Romanik
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received October 12, 2011, in final form November 15, 2011
The critical behavior of a 3D Ising-like system is studied at the microscopic level of consideration. The free en-
ergy of ordering is calculated analytically as an explicit function of temperature, an external field and the initial
parameters of the model. Within a unified approach, both Gibbs and Helmholtz free energies are obtained and
the dependencies of them on the external field and the order parameter, respectively, are presented graphi-
cally. The regions of stability, metastability, and unstability are established on the order parameter–temperature
plane. The way of implementation of the well-known Maxwell construction is proposed at microscopic level.
Key words: Ising-like model, critical behavior, external field
PACS: 05.50.+q, 05.70.Ce, 64.60.F-, 75.10.Hk
1. Introduction
Basic principles of phenomenological theory for second order phase transitions (PT) were formu-
lated by Landau in late 30–40s [1, 2], originally to describe superconductivity. The key assumption
of the theory is that in the vicinity of the critical point, the free energy can be expanded in a
power series in the order parameter, the equilibrium value of which is found to obey the minimum
condition for free energy. As is now known [3], this expansion is well defined in the neighborhood
of Tc (critical value of temperature) except for a narrow interval determined by the Ginzburg
criterion [4]. In this interval, the crucial role is played by the order parameter fluctuations while
Landau’s theory is essentially a mean-field theory. Nevertheless, it is of great use as a qualitative
tool for understanding the nature of PT.
The above mentioned power series expansion is often called Landau’s free energy. It turns
out that similar quantity can also be derived (not constructed) in a theory considering PT at a
microscopic level. Indeed, in the collective variables (CV) method proposed in [5, 6] to describe
the critical behavior of spin systems, such a quantity appears in a natural way. In particular, a
microscopic analogue of Landau free energy was calculated in [7], where the Ising-like system was
considered in zero external field.
The purpose of the present paper is to extend the results obtained earlier by the CV method for
physical characteristics of 3D Ising-like model to the presence of an external field and look at what
is going on when the field changes sign. As a result, we obtain the Gibbs free energy, Helmholtz free
energy, and the order parameter and establish the regions of stability, metastability, and unstability
for the system under investigation.
2. Method
In our research we consider the system of N Ising spins placed on sites of a simple cubic lattice
of the spacing c. The Hamiltonian of such a system in an external field is well known
H = −
1
2
∑
i,j
Φ(ri,j)σiσj −H
∑
i
σi . (2.1)
c© M.P. Kozlovskii, R.V. Romanik, 2011 43002-1
http://dx.doi.org/10.5488/CMP.14.43002
http://www.icmp.lviv.ua/journal
M.P. Kozlovskii, R.V. Romanik
Here, Φ(ri,j) is a short-range interaction potential between spins located at the i-th and j-th sites,
the spin variables σi take on values ±1, H is the external field. We do not restrict the summation
in (2.1) to the nearest neighbors. To this end, Φ(r) = const × exp (−r/b) can be chosen as the
interaction potential with an effective range b.
It is also known that this problem has not yet been solved. Universal critical characteristics of
three-dimensional (3D) systems are successfully described in a variety of approaches. Among them
it is worth mentioning the field theoretical and renormalization group (RG) methods [8], which also
enable one to calculate nonuniversal characteristics (critical amplitudes, critical temperatures),
but without taking into account the dependency on initial parameters of the Hamiltonian. An
alternative method for theoretical investigation of the model under consideration is the method of
CV [5], which has an advantage of being a successive microscopic approach, which, in turn, makes
it possible for thermodynamic functions and physical characteristics of the system to be explicitly
obtained as functions of temperature, the field, and microscopic parameters of the model.
In the framework of the “ρ4-model” approximation the functional representation for the parti-
tion function in terms of CV ρk is as follows:
Z = Z0
∫
(dρ)N0 exp
[
a1
√
N0ρ0 −
1
2
∑
k∈B0
d(k)ρkρ−k −
−
a4
4!
N−1
0
∑
ki∈B0
ρk1
. . . ρk4
δk1+...+k4
]
. (2.2)
Here, the quantity d(k) contains the Fourier transform of the interaction potential
d(k) = a2 + βΦ(0)Φ̄− βΦ(k). (2.3)
The explicit expressions for an can be found in [9] (equation (1.25)) , N0 = N/s30 where the
quantity s0 defines the region of validity for the parabolic approximation of Fourier transform of
the interaction potential (for details, see [10]). The summation in (2.2) is performed over wave
vectors of the first Brillouin zone corresponding to a reciprocal effective lattice with the lattice
constant cs0.
In the work [11] the method for calculation of the partition function (2.2) of the model near
the second order PT point was generalized to the case of the presence of the fixed external field of
an arbitrary magnitude. Unlike the work [12], here no assumption concerning either the strength
or the weakness of the applied field is made and, therefore,no perturbation series are used. The
calculation procedure is based on Kadanoff’s idea of constructing effective spin blocks [13]. We
divide the phase space of the CV ρk into layers, the RG parameter being s, and average the
Fourier transform of the potential in each n-th layer. The first step in calculating equation (2.2) is
to integrate over ρk with k ∈ [kmax, kmax/s], kmax = π/cs0. The result of integration proves to be
represented in the form of the initial expression, equation (2.2), with renormalized coefficients a
(1)
1 ,
d1(k), and a
(1)
4 . If we perform a step-by-step integration of the partition function over np layers,
we arrive at
Z = Z0
[
Q(d)
]N0
( np
∏
n=1
Qn
)
ZLGR . (2.4)
The partial partition functions Qn of the n-th level are characterized by a set of coefficients a
(n)
1 ,
dn(0), a
(n)
4 , for which the recurrence relations (RR) hold (the work [9] is devoted to this problem).
The quantity np is called the exit point from the critical regime of the order parameter fluctuations.
It defines the number of iterations at which the system is still in the scaling region.
In [12], two regions in (h, τ)-plane were distinguished, which are of weak and of strong field. The
calculations in the regions were performed using different forms of exit points. These quantities
were subject to certain conditions which we do not discuss in the present work. If the field was
considered to be strong, the formula np = nh = − ln h̃/ lnE1 − 1 was used, while in the case
of a weak field another expression np = mτ = − ln τ̃ / lnE2 − 1 was taken. Hence, the question
43002-2
Free energy of an Ising-like model
immediately arises what should be done at intermediate values of a field. The problem is solved
by constructing a new expression for the exit point, proposed in [11]:
np = −
ln
(
h̃2 + h
(±)
c
2)
2 lnE1
− 1, (2.5)
where some temperature fields h
(+)
c = |τ |p0 and h
(−)
c = |τ1|
p0 are introduced, p0 = lnE1/ lnE2
is the so-called “gap exponent” [14], the signs “+” and “–” are related to T > Tc and T < Tc
respectively, and
h̃ =
s
3/2
0
h0
h, h = βH, τ̃ =
ck1
f0
τ, τ1 = −En0
2 τ, τ =
T − Tc
Tc
. (2.6)
The quantities E1 and E2 are the eigenvalues of the matrix of the RG transformation linearized
near the fixed point of RR.
In the limiting cases, as h → 0 or as τ → 0, equation (2.5) takes on the form of mτ or nh,
respectively, and therefore can be applied to the case of an arbitrary field. Nevertheless, it is worth
mentioning that the inequality h̃ ≫ hc defines a strong field, and h̃ ≪ hc defines a weak field.
Note also that there is an arbitrariness in choosing np, which is discussed in [10, 11] in more detail.
What is important is that for n > np one should be able to integrate the partition function with
Gaussian measure density.
Table 1. Numerical values of the parameters used in present calculations.
b/c βcΦ(0) s0 f0 h0 ck1 φ0 Φf n0
0.3 1.6411 2.0 0.5 0.760 1.1762 0.5938 0.105 0.5
We will adhere to the approach adopted in [15], where the coefficients ck1 , f0, h0 are also defined
(see equations (4.4), (4.17), (4.23), and (4.49) in [15]), and take E1 = 24.551, E2 = 8.308. The
information on the physical meaning of coefficient n0 can be found in [16]. Numerical values of all
coefficients used to obtain the final graphical results are presented in table 1. It is worth stressing
that having fixed the value of RG parameter s = s∗ (s∗ = 3.5977 in the present calculation), there
remains only one initial parameter, which is the ratio of the effective range of interaction b to the
lattice constant c. All the other coefficients can be expressed via b/c [10, 17].
3. Results and discussion
Based on the above mentioned method, in the work [11] as well as in [16] free energy of 3D
Ising-like model was calculated as a logarithm of the partition function (2.4) multiplied by −kT
and presented in the form of several contributions
F (τ, h) = Fa + F (±)
s + F
(±)
0 . (3.1)
Here, the term Fa is the analytical part of free energy and does not affect the critical behavior of
the system. The last two terms in r.h.s. of (3.1) have non-analytical dependence on temperature
and on the external field. The explicit expressions for F
(±)
s and F
(±)
0 are
F (±)
s (τ, h) = −kTNγ(±)
s
(
h̃2 + h(±)
c
2
)
d
d+2
, (3.2)
and
F
(±)
0 (τ, h) = −kTNE0(σ±). (3.3)
43002-3
M.P. Kozlovskii, R.V. Romanik
The quantity γ
(±)
s from (3.2) includes contributions from the critical regime of the order parameter
fluctuations and from the limiting Gaussian regime (LGR)(for details, see [11, 16]). The mentioned
critical regime is characterized by the RG symmetry.
The quantity E0(σ±) is the contribution from the collective variable ρ0. As is known from the
theory of CV [5], the mean value of ρk=0 is connected with the order parameter and consequently
the quantity F0 is the free energy of ordering [7]. If there is no external field, F0 is analogous to
Landau’s free energy in phenomenological theory of second order PT [7]. Since we work in the
ensemble of N particles, with field h and temperature τ being independent variables, this analogy
is now not direct. Following Stanley [14], and based on convincing arguments of the work [18], we
associate the partition function of our model with Gibbs free energy. An appropriate thermody-
namic potential is Gibbs free energy provided the magnetic field and temperature are independent
(“natural”) variables. If the role of independent variables is played by an order parameter (here
magnetization per spin) and temperature, then the thermodynamic potential associated with the
partition function is Helmholtz free energy. Hence, Landau’s free energy is rather Helmholtz than
Gibbs one. The expressions (3.1)–(3.3) in turn present Gibbs free energy for the considered model,
although this was not specified in earlier works, particularly in [11, 16]. Therefore, in order to ob-
tain a microscopic analogue of Landau’s energy one should perform the Legendre transformation.
Let A0 denote this analogue. Then,
A0(τ,M) = F0 +M0H, (3.4)
where M0 is the magnetization of the system under consideration such that
M0 = −
1
N
∂F0
∂H
=
∂E0
∂h
. (3.5)
The expression for E0(σ±), which was found in [11] (see equation (4.4) there) for T > Tc and
in [16] (see equation (4.3) there) for T < Tc, reads
E0(σ±) = hσ± −
1
2
dnp+2(0)σ
2
± −
s30s
3(np+2)
4!
a
(np+2)
4 σ4
± . (3.6)
The quantity σ± was found from condition
∂E0(σ±)
∂σ±
= 0 (3.7)
in the form
σ± = σ
(±)
0 s−(np+2)/2. (3.8)
This results in the cubic equation for σ
(±)
0
σ3
0 + pσ0 + q = 0 (3.9)
with coefficients
p = 6s−3
0 rnp+2/unp+2 ,
q = −6s
−9/2
0 s5/2
h0
unp+2
h̃
(
h̃2 + h
(±)
c
2)1/2
, (3.10)
where the denotations rn and un are connected with dn(0) and a
(n)
4 through dn(0) = s−2rn,
a
(n)
4 = s−4un, and according to [11, 16] are expressed as
rnp+2 = βcΦ(0)f0
(
−1± τ̃E
np+2
2
)
,
unp+2 = [βcΦ(0)]
2φ0
(
1± τ̃E
np+2
2 Φf
)
. (3.11)
43002-4
Free energy of an Ising-like model
(a)
σ
03
02
01
0
−2
10τ,
1.0
0.5
−1.0
−0.5
0.0
−1.0
1.0−0.5
1.5
0.0 0.5
−1.5
(b)
−410h,
σ
03
02
01
0
1
1
0
−1
−1 0
Figure 1. (Color online) Solutions of cubic equation (3.9) (a) as functions of temperature for
h = 10−4; (b) as functions of field for τ = ±10−3. The curves marked by “0” correspond to the
real solutions at τ > 0; marked by “01”, “02”, and “03” denote different solution at τ < 0. In this
respect, we will refer to the solutions as σ
(+)
0 and σ
(−)
0i (i = 1, 2, 3).
Note, that the coefficients p, q should be marked by the superscript ± (as should be done for np
from (2.5)), but where it does not cause confusion, we drop this superscript out. One should just
remember about a difference in temperature scales between the cases of T > Tc and of T < Tc.
Equation (3.9) can be solved by Cardano’s method. The solutions are presented graphically
in figure 1. From the plot we see that for a given value of field there exists some temperature
τ0 < 0 such that equation (3.9) has three real solutions for τ 6 τ0 and one real solution for τ > τ0.
This quantity, τ0, corresponds to the condition Q = 0 where Q is the discriminant of the cubic
equation (3.9):
Q = (p/3)3 + (q/2)2 (3.12)
with q and p from (3.10). When h = 0, then τ0 = 0. If h 6= 0, then τ0 is determined by setting r.h.s.
of (3.12) equal to zero. In [16], τ0 was found numerically and the plot of τ0 versus h was drawn.
As we will see in what follows, the solution τ0 = τ0(h) to the equation Q = 0 defines a curve in
“order parameter–temperature” plane which can be identified with the spinodal of a fluid.
Taking into account (2.5), and (3.8), the quantity E0(σ) takes on
E0(σ) = he
(±)
0
(
h̃2 + h(±)
c
2
)
1
2(d+2)
− e
(±)
2
(
h̃2 + h(±)
c
2
)
d
d+2
, (3.13)
where we have introduced the following notation
e
(±)
0 = σ
(±)
0 s−1/2,
e
(±)
2 =
1
2
σ
(±)
0
2
s−3
(
rnp+2 +
1
12
unp+2s
3
0σ
(±)
0
2
)
. (3.14)
In the previous works [10, 11, 15, 16] the system was considered in the external field h > 0.
Analytical results for (Gibbs) free energy [11, 16], the order parameter [10, 16], and the suscepti-
bility [10] were obtained. The purpose of this paper is to extend the method to the region h < 0
and to take into account all solutions to the equation (3.9). We look at Gibbs free energy F0 and
at the contribution from it to the order parameter M0 of equation (3.5). Thereupon, Helmholtz
free energy is calculated.
The main results of this paper are presented in figures 2, 3, and 4. In figure 2 we can see: (a)
Gibbs free energy F0 as a function of the external field in low temperature region (τ = −0.001);
(b) a plot of F0 versus M0. Each of the points a, b, c, . . . represents a certain specific state of
43002-5
M.P. Kozlovskii, R.V. Romanik
(a)
f
e
d c
b
a
−410h,
−6
,100F
0
−5
10
−10
−1 (b)
f
e
dc
b
a
0M
−6
,100F
−5
0.20.0−0.2
−10
0
(c)
03
02
01
−410h
σ
1
1
0
−1
0−1
(d)
M
−4
10h,
−0.2
−0.5
0.5
0.0
0.0
0.2
Figure 2. (Color online) A set of 4 pictures is placed for the sake of clarity. All are drawn for
τ = −0.001. The top left is the field dependence of Gibbs free energy F0. The top right shows F0
against M0. The bottom left presents the solutions of the cubic equation (3.9). The bottom right
is the equation of state for the system under consideration. The values of energies are normalized
by dividing by kTN. The points a to f represent some particular states of the system in different
coordinates. See the text for details.
h<0
h>0
h>0
h>0
h<0
h<0
−0.001
τ
0M
20
19
18
17
16
10
9
8
7
6
15
14
13
12
11
5
4
3
2
1
−0.1
0.1
0.0
Figure 3. (Color online) The plot shows temperature dependence of magnetization calculated
at different solutions to equation (3.9). Thick line is magnetization in zero external field and
corresponds to the coexistence curve (binodal). In negative external field, the solution σ01 gives
rise to Curves 1–5 on the magnetization-temperature plane. In positive external field, σ03 gives
rise to Curves 6–10. Curves 11–20, which correspond to σ02, can be interpreted as non-physical.
Thick dashed curve corresponds to the saturation curve (spinodal). In order to recover that, one
should compute magnetization corresponding to σ01 at h < 0 and τ0 = τ0(|h|) (upper branch)
and corresponding to σ03 at h > 0 and τ0 = τ0(h) (lower branch). The magnitudes of field, at
which the results are drawn, are |h| = 0, 10−7, 10−6, 5 · 10−6, 10−5, 2 · 10−5.
43002-6
Free energy of an Ising-like model
0M
−5
,100A
τ>0
τ=0
τ<0
fedcba
0
0.0
−1
−0.2
2
1
0.2
3
Figure 4. (Color online) Helmholtz free energy A0 as a function of order parameter for three
different temperatures T < Tc (τ = −0.001), T = Tc (τ = 0), and T > Tc (τ = 0.001). The
value is normalized by dividing by kTN .
the system in different coordinates. Especially, points a, c, d, and f correspond to such a relation
between the field h and temperature τ that the equality τ = τ0(h) holds. Thus, in each of these
we have Q = 0. Points b and e are associated with first order phase transition and correspond
to h = 0 and τ = −0.001. The isotherm in figure 2 (b) was drawn with the help of parametric
representation {F0 = F0(h), M0 =M0(h)}.
Recall that Gibbs free energy is a concave function of field [14]. Therefore, from figure 2 (a) we
can conclude that the solution σ
(−)
02 gives rise to a non-physical result. It corresponds to the region
of unstability (c−d part of the curve in figure 2 (a), (b)). This is not surprising if we note that σ
(−)
02
maximizes F0. Solutions σ
(−)
01 at h < 0 and σ
(−)
03 at h > 0 correspond to the region of metastable
states (d − e and b − c parts of the curve respectively). In the case of liquid-gas PT it would be
the region between the binodal and the spinodal curves. Hence, we regard these results important
from the perspective of using this method for description of the critical behavior in simple fluids.
Figure 2 (c) to some extent repeats figure 1 (b) and is placed here for convenience of repre-
sentation. The quantity σ0 is related to the scaling function of magnetization via formula (4.8) in
work [16]. Finally, in figure 2 (d) the equation of state computed by means of (3.5) is presented at
τ = −0.001.
A more detailed representation of the equation of state is shown in figure 3. The derivatives of
Gibbs free energy with respect to field are presented as functions of temperature, different solutions
to equation (3.9) being taken into account. As we can see, such an accounting allows us to establish
spinodal and binodal curves of the model and, in this respect, to find the stable, metastable, and
unstable regions on “order parameter–temperature” plane.
The order parameter dependence of free energy A0 is presented in figure 4 for three different
temperatures T > Tc, T = Tc and T < Tc. Here again a, b, c, . . . denote the states as in figure 2. We
can see that Helmholtz free energy has two minima below Tc. The equilibrium values for the order
parameter are defined by these minima. Knowing their positions at different temperatures, we can
recover the coexistence curve. Points b and e coincide with the minima at τ = −0.001. Note that
based on the principle of minimum Gibbs free energy and from figure 2 (a), we can conclude that
with the field changing from h < 0 to h > 0 the system will tend to move along a− b− e− f part
of the curve. Applied to Helmholtz free energy, it means that the system will tend to jump from
state b to state e. Therefore, one may supplement figure 4 with a double-tangent construction. As
43002-7
M.P. Kozlovskii, R.V. Romanik
a consequence, on M0 − h plane we have a horizontal segment for M0 at h = 0 (see figure 2 (d)),
which corresponds to a Maxwell construction derived at microscopic level. On the other hand,
based on the maxima of Gibbs free energy with respect to M0 (see figure 2 (b)), we are able to
construct a saturation curve. Let us just remember that dependence F0 on the order parameter is
formal due to M0 being not “natural” variable for F0. However, the same results for the binodal
and the spinodal can immediately be obtained if one appropriately chooses the solutions to (3.9).
The forms of isotherms in figure 4 are similar to those in Landau’s theory [3]. However, the
peculiar feature is that in the present work the free energy is an explicit function of temperature,
external field and microscopic parameters (in the present case a ratio b/c) of the model, with a
non-analytical dependence on its arguments. This gives rise to a non-classical critical behavior of
the system, i.e., to the critical exponents taking on non-classical values. These values for the most
important exponents are reported in table 2. There are also collected the values obtainable by
CV method in higher, ρ6 approximation as reported in Conclusions of the work [7]. Note that in
our calculation we have neglected the critical exponent η responsible for the behavior of the pair
correlation function at T = Tc. Accounting for corrections to scaling is beyond the scope of the
present paper as well. In table 2 the classic values for the critical exponents are presented as well
as the results of other authors. We stick to the following notation. The temperature behavior of
magnetization is governed by β, of the heat capacity by α, of susceptibility by γ, of the correlation
length by ν. The field behavior of magnetization is governed by δ, M ∼ |h|1/δ. We have calculated
some of the exponents with the help of scaling laws. Based on earlier results [15], we are also able
to estimate the critical exponents ϕ and ψ that describe the field behavior of the heat capacity,
C ∼ |h|−ϕ, and of the entropy, S ∼ |h|ψ , respectively. Their numerical values are ϕ = 0.122 and
ψ = 0.539, which differ from ϕ = 0 and ψ = 2/3 in mean-field theory [14].
Table 2. Numerical values of the critical exponents. CV, collective variables method; MF, mean-
field values; HT, high-temperature expansion results; FT, field theory approach.
CV, CV, MF HT [19, 20] FT [21]
ρ4-model ρ6-model [7]
α 0.185 0.088 0 0.110(1) 0.109(4)
β 0.302 0.319 1/2 0.3263(4) 0.3258(14)
γ 1.210 1.275 1 1.2373(2) 1.2396(13)
ν 0.605 0.637 1/2 0.6301(2) 0.6304(13)
δ 5 5 3 4.792 4.805
It should also be mentioned that the scaling properties for physical characteristics of the con-
sidered model are discussed in earlier works. In section 3 of the work [10] the scaling functions for
(Gibbs) free energy, for the order parameter, and for susceptibility were calculated.
4. Conclusions
In this work, an analytical expression for Gibbs free energy of Ising-like system is obtained and
investigated near the second order phase transition. The emphasis is made on the presence of the
external field and on the situation where the field changes its direction. Some well-defined concepts
of phenomenological theory of phase transitions - e.g. Landau’s energy, the Maxwell construction,
the double-tangent construction - are derived and reexamined at microscopic level. We hope that
the approach described and the results presented should be useful in further investigations of
critical behavior in 3D Ising-like systems with analytical methods; in particular, in order to obtain
the coexistence curve and the spinodal curve for a system at least in the vicinity of critical point.
There exists a general belief [14, 20, 22] that simple fluids belong to the Ising universality class.
Hence, we regard these results important from the perspective of using this method for description
of the critical behavior in simple fluids.
43002-8
Free energy of an Ising-like model
References
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http://dx.doi.org/10.1007/BF02740597
http://dx.doi.org/10.1103/PhysRevB.66.134410
http://dx.doi.org/10.5488/CMP.13.43004
http://dx.doi.org/10.5488/CMP.12.2.151
http://dx.doi.org/10.1103/PhysRevB.73.174406
http://dx.doi.org/10.1016/S0304-8853(02)01286-6
http://dx.doi.org/10.1103/PhysRevB.72.014442
http://dx.doi.org/10.1103/PhysRevB.83.054433
http://dx.doi.org/10.1016/S0370-1573(00)00126-5
http://dx.doi.org/10.1103/PhysRevB.59.14533
M.P. Kozlovskii, R.V. Romanik
Вiльна енергiя Гiббса та вiльна енергiя Гельмгольца для
тривимiрної iзiнгоподiбної моделi
М.П. Козловський, Р.В. Романiк
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
В данiй роботi на мiкроскопiчному рiвнi розгляду вивчається критична поведiнка тривимiрної iзiнго-
подiбної системи. Аналiтично обчислюється вiльна енергiя впорядкування як функцiя температури,
зовнiшнього поля i початкових параметрiв моделi. В межах запропонованого пiдходу отримано ви-
рази для вiльної енергiї Гiббса i Гельмгольца, їх залежностi вiд поля i параметра порядку приведенi
також графiчно. На площинi параметр порядку–температура знайденi областi стабiльностi, метаста-
бiльностi та нестабiльностi. Запропоновано спосiб реалiзацiї правила Максвелла на мiкроскопiчно-
му рiвнi.
Ключовi слова: модель Iзiнга, критична поведiнка, зовнiшнє поле
43002-10
Introduction
Method
Results and discussion
Conclusions
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