Thermodynamic and kinetic description of the second order phase transitions
Thermodynamic and kinetic description of phase transitions for the model of ferroelectrics based on the kinetic equation for the distribution function of values of the “order parameter”, coordinates and time is considered. For one-domain ferroelectrics, the self-consistent approximation for the fi...
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Klimontovich, Yu.L. 2017-06-13T13:13:25Z 2017-06-13T13:13:25Z 2000 Thermodynamic and kinetic description of the second order phase transitions / Yu.L. Klimontovich // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 393-416. — Бібліогр.: 18 назв. — англ. 1607-324X DOI:10.5488/CMP.3.2.393 PACS: 64.60.-i, 64.60.Fr, 64.60.My, 65.50.tm https://nasplib.isofts.kiev.ua/handle/123456789/121030 Thermodynamic and kinetic description of phase transitions for the model of ferroelectrics based on the kinetic equation for the distribution function of values of the “order parameter”, coordinates and time is considered. For one-domain ferroelectrics, the self-consistent approximation for the first moment is used. The kinetic equation is reduced to the relaxation Ginsburg- Landau equation. The susceptibility is governed by the Curie law and the heat capacity has the jump. Calculations are carried out for one-domain and polydomain ferroelectrics. In the first case, the self-consistent approximation for the first moment is used. In the second case, the self-consistent approximation for the second moment is carried out. In the last case, there is the jump of the susceptibility. The heat capacity is governed by the Curie law. It is also shown that the Ornstein-Zernike formula is valid not for the space correlator of fluctuations but only for the temporal spectral density of the space correlator at zero frequency. In the kinetic theory of the phase transition, all physical characteristics at the critical point have got finite values. Thus, the problem of the “infinities” is absent. Проведено опис термодинаміки і кінетики фазових переходів моделі сегнетоелектриків на основі кінетичного рівняння для функції розподілу, залежної від “параметра порядку”, координат і часу. Для однодоменних сегнетоелектриків застосовано самоузгоджений підхід до розрахунку першого момента. Кінетичне рівняння зводиться до релаксаційного рівняння Гінзбурга-Ландау. Сприйнятливість описується законом Кюрі, а теплоємність має стрибок. Розрахунки проведено для однодоменних та полідоменних сегнетоелектриків. У першому випадку використовується самоузгоджене наближення для першого моменту. У другому випадку самоузгоджене наближення робиться для другого моменту. В цьому випадку відзначається стрибок сприйнятливості, а теплоємність описується законом Кюрі. Показано, що формула Орнштейна-Церніке справедлива не для просторового корелятора флуктуацій, а лише для спектральної густини цієї кореляційної функції при нульовій частоті. В кінетичній теориї фазових переходів усі фізичні величини мають скінчені значення в критичній точці. Таким чином, немає проблеми “розбіжностей”. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Thermodynamic and kinetic description of the second order phase transitions Термодинамічний та кінетичний опис фазових переходів другого роду Article published earlier |
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Thermodynamic and kinetic description of the second order phase transitions |
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Thermodynamic and kinetic description of the second order phase transitions Klimontovich, Yu.L. |
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Thermodynamic and kinetic description of the second order phase transitions |
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Thermodynamic and kinetic description of the second order phase transitions |
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Thermodynamic and kinetic description of the second order phase transitions |
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Thermodynamic and kinetic description of the second order phase transitions |
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thermodynamic and kinetic description of the second order phase transitions |
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Klimontovich, Yu.L. |
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Klimontovich, Yu.L. |
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Термодинамічний та кінетичний опис фазових переходів другого роду |
| description |
Thermodynamic and kinetic description of phase transitions for the model
of ferroelectrics based on the kinetic equation for the distribution function
of values of the “order parameter”, coordinates and time is considered.
For one-domain ferroelectrics, the self-consistent approximation for the first
moment is used. The kinetic equation is reduced to the relaxation Ginsburg-
Landau equation. The susceptibility is governed by the Curie law and the
heat capacity has the jump.
Calculations are carried out for one-domain and polydomain ferroelectrics.
In the first case, the self-consistent approximation for the first moment is
used. In the second case, the self-consistent approximation for the second
moment is carried out. In the last case, there is the jump of the susceptibility. The heat capacity is governed by the Curie law.
It is also shown that the Ornstein-Zernike formula is valid not for the space
correlator of fluctuations but only for the temporal spectral density of the
space correlator at zero frequency.
In the kinetic theory of the phase transition, all physical characteristics at
the critical point have got finite values. Thus, the problem of the “infinities”
is absent.
Проведено опис термодинаміки і кінетики фазових переходів моделі
сегнетоелектриків на основі кінетичного рівняння для функції розподілу, залежної від “параметра порядку”, координат і часу.
Для однодоменних сегнетоелектриків застосовано самоузгоджений
підхід до розрахунку першого момента. Кінетичне рівняння зводиться до релаксаційного рівняння Гінзбурга-Ландау. Сприйнятливість
описується законом Кюрі, а теплоємність має стрибок.
Розрахунки проведено для однодоменних та полідоменних сегнетоелектриків. У першому випадку використовується самоузгоджене
наближення для першого моменту. У другому випадку самоузгоджене наближення робиться для другого моменту. В цьому випадку відзначається стрибок сприйнятливості, а теплоємність описується законом Кюрі.
Показано, що формула Орнштейна-Церніке справедлива не для просторового корелятора флуктуацій, а лише для спектральної густини
цієї кореляційної функції при нульовій частоті.
В кінетичній теориї фазових переходів усі фізичні величини мають
скінчені значення в критичній точці. Таким чином, немає проблеми
“розбіжностей”.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121030 |
| citation_txt |
Thermodynamic and kinetic description of the second order phase transitions / Yu.L. Klimontovich // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 393-416. — Бібліогр.: 18 назв. — англ. |
| work_keys_str_mv |
AT klimontovichyul thermodynamicandkineticdescriptionofthesecondorderphasetransitions AT klimontovichyul termodinamíčniitakínetičniiopisfazovihperehodívdrugogorodu |
| first_indexed |
2025-11-25T20:35:28Z |
| last_indexed |
2025-11-25T20:35:28Z |
| _version_ |
1850523710574821376 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 2(22), pp. 393–416
Thermodynamic and kinetic description
of the second order phase transitions
Yu.L.Klimontovich∗
Department of Physics, M.V.Lomonosov Moscow State University,
Vorob’evy gory, 119899 Moscow, Russia
Received November 19, 1999
Thermodynamic and kinetic description of phase transitions for the model
of ferroelectrics based on the kinetic equation for the distribution function
of values of the “order parameter”, coordinates and time is considered.
For one-domain ferroelectrics, the self-consistent approximation for the first
moment is used. The kinetic equation is reduced to the relaxation Ginsburg-
Landau equation. The susceptibility is governed by the Curie law and the
heat capacity has the jump.
Calculations are carried out for one-domain and polydomain ferroelectrics.
In the first case, the self-consistent approximation for the first moment is
used. In the second case, the self-consistent approximation for the second
moment is carried out. In the last case, there is the jump of the susceptibil-
ity. The heat capacity is governed by the Curie law.
It is also shown that the Ornstein-Zernike formula is valid not for the space
correlator of fluctuations but only for the temporal spectral density of the
space correlator at zero frequency.
In the kinetic theory of the phase transition, all physical characteristics at
the critical point have got finite values. Thus, the problem of the “infinities”
is absent.
Key words: phase transitions, self-consistent approximations, problem of
“infinities”, coherent fluctuations, thermodynamic functions, critical point
PACS: 64.60.-i, 64.60.Fr, 64.60.My, 65.50.tm
1. Introduction
After Gibbs, Van der Waals, Ornstein-Zermike and Weiss in the Landau paper
[1], (see also in [2]) the general phenomenological theory of phase transitions was
developed. It also serves as the basis for the modern fluctuation theory of phase
transitions [2–7]. In this theory, the behaviour of thermodynamic functions in the
critical region is characterized by eight critical indexes which are connected by means
of five general relations. The sixth relation based on “the scale invariance hypothesis”
∗E-mail: ylklim@hklim.phys.msu.su
c© Yu.L.Klimontovich 393
Yu.L.Klimontovich
was formulated by Kadanoff, Patashinskii-Pokrovskii. In Wilson theory, the special
perturbation theory method was developed allowing one to calculate the critical
indexes in a better agreement with experimental data. However, the susceptibility
and the correlation radius at the critical point, as well as in the Landau theory, are
infinite. Only the “rate” of their increasing to infinity varies. Thus, the question
about the thermodynamical description of a critical point remains open.
In recent years, in the theory of phase transitions in polymers and spin glasses,
many new questions arose, to which only qualitative answers can be given now (see
in [8] and [9]). In this situation, the development of non-traditional methods for the
description of the phase transitions is natural.
In the present paper, the description of the second order phase transitions is
carried out based on the kinetic equation for the distribution function of values
of internal parameter (in non-symmetrical phase it plays the role of “the order
parameter”), coordinates, and time.
Depending on the structure of the system under consideration (monodomain
or polydomain) two possible cases of reducing the kinetic equation to the equa-
tions for the moments of distribution function are investigated. In the first case,
the self-consistent approximation for the first moment is used. Thus, the kinetic
equation is reduced to the relaxation Ginsburg-Landau equation, but the latter one
possesses two independent dissipative terms. This equation serves as an example
of the reaction-diffusion Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equation
for the first moment of the distribution function. In this case, the variation of the
susceptibility in the critical region is governed by the Curie law, and there is a jump
of the heat capacity.
For polydomain systems, the first moment is negligible while the second moment
approximation is more effective. In this case, there is also the equation of FKPP
type, but it has a different structure. For such systems, there is a jump for the
susceptibility and for characteristics which are connected with it, namely, for the
correlation radius and for the correlation time. The thermal capacity has no jump.
In the critical region, it is governed by the Curie law, and it is finite at the critical
point.
Based on the kinetic equation, the calculation of the fluctuations of moments of
the distribution function is also carried out. The results differ greatly from those
received based on the Landau theory. In particular, it is shown that the one-time
space correlations of the Ornstein-Zernike type do not exist. The formula similar to
the Ornstein-Zernike one takes place only for the temporal spectral density of the
space correlator at zero frequency.
The kinetic derivation of the Ornstein-Zernike formula does not seem to be nec-
essary. Really, it can be obtained, but for certain assumptions (!) regarding the
structure of the effective Hamilton function, based on the appropriate “canonical
Gibbs distribution”. The kinetic description allows, however, to remove these re-
strictions and to receive additional information about temporal evolution of the
fluctuations in the critical region.
In the kinetic theory of the second order phase transitions, the “problem of
394
Description of the second order phase transitions
infinities” of the thermodynamic functions at the critical point does not exist. There
is the opinion that in the vicinity of the critical point the thermodynamic description
is violated. This view of the critical state is based, to a great extent, on the Lee-
Yang theorem, according to which, in the thermodynamic limit, the functions such
as susceptibility and correlation radius, take infinite values at the critical point.
The Lee-Yang theorem is, of course, very important, as it shows that critical
points are singular. However, in real systems having finite volumes, all the thermo-
dynamic functions are finite. For example, it is obvious, that the correlation radius
cannot be larger than the volume of the system. The occurrence of infinity in the
theory of phase transitions is the consequence of a certain choice of the succession
of two limiting transitions: the transition to the critical point T → T c and the ther-
modynamic limit (N → ∞ and V → ∞, but N/V finite). Usually the first one
is the thermodynamic limit. The change of the order of limiting transitions, as it
was pointed in the book [10], gives an opportunity to avoid the difficulty mentioned
above.
It will be shown, that the finiteness of thermodynamic functions at the critical
point is one of the consequences of the kinetic theory of phase transitions with
allowance for structures “of continuous medium”. Relaxation times also take finite
values at the critical point.
Firstly, we compare the kinetic description of phase transitions in the Landau
theory as well as based on the statistical theory of open systems [11]. In order
to represent the results in the clearest form, we use the elementary model of the
ferroelectric phase transition (see [12,13]).
2. The relaxation Ginsburg-Landau equation (RGLE)
Let X(R, t) be the local value of a relative displacement of crystal lattices in
ferroelectrics. In the Landau theory for the nonsymmetrical phase it will play the
role of the local order parameter.
According to Landau theory (see in [14]) we introduce (by definition!) the ef-
fective Hamilton function. For the ferroelectrics model we can represent it in the
form:
Heff = nmω2
0
∫
{
1
2
X2
[
(1− af) +
1
2
bX2
]
+
g
2
(
∂X
∂R
)2
}
dR. (1)
Here n is the mean density of the particles. Inclusion of this factor allows us to
introduce the effective Hamilton function on one particle. m is the mass of atom,
mω2
0 is the hardness, and ω0 is the proper frequency of the vibration of atoms.
The parameter af(T ) defines the effect of the Lorentz effective field on the hard-
ness. This parameter depends on the temperature. According to Landau theory in
the region of the critical point, this dependence is defined by the expression
1− af =
T − Tc
Tc
≡ τ. (2)
395
Yu.L.Klimontovich
Here Tc is the critical temperature. Thus, the hardness is zero at the critical point.
Below the critical point in the nonsymmetrical phase it is negative. The Lorentz field
is determined by the small-scale correlations, which are taken into account here as
the phenomenological parameter af .
To get a qualitative notion regarding the behaviour of the system for all temper-
atures one may represent the function af(T ) as follows:
1− af = tanh
T − Tc
∆T
. (3)
Here ∆T is the “temperature width” of the transition region.
The parameter g in the expression Heff characterizes the degree of nonlocality,
and, therefore, is nondissipative one.
In order to simplify the comparison with formulas of the Landau-Lifshits course
[1976], we write the correspondence rules:
α(T − Tc)η
2 ⇔ nmω2
0
X2
2
T − Tc
Tc
, bη4 ⇔ nmω2
0b
X4
4
,
gη2 ⇔ nmω2
0
g
2
X2, ηh⇔ nmω2
0Xh. (4)
The kinetic equation for the local order parameter in the Landau theory (see in [14])
is the result of the functional differentiation of the effective Hamilton function:
∂X(R, t)
∂t
= − γ
nmω2
0
δHeff
δX(R, t)
. (5)
Here γ is the arbitrarily introduced dissipative coefficient. As a result, we obtain the
relaxation Ginsburg-Landau equation (RGLE)
∂X(R, t)
∂t
= − γ
[
T − Tc
Tc
+ bX2(R, t)
]
X(R, t) + γg
∂2X(R, t)
∂R2
. (6)
Note, that the assumption (5) can be justified only in some special cases. It is
impossible, in general, to find the kinetic equation by the functional differentiation
of the effective Hamilton function Heff .
Note also, that the initial expression (1) for the effective Hamilton function
corresponds to the model of continuous medium. In this case, the introduction of
the gradient of macroscopic variables into the Heff and the use of the corresponding
“canonic Gibbs distribution” contradicts the theory of irreversible processes.
The RGLE has the form of reaction-diffusion equation but the diffusion coeffi-
cient is defined by the expression D = γg (!?).
3. Landau and Boltzmann distribution functions
Consider the case when the space diffusion process in RGLE is the fastest. Then,
for the homogeneous state, the effective Hamilton function has the form:
Heff ≡ Nheff , heff =
mω2
0
2
X2
[
(1− af) +
1
2
bX2
]
. (7)
396
Description of the second order phase transitions
Here we introduced the effective Hamilton function heff for one particle.
The expression (7) for the system of bistable elements defines the Landau ef-
fective Hamilton function in one-mode approximation. Through it, the “Landau
distribution function” is defined by the following expression:
fL(X, af , T ) = exp
F (af , T )−Heff(X, af)
kBT
. (8)
Consider the free energy per one particle F (af , T ) = Nψ(af , T ). Then the Landau
distribution function fL(X, af , T ) may be represented in the following form:
fL(x, af , T ) = exp
N
(
ψ(af , T )− heff(x, af)
)
kBT
. (9)
Compare it with the appropriate Boltzmann distribution:
fB(x, af , T ) = exp
ψ(af , T )− heff(x, af , )
kBT
. (10)
It differs from the Landau distribution function by factor N being absent in the
exponent. For this reason, for the Boltzmann distribution, the fluctuations are finite
for the thermodynamic limit as well.
For the Landau distribution, for the thermodynamic limit, the fluctuations are
zero. In this sense the Landau distribution is coherent (or self-consistent). Thus, in
the Landau theory, the “coherence” is already introduced in the initial definition of
the effective Hamilton function (1).
Via the parameter af , the Hamilton function Heff depends on temperatures and,
as a consequence, the additional thermodynamic “thermal force” appears [15]:
Fth(T ) = −
∫
∂Heff(X, T )
∂T
f(X, af , T )dX. (11)
This leads to the change of the thermodynamic relations. In particular, the famous
Helmholtz relation now has the following form:
∂F (T )
∂T
= −Seff(T )− Fth(T ). (12)
Here F (T ) is the Gibbs free energy, the effective entropy Seff(T ) being defined by
the Boltzmann distribution fB(X, af , T ). However, there are other new questions
arising in the phase transitions theory. Let us illustrate them using as an example
the temperature dependence of the heat capacity.
4. Thermodynamic functions
In the Landau theory thermodynamic functions are defined through the most
probable values Xm.p. of the distribution function (9):
Xm.p. = 0, τ > 0; Xm.p. = ±
√
|τ |b, τ 6 0. (13)
397
Yu.L.Klimontovich
At the critical point the jump of the heat capacity takes place. This jump may be
expressed through the dimensionless parameter of nonlinearity:
∆C =
1
2
kBN
(
1
X2
Tc
b
− 1
)
>
1
2
kBN, X2
Tc
=
kBT
mω2
0
. (14)
One can see, that in the Landau theory the peak of the function C(T ) is absent. In
the approximation considered the heat capacity is defined not by the Boltzmann-
Gibbs entropy but by the thermal force. Thus, the Helmholtz relation (12) is replaced
by the following expression:
∂F (Xm.p., T )
∂T
= − Fth(Xm.p., T ), Seff(Xm.p., T ) = 0. (15)
The Boltzmann-Gibbs entropy is not taken into account at this level of the descrip-
tion. For this reason it is impossible to reveal the peak of the function, which defines
the λ-point. In order to obtain a more general result, the following should be taken
into account.
In the Gibbs theory it is assumed by definition that the thermodynamic functions
are the mean values (the first moments) of the corresponding dynamical variable
functions. As an example, the internal energy is represented as the mean value of
the Hamilton function U = 〈H〉. There are, however, other possibilities of defining
the thermodynamic functions. For example, the internal energy can be defined via
the most probable value for the Gibbs distribution of energy values U = Hm.p. .
These two definitions are equivalent only under the following condition:
lim
N→∞
1
N
(〈H〉 −Hm.p.) = 0. (16)
In the theory of phase transitions for Landau and Gibbs distribution functions
this condition is, in general, not satisfied. Really, in the nonsymmetrical phase the
average value 〈X〉 = 0, but the most probable value differs from zero and is defined
by the expression (13). Thus, the definition of the thermodynamic functions for
phase transitions is not unique and there is a possibility for choice. We shall make
use of it to calculate the temperature dependence of the heat capacity C(T ) in the
critical region.
In the paper [15] the calculation of the function C(T ) was conducted based on the
generalized Helmholtz relation (12). In the Gaussian approach the following result
was obtained:
C =
1
2
kBN
T 2
c
(T − Tc)2
, T > Tc, (17)
C =
1
2
kBN
T 2
c
(T − Tc)2
+
1
2
kBN
1
X2
Tc
b
, T < Tc. (18)
The second term in the right-hand side coincides with the Landau jump of the
heat capacity. There is, however, an additional singular term, which describes the
nonmonotonic behaviour of the heat capacity near the point λ. The value C(T =
Tc) ∼ 1/(bX2
T ) is finite and has the same order as the Landau jump.
398
Description of the second order phase transitions
Such a calculation of the function C(T ) corresponds to the Gibbs definition of
thermodynamic functions, since all the terms in the right-hand part of the formula
(12) are the mean values (the first moments) of the Boltzmann distribution (10).
However, values 〈X〉 , Xm.p. in nonsymmetrical phase are essentially different.
It is possible to use “another” method for calculation, when the entropy is defined
by Boltzmann distribution, while the thermal force is defined by Landau distribution.
For this case we single out the value of the effective Hamilton function for X = Xm.p.
in the Boltzmann distribution:
heff(X) = heff(Xm.p.) + h′eff(X −Xm.p.); h′eff(0) = 0. (19)
Since heff(Xm.p.) is the thermodynamic characteristic, the thermal force in the for-
mula (12) can be replaced by the derivative of the function heff(Xm.p.) with respect
to temperature. As a result, the generalized Helmholtz relation will be presented in
the following form:
∂F (T )
∂T
= −Seff(T )− Fth(Xm.p.); Fth(Xm.p.) = −N ∂heff(Xm.p.)
∂T
. (20)
The entropy is defined by the Boltzmann distribution with the Hamilton function
h′eff(X−Xm.p.); h
′
eff(0) = 0. The heat capacity is now determined by the expression:
C(T ) = T
∣
∣
∣
∣
dSeff(T )
dT
∣
∣
∣
∣
+ T
∂Fth(xm.p.)
∂T
. (21)
There is the module of the derivative because the heat capacity is positive, but in
the region of the λ-point the entropy increases with the decrease of temperature. As
a result, instead of equations (17) and (18) we have the following expressions:
C =
1
2
kBN
Tc
T − Tc
, T > Tc, (22)
C =
1
2
kBN
Tc
Tc − T
+
1
2
kBN
1
X2
Tc
b
, T < Tc. (23)
Thus, the singular part of the heat capacity is defined by the Curie law.
5. Derivation of the Ornstein-Zernike formula based on the
Ginsburg-Landau equation
Let’s return to the relaxation Ginsburg-Landau equation (RGLE) (6), which
was obtained based on the expression (1) for the effective Hamilton function by the
“recipe” (5). Equation (6) is the example of the reaction-diffusion equation, which
serves as the basic equation for the description of a different kind of spatio-temporal
dissipative structures and, therefore, plays the role of basic equations in the theory
of nonequilibrium phase transitions. Equilibrium phase transitions differ by the fact
that the temperature plays the role of control parameter for them.
399
Yu.L.Klimontovich
The RGLE (6) being an example of the reaction-diffusion equation can be ob-
tained independently of the recipe of its derivation which was represented above.
It can serve for the description of equilibrium phase transitions. For this, however,
instead of equation (6) with only one dissipative coefficient γ, it is necessary to use
a more general equation (see the next section).
We shall calculate the spatio-temporal fluctuations by two different methods
which lead to essentially different results and, consequently, to the two different
expressions for one-moment correlator of fluctuations. We shall discuss the reason
for such double-natured results and give a physical basis for a unique choice of the
result.
First, we use the method based on the fluctuation-dissipation relation (FDR)
(see paragraph 101 in [14]). According to this method, introduce an additional term
in the effective Hamilton function (1) which is defined by the local external force
X(R, t)h(R, t) as consequence, in the right-hand part of the (RGLE) (6) an addi-
tional term γh(R, t) appears.
If the external force is not present this equation has a partial homogeneous
solution (13). Let δX be a small deviation from it for temperatures T > T c. Then
the corresponding susceptibility is defined by the following expression:
χ(ω, k) =
γ
−iω + γ T−Tc
Tc
+ γgk2
. (24)
The spatio-temporal spectral density of fluctuations δX in the Landau theory is
defined by FDR via the imaginary part of the susceptibility (see paragraph 101 in
[14])
(δXδX)ω,k =
2
ω
Imχ(ω, k)
X2
T
n
, X2
T =
κBT
mω2
0
. (25)
From the above formulas it follows that for the symmetrical phase the spectral
density of fluctuation is
(δXδX)ω,k =
2γ
ω2 + (γ T−Tc
Tc
+ γgk2)2
X2
T
n
. (26)
By carrying out the integration with respect to frequency we find the appropriate
spatial spectral density
(δXδX)k =
1
1 + r2ck
2
〈(δX)2〉
n
,
〈
(δX)2
〉
=
X2
T
T − Tc/Tc
. (27)
and after the Fourier transformation we obtain the Ornstein-Zernike (OZ) type
formula for the space correlation function
〈δXδX〉r =
1
n
〈
(δX)2
〉 1
4πr2cr
exp
(
− r
rc
)
, r = |R−R′|. (28)
The definition for the correlation radius used here is as follows
r2c = gχ(ω = 0, k = 0) = g
Tc
T − Tc
. (29)
400
Description of the second order phase transitions
Recall the Ginsburg-Levanuk parameter, which characterizes the relative role of
fluctuations. Since the most probable distance for correlator (28) is r = rc it is
possible to characterize the role of fluctuations by the ratio of the correlator (28)
for r = rc to the square of the order parameter (13). It is defined by the Ginsburg
number
Gi =
〈δXδX〉rc
X2
m.p.
∼ X2
Tb
ng3/2
(
Tc
|T − Tc|
)1/2
≪ 1. (30)
From this, the condition of applicability of the Landau theory follows
|T − Tc|
Tc
≫ (X2
Tc
b)2
n2g3
, T − Tc ≪ 0. (31)
It is based on the formula O-Z (28). However, the O-Z formula was obtained using
another way. The derivation which was represented here gives up some unsolved
questions.
First, the calculation of fluctuations is carried out using RGLE (6). The latter
was obtained by the “recipe” (5). It contains only one dissipative factor γ. As a
result, the spectral line (26) is “nonsymmetric”, i.e., the width of the spectral line
is defined by the sum of two dissipative factors, the latter being not represented in
the structure of the numerator.
Second, it is accepted that FDR (25) is possible to be expressed through the
response to either an appropriate external force, or to an appropriate Langevin ran-
dom source. In some cases both of these possibilities are equivalent. The considered
case is, however, not the general one.
Really, the Langevin internal sources reflect the properties of the system under
consideration. Their moments in the kinetic theory of fluctuations (see in [10,11,14])
are determined by the structure of “collision integrals” and don’t coincide with that
corresponding to the Landau theory. Let us show this.
Designate the Langevin source in RGLE (6) as y(R, t) and express the spectral
density (26) via the spectral density of the source (yy)ω,k:
(δXδX)ω,k =
(yy)ω,k
ω2 + (γ T−Tc
Tc
+ γgk2)2
. (32)
We see, that for the coincidence of the formulas (26) and (32), the spectral density
of the Langevin source in RGLE (6) must be defined by the following expression:
(yy)ω,k = 2γ
X2
T
n
. (33)
Thus, the intensity of noise is defined here only by the dissipative factor γ intro-
duced formally in equations (5) and (6). In formula (33) the “diffusion noise”, the
intensity of which is proportional to the square of the wave vector k2, is not taken
into account. This makes one doubt about the derivation of O-Z formula (28) which
was presented above. We shall see that the diffusion noise, if it is taken into account,
radically changes the behaviour of fluctuations in the vicinity of the critical point.
401
Yu.L.Klimontovich
Below the expression for the intensity of the Langevin source in the reaction-diffusion
equation (see in [11,16]), and, as a consequence, in RGLE will be presented.
Correlation radius (29) takes an abnormally large value only in the critical region
of temperatures. Far from the critical point either for high, or for low temperatures
its value (in any case, based on the Ginsburg-Landau equation) decreases up to
molecular scales. Correlations of such a kind cannot ensure the spatio-temporal
coherence in nonsymmetrical phase.
In accordance with the width of the spectral line (26), the appropriate relaxation
time (see paragraph 101 in [14]) is determined
τcor =
1
γ T−Tc
Tc
+ γgk2
. (34)
For k 6= 0 it stays finite even at the critical point.
The Ginsburg number Gi is the correlation parameter for a nonsymmetrical
phase. Below we shall introduce the correlation parameter K for all temperatures.
Before doing this, however, we shall conduct the alternative calculation of the spatio-
temporal correlator based on the kinetic equation.
6. Calculation of fluctuations based on the kinetic equation
For kinetic description of the second order phase transitions we shall use, fol-
lowing chapter 18 in [Klimontovich, 1993], and [Klimontovich, 1995], the kinetic
equation for the distribution function f(X,R, t) of not only internal variable X , but
also of coordinates R of bistable elements:
∂f(X,R, t)
∂t
=
∂
∂X
[
D(x)
∂f
∂X
+
1
mγ
∂heff(X, af)
∂X
f
]
+D
∂2f
∂R2
. (35)
This equation can be obtained using the general method of deriving the kinetic
equations (see in [11,16]). It contains two dissipative terms which are defined by
redistribution of bistable elements in space R with the diffusion coefficient D, and
redistribution in the space X with the diffusion coefficient D(x), respectively. For
simplicity it is easier to suppose that the two diffusion coefficients are equal to each
other. In general case there is, of course, an additional parameter D/D(x).
If relaxation time τD = L2/D is much shorter than the other relaxation times
the equation (35) can be replaced by the one for the distribution function f(X, t):
∂f(X, t)
∂t
=
∂
∂X
[
D(x)
∂f
∂X
+
1
mγ
∂heff(X, af)
∂X
f
]
. (36)
Its equilibrium solution for the homogeneous distribution coincides with the
Boltzmann distribution (10). Note, that the Fokker-Planck equation (36) is used
in the theory of phase transitions (see chapter 13 in [5], and chapter 21 in [10]).
Here, however, the kinetic equation (35), being one of the basic equations of the
statistical theory of open systems (chapter 18 in [11], and [16]) serves as the basis of
402
Description of the second order phase transitions
the kinetic theory of phase transitions. It can also serve as the basis of the kinetic
theory of fluctuations.
As is well-known, calculation of kinetic fluctuations is carried out (see in [10,11,
14]), using the two equivalent methods. Practically it is more convenient to use the
Langevin method when the random source is introduced into the kinetic equation
(35):
∂f(X,R, t)
∂t
=
∂
∂X
[
D(x)
∂f
∂X
+
1
mγ
∂heff(X, af)
∂X
f
]
+D
∂2f
∂R2
+ y(X,R, t). (37)
Let’s return to the “dynamical” kinetic equation (35). In the first-moment ap-
proximation, the distribution function f(X,R, t) may be represented in the following
form:
f(X,R, t, ) = δ(X −X(R, t, )), 〈X〉 = X(R, t). (38)
In this case, the equation (35) is reduced to the reaction-diffusion equation (FKPP)
for the first moment, i.e., to the function X(R, t):
∂X(R, t)
∂t
= −ω
2
0
γ
[
T − Tc
Tc
+ bX2(R, t)
]
X(R, t) +D
∂2X(R, t)
∂R2
. (39)
It differs from RGLE (6) by the structure of dissipative terms.
With the help of the equation (39) it is possible to describe, for example, the
structure of domain walls in ferroelectrics (see in [13]). In the stationary one-
dimensional state (R is parallel to y) the solution is defined by the following ex-
pression:
X(y) = −
〈
Tc − T
Tcb
〉
1
2
tanh
y
d
. (40)
The width of the wall is defined by the formula:
d = XT
(
2
Tc
Tc − T
)
1
2
, X2
T =
kT
mω2
0
. (41)
The ratio of the thickness of the wall d to the “amplitude” for X(y = −∞) is defined
by the following expression:
d
X(y = −∞)
= (2X2
T b)
1/2 Tc
Tc − T
. (42)
While approaching a critical point this ratio is increased by the Curie law. In the
opposite case (T ≪ Tc) it is proportional to the small dimensionless parameter
(X2
T b)
1/2.
At the critical point, the formula (42) loses its sense. To obtain the corresponding
result at the critical point we must use the kinetic equation (37) with the Langevin
source (see (19.6.4) in [11]), the appropriate moments of which are defined by ex-
pressions (see (19.4.14) in [11]), or (9.2) in [16]).
403
Yu.L.Klimontovich
From this equation (38) for pulsing function in the approximation of the first
moment we obtain the equation FKPP (39), which corresponds to RGLE (6), but
now with the Langevin source already. Thus, the noise intensity is defined as a sum
of “reaction” and diffusion terms.
We solve the obtained Langevin equation in the linear approximation with re-
spect to fluctuations δX(R, t) =
∫
Xδf (X,R, t) dX . By solving this equation for
T > Tc, we get the following expression for spatio-temporal spectral density of
fluctuations δX(R, t) :
(δXδX)ω,k =
2
(
γ(x) +Dk2
)
ω2 +
(
γ(x) +Dk2
)2 · 1
n
〈
(δX)2
〉
. (43)
We denote the “reaction” dissipative coefficient as γ(x) at T > Tc and dispersion as
〈(δX)2〉 for the Boltzmann distribution according to Gaussian approach
γ(x) =
T − Tc
Tc
ω2
0
γ
,
〈
(δX)2
〉
=
kBT
mω2
0
Tc
T − Tc
≡ X2
T
Tc
T − Tc
. (44)
Spectral line (43) is more symmetric than spectral line (26) because both dissipative
terms in it are taken into account on equal rights. As a consequence, the spatial
density of fluctuations and the spatial correlation functions are defined by other
formulas:
(δXδX)k =
1
n
〈
(δX)2
〉
; 〈δXδX〉R,R′ =
1
n
〈
(δX)2
〉
δ(R− R′). (45)
Thus, the spatial correlation function is not defined now by the O-Z formula. It
differs from zero only in the volume of a point of continuous medium (in physically
infinitesimal volume Vph), as the function δ(R − R′)|R=R′ = V −1
ph . As a result, for
one-point correlator 〈(δx)2〉 we have the following expression:
〈δXδX〉R=R′ =
1
Nph
〈
(δX)2
〉
. (46)
Here Nph = nVph is a number of particles in a physically infinitesimal volume.
Formula (46) shows, that the dispersion of fluctuations which are smoothed over
the volume of a “point” of “continuous medium” is Nph times less than for the
Boltzmann distribution. Thus, the spatial correlator now has the form (45).
The O-Z formula, however, is important for the kinetic theory as well. Now it is
connected not with the integral for the spectral line at the frequency ω, but with
the spectral line for zero frequency. Let’s show this.
From equation (43) the equality follows:
(δXδX)ω=0,k γ(x) =
2
1 + r2ck
2
· 1
n
〈
(δX)2
〉
. (47)
Here, the designation (41), which was introduced earlier for the correlation radius,
is used. We shall carry out the integration with respect to the wave number in it.
404
Description of the second order phase transitions
As a result, we get the following expression:
(δXδX)ω=0,r γ(x) =
1
n
〈
(δX)2
〉 1
2πr2cr
exp
(
− r
rc
)
; r2c = X2
T
Tc
T − Tc
. (48)
The right-hand part in it coincides with the O-Z formula (28) for the spatial corre-
lator of fluctuations.
Thus, the O-Z formula defines the spatial correlation of the temporal spectral
density at zero frequency. For this reason, the meaning of the formula O-Z in the
theory of phase transitions is changed.
The results obtained are based on the solution of the kinetic equation with a
Langevin source. The O-Z formula can be also obtained directly using the “Gibbs
distribution” with the functionHeff . However, this distribution is not the equilibrium
solution of the kinetic equation, but it “was introduced by definition”. In a separate
paper it will be shown that temporal evolution to the O-Z distribution, as well
as the Debye distribution in the plasma theory, is described by the equations of
the structure being different from that in the theory of the second order phase
transitions.
To conclude this section, we shall compare some results of calculating the fluc-
tuations by two different methods, which were presented above. The comparison of
results (28) and (48) allows one to introduce a new correlation parameter K
(ω=0)
n ,
which is an analogy of the nonsymmetrical phase of Ginsburg parameter Gi:
K(ω=0)
n =
(δXδX)ω=0,rc γ(x)
X2
m.p.
∼ X2
Tc
b
nr3c
(
Tc
|T − Tc|
)2
. (49)
Below, we shall give an example of a unified definition of the correlation parameter
for all temperatures.
It is possible to consider the spectral density (43) as the fluctuation-dissipation
relation (FDR). In order to compare it with the FDR (25), we first write the ex-
pression for a response to a Langevin source. It has the following form:
χ(y)(ω, k) =
1
−iω + γ(x) +Dk2
. (50)
Instead of the FDR (25) we now have (for T − Tc > 0) the following expression:
(δXδX)ω,k =
1
ω
Imχ(ω, k)(yy)ω,k. (51)
Here the designation for the spectral density of the Langevin source is introduced:
(yy)ω,k = 2
(
γ(x) +Dk2
) 〈(δX)2〉
n
. (52)
The introduction of it in equation (51) results in the formula (43) for the spatio-
temporal spectral density. Thus, the form of FDR of the model considered here is
essentially different from the form of FDR (25) in the Landau theory.
405
Yu.L.Klimontovich
The results obtained are valid only in the linear approximation over fluctuations
and are consequently justified only far from the critical point. To obtain the results
for the critical point it is necessary to use a more general solution of the kinetic
equation (35) .
Based on the equation (35 ) we obtain a chain of equations for the moments 〈X n〉
and, therefore, the “problem of closure” appears. In the self-consistent approxima-
tion, when 〈Xn〉 = 〈X〉n, we get a closed equation for the first moment 〈X〉. The
RGLE (39) serves as an example. Now consider other possible methods of closure.
7. Approximation of the second moment. Polydomain
ferroelectrics
Consider an ensemble of ferroelectrics in which the two opposite directions of the
polarization vector in the nonsymmetrical phase are equally probable. Its realiza-
tion is possible if the observation time is much longer than the polarization vector
switching time.
In such an ensemble the first moment 〈X(R, t)〉 = 0 and it is more natural to use
the self-consistent approximation for the second moments 〈X 2m〉 = 〈X2〉n . We shall
show that it also leads to the reaction-diffusion equation, but now for the function
〈X2〉 ≡ 〈E〉 ≡ E(R, t). To this end, instead of expression (38) we shall use the
following distribution function:
f(X,R, t) =
1
2
(
δ(X −
√
〈E〉) + δ(X +
√
〈E〉)
)
, (53)
for which all odd moments are equal to zero, and the second moment 〈X 2〉 ≡ 〈E〉 ≡
E(R, t). The appropriate distribution function of energy E is defined by the following
expression:
f(E,R, t) =
∫
δ(E −X2)f(X,R, t)dX = δ(E −E(R, t)). (54)
The equation for the function E(R, t) is obtained using the kinetic equation (39).
In this case it has the following form:
∂E(R, t)
∂t
= 2
[
D(x) − Γ
(
T − Tc
Tc
+ bE(R, t)
)
E(R, t)
]
+D
∂2E(R, t)
∂R2
. (55)
Now we use the stationary solution for the homogeneous state. Under the condi-
tion accepted above D(x) = D = kBT/mγ, and 〈X2〉 is determined by the following
expression:
〈
X2
〉
+
T − Tc
Tcb
〈
X2
〉
=
X2
T
b
, E(R, t) ≡
〈
X2
〉
. (56)
Using the dimensionless parameter
X2
T b≪ 1, (57)
406
Description of the second order phase transitions
let’s write the solution of the last equation for the three selected states:
〈
X2
〉
=
X2
T
Tc
T−Tc
, T > Tc ,
X2
T
(
1
X2
T
b
)1/2
, T = Tc ,
Tc−T
Tcb
, T < Tc .
(58)
We see that the second moment plays the role of the order parameter for the non-
symmetrical phase. At the critical point it takes a finite value. Above the critical
point the function 〈X2〉 coincides with the dispersion for the Boltzmann distribution
(10).
Thus, the self-consistent approximation of the second moment again results in
the RGLE (39), i.e., in the reaction-diffusion equation. However, now the “reaction”
term has another structure. One of the consequences is that at the critical point the
order parameter differs from zero. As we shall see below, the character of spatio-
temporal correlations is also varied.
Consider now a system which consists of a large number of domains. Suppose
that in the volume of measurement the number of domains is great. In this case it
is possible to use the self-consistent approximation already in the expression for the
hardness of atomic oscillators. The latter means that in the kinetic equation (35)
(in the member with elastic force) the following substitution is carried out:
1
mγ
∂heff(X, af)
∂X
= Γ
(
T − Tc
Tc
+ bX2
)
X → Γ
(
T − Tc
Tc
+ b
〈
X2
〉
)
. (59)
In this approximation the kinetic equation (37) has the following form:
∂f
∂t
+
∂
∂X
[
D(x)
∂f
∂X
+ Γ
(
T − Tc
Tc
+ bE(R, t)
)
Xf
]
+D
∂2f
∂R2.
. (60)
Together with equation (55) it consists of the closed system of equations for the
function f(X,R, t) and average energy E(R, t).
We shall see that in the course of evolution to the equilibrium state there are
“fast” and “slow” processes. The evolution of the average energy is a fast process
described by the closed equation (55), while the kinetic equation (60) will be used to
describe only slow processes. In the latter equation it is possible to use the solution
of equation (56) for the function E(R, t). While describing the “slow” processes by
the kinetic equation (60), we can represent it in the following form:
∂f
∂t
=
∂
∂X
[
D(x)
∂f
∂X
+
D(x)
〈X2〉Xf
]
+D
∂2f
∂R2.
. (61)
The equilibrium solution of this equation is represented by the Gaussian distribution
f(X) = (2π 〈E〉)−1/2 exp
(
− X2
2 〈X2〉
)
. (62)
Now we can proceed to the calculation of fluctuations.
407
Yu.L.Klimontovich
8. Fast and slow fluctuations under phase transitions
Now we carry out the calculation of fluctuations, and show that the equa-
tions (55) and (61) describe two kinds of fluctuations, namely “fast” and “slow”.
Using equation (55) with the corresponding Langevin source δy(E)(R, t) receive,
in the linear approximation, the equation for fluctuation of energy δE(R, t). Write
this equation for Fourier components:
∂δE(ω, k)
∂t
+
1
τ(E)
δE(ω, k) = δy(E)(ω, k). (63)
Here the designation for relaxation time of energy fluctuations and an appropriate
width of a spectral line are introduced:
1
τ(E)
≡ ∆(E)(k) = 2Γ
(
T − Tc
Tc
+ 2b
〈
X2
〉
)
+Dk2. (64)
The average energy 〈X2〉 ≡ 〈E〉 is defined by the solution of equation (56) for all
values of temperature. As an example, for temperatures being lower than the critical
one the relaxation time is defined by the following expression:
τ(E) =
1
2ΓTñ−T
Tc
+Dk2
, Γ =
ω2
0
γ
. (65)
We received the expression being the analogue of equation (34) in Landau theory.
However, in the approximation of the second moment we have a general expression
(64) for relaxation time, which is valid for all values of temperature.
Consider the expression for the response to a Langevin source δy(E)(R.t), which
follows from equation (55) with an appropriate Langevin source and can be deter-
mined through the width of a spectral line of energy fluctuations (64):
χ(E)(ω, k) =
1
−iω +∆(E)(k)
. (66)
The response is also determined for all values of temperature in a critical region. In
particular, for a static response in an unbounded medium (k = 0) we have:
χ(E)(ω = 0, k = 0) =
1
2Γ
(
T−Tc
Tc
+ 2b 〈X2〉
) , Γ =
ω2
0
γ
. (67)
Far from the critical point the response is defined by the Curie law, but at the
critical point
χ(E)(ω = 0, k = 0) =
1
4Γ (X2
T b)
1/2
. (68)
Together with the fast fluctuations there exist slow ones. During their calculation
it is possible to assume that the process determined by “fast” fluctuations is already
408
Description of the second order phase transitions
established. It gives the ground to use the stationary solution of equation (55) (equa-
tion (56)) for the average value of “energy”.
For calculating the fast fluctuations we shall use the kinetic equation (61) with
the appropriate Langevin source. From it the equation for the function X(R, t)
follows:
∂X(R, t)
∂t
+
D(x)
〈X2〉X(R, t) = D
∂2X(R, t)
∂R2
+ y(x)(R, t). (69)
Average value 〈X(R, t)〉 = 0 and, therefore, δX(R, t) = X(R, t). As a result, the
equation for the Fourier components of function X(R, t) has the following form:
(
−iω +
1
τ(X)
)
X(ω, k) = y(X)(ω, k). (70)
Here, as well as in equation (63), the designation for the relaxation time of fluctua-
tions X(R, t) and the appropriate width of a spectral line is introduced:
1
τ(X)
≡ ∆(X) =
D(X)
〈X2〉 +Dk2 ≡ D
〈X2〉(1 + r2ck
2). (71)
We take into account the equality D(X) = D and enter the designation for the
square of the correlation radius r2c = 〈X2〉 . The intensity of Langevin source in
equation (63) is defined by expression (19.6.4) in [10], or (9.2) in [11]:
(y(X)y(X))ω,k = 2∆(X)
〈X2〉
n
. (72)
The appropriate response to y(ω, k) is defined by the following expression:
χ(X)(ω, k) =
1
−iω + D
〈X2〉
(1 + r2ck
2)
, (73)
and, therefore,
χ(X)(ω = 0, k = 0) =
〈X2〉
D
≡ 1
Γ
〈X2〉
X2
T
. (74)
Using the solution of equation (56), all of the obtained characteristics can be deter-
mined at the critical point as well. Their behaviour in the symmetrical phase (which
is higher than the critical point) is qualitatively the same as in the Landau theory. In
the nonsymmetrical phase, however, (which is lower than the critical temperature)
it is essentially different. Let’s show it.
As in the formula (58), we shall distinguish three characteristic states:
χ(X)(ω = 0, k = 0) =
1
Γ
Tc
T−Tc
, T > Tc ,
1
Γ
(
1
X2
T
b
)1/2
, T = Tc ,
1
Γ
Tc−T
Tc
1
X2
T
b
, T < Tc .
(75)
One can see that the susceptibility increases according to the Curie law only as one
approaches the critical point on the side of a symmetrical phase. As one approaches
409
Yu.L.Klimontovich
the critical point on the side of low temperatures, the susceptibility, on the contrary,
decreases.
Thus, it is possible to speak about the existence of the “jump of susceptibility”.
To reveal it we define the temperature dependence of rigidity not by the Landau
formula (2), but by the formula (3). It roughly describes the behaviour of differ-
ent characteristics of phase transitions for all the values of temperature. Then the
susceptibility jump will be defined by the expression:
χ(X)|Tc−T≫∆T − χ(X)|T−Tc≫∆T =
1
Γ
(
1
X2
Tc
b
− 1
)
. (76)
Note, that it has the similar structure as the formula (14) for the jump of the heat
capacity in the Landau theory. Similar behaviour of a dielectric susceptibility is
observed in some types of ferroelectrics (page 88 in [13]).
Below we determine the temperature dependence of the heat capacity for the case
considered here. Before this, however, it is necessary to carry out the appropriate
determination of relaxation times, screening radius (thickness of domain walls) and
correlation radius.
Return to the formula (71). Consider the case of an unbounded system (k = 0).
In this case the relaxation time is defined by the expression
τ(X) =
〈X2〉
D
≡ 1
Γ
〈X2〉
X2
T
= χ(X)(ω = 0, k = 0) (77)
for all the values of temperature. Using (75) it is possible to determine the relaxation
time for the three characteristic states.
Thus, as one approaches the critical point from the side of high temperature,
the relaxation time increases according to the Curie law, it is finite at the critical
point, and increases on a measure of downturn of temperature. By analogy with the
formula (18) it is possible to define the “jump of a relaxation time”:
τ(X)|Tc−T≫∆T − τ(X)|T−Tc≫∆T =
1
Γ
(
1
X2
Tc
b
− 1
)
≫ 1
Γ
. (78)
The latter confirms the assumption stated above concerning the existence of fluc-
tuations with large relaxation time, i.e., “slow” (or coherent) fluctuations. Remind,
that in the Landau-Khalatnikov theory the relaxation time in the symmetrical phase
decreases according to the Curie law (see [14]).
Finally, the correlation radius (or the width of the domain wall) is defined by
the expression
r2c =
D
Γ
〈X2〉
X2
T
=
〈
X2
〉
= Dτ(X). (79)
In accordance with this it is possible to define (qualitatively) the “jump of square
of a correlation radius” as well.
Thus, the square of the correlation radius for all T < Tc is much larger than the
square of the displacement by the molecular diffusion, as well as the points of the
existence of spatial coherence for temperatures are lower than the critical one.
410
Description of the second order phase transitions
Using equation (70) and formula (72) we have the expression for the spatio-
temporal spectral density
(XX)ω,k =
2∆(X)
ω2 + (∆(X))2
〈X2〉
n
. (80)
The width of the lines ∆(X) and the dispersion 〈X2〉 are determined by formulas
(71) and (56) for all values of temperature. The integration with respect to ω gives
the appropriate expression for the spatial spectral density:
(XX)k =
〈X2〉
n
. (81)
Since the right-hand part does not depend on k, the spatial correlator and the
dispersion are defined by the following expressions:
〈XX〉R−R′ =
〈X2〉
n
δ(R−R′); 〈XX〉R=R′ =
〈X2〉
Nph
. (82)
〈X2〉 is defined by the solution of equation (56) . The coincidence with the corre-
sponding formulas (45) and (46) for “fast” fluctuations takes place only for T > Tc.
Here, the Ornstein-Zernike formula is also valid only for the temporal correlator
at zero frequency:
(δXδX)ω=0,|R−R′|∆(X) =
〈X2〉
n
1
2πr2c |R− R′| exp
(
−|R− R′|
rc
)
. (83)
The structure of this formula is similar to the one of formula (48). However, the
temperature dependence of the correlation radius coincides only for T > Tc, while
for T < Tc the correlation radius in the formula (48), i.e., for “fast” fluctuations,
decreases as the temperature is decreased in accordance with the Curie law. For
“slow” fluctuations the correlation radius in accordance with the expression (79)
remains a macroscopic characteristic for low temperatures T ≪ Tc as well. The latter
means that in the nonsymmetrical phase the interaction at temperature T ≪ Tc is
collective.
The result (83) allows one to introduce the dimensionless correlation parameter
for all values of temperature:
K(ω=0) =
(δXδX)ω=0,rñ∆(x)
〈X2〉 =
1
nr3c
, r2c =
〈
X2
〉
. (84)
In the symmetrical (index “s”) phase (T > Tc):
K(ω=0) = K(ω=0)
s =
1
nr3c
, r2c = X2
T
Tc
T − Tc
(85)
and, consequently, coincides with the result for “fast” fluctuations. At the critical
point the correlation parameter is defined by the following expression:
K(ω=0) (T = Tc) =
1
nr3c
, r2c = X2
T
(
1
X2
Tb
)1/2
. (86)
411
Yu.L.Klimontovich
Finally, for “slow” fluctuations in a nonsymmetrical phase we receive the following
expression:
K(ω=0) = K(ω=0)
n =
1
nr3c
, r2c = X2
T
Tñ − T
Tc
1
X2
Tb
. (87)
And as above, for the qualitative representation of evolution of the correlation
parameter during the phase transition we define the temperature dependence of
rigidity not by the Landau formula (2), but by the formula (3). Then, by analogy
with formulas (76) and (78), it is possible to define the jump of the correlation radius
in the course of transition from a symmetrical phase to a nonsymmetrical one.
Thus, during the phase transition (in the process of temperature decrease), the
correlation parameter decreases monotonously. The latter means the reduction of
the role of fluctuations and, consequently, the increase of the degree of coherence
during the transition to a nonsymmetrical phase.
The latter shows, that in the process of temperature decreasing, the condition of
applicability of the self-consistent approximation is improved. When the transition
region is narrow enough, one may speak of the jump of correlation parameters.
Since the maximum value of the distribution function (62) is equal to zero, then
the heat capacity during the phase transition is defined by the formula
C(T ) = T
∣
∣
∣
∣
dSeff(T )
dT
∣
∣
∣
∣
, Seff(T ) =
1
2
kBN
(
ln(2π
〈
X2
〉
+ 1
)
. (88)
As a result, we have the following expression for the heat capacity:
C(T ) =
1
2
kBNT
1
〈X2〉
∣
∣
∣
∣
d 〈X2〉
dT
∣
∣
∣
∣
, (89)
which is valid for all values of temperature. From this expression, in particular,
it follows, that as one approaches the critical point the heat capacity increases
according to the Curie law:
C(T ) =
1
2
kBN
Tc
|T − Tc|
, (90)
and at the critical point
C(T = Tc) =
1
2
kBN
1
√
X2
Tb
. (91)
One can see, that now the jump of the heat capacity is equal to zero and the function
C(T ) is the symmetrical function of temperature with a finite value at the critical
point.
Thus, for polydomain systems, the jump takes place not for the heat capacity,
but for a susceptibility and for characteristics connected with it, for example, for the
correlation radius. The similar behaviour in ferroelectrics is described on page 88
in [13]. It is interesting that the similar behaviour of the susceptibility and the
heat capacity was observed in the course of the phase transitions in the compound
Ni52.3Mn23.7Ga25 (the private communication with A.N.Vasil’ev).
412
Description of the second order phase transitions
9. Conclusion
9.1. Some final remarks
Thus, we have seen in the theory of phase transitions based on the kinetic equa-
tion (35), the characteristics of the phase transitions are finite for all values of
temperature. Therefore, the “problem of infinities” of the susceptibility, the heat
capacity, and the correlation radius at the critical point is solved.
In the kinetic theory of phase transitions there are two limit cases. In the first
of them, the “order parameter” is connected with the first moment and calculations
of either dynamical or statistical characteristics are carried out based on the RGLE
or the corresponding reaction-diffusion equations. To calculate the fluctuations it is
necessary to determine the corresponding Langevin sources.
In the second case, such as in polydomain systems, when the first moment is equal
to zero, the approximate equation for the second moment follows from the kinetic
equation. Below the critical point it plays the role of “order parameter”. In this
case the “nonstandard behaviour” of the main characteristics of phase transitions,
i.e., the susceptibility and the characteristics connected with it, exhibit a jump. The
heat capacity in this case has no jump being symmetrical and taking a finite value
at the critical point.
9.2. Physically infinitesimal scales
Space correlators in both cases are δ-correlated. The “width” of δ-function is
defined by a physically indefinite scale, determining the size of “points of continuous
medium”. The characteristic size of a domain L is macroscopic for all temperatures
T < Tc. The width of wall d is macroscopic only in the critical region.
Everything stated above allows one to use a model of continuous medium for
the description of phase transitions. Then, following [10], and [11], the physically
infinitesimal scale for the critical region can be defined by the following relations:
lph ∼ d
N1/5
, N ∼ nd3. (92)
Here the designation for the number of particles in volume d3 is introduced.
Remind that simultaneous spatial correlators are proportional to the function
δ(R− R′). For this reason the relative dispersion of kinetic fluctuations is small by
virtue of the fact, that in the “point” of continuous medium there are many particles
(the equation (46)):
〈δXδX〉R=R′
〈(δX)2〉 =
1
Nph
=
1
nl3ph
∼ 1
N2/5
≪ 1. (93)
The dispersion 〈(δX)2〉 is defined by the Boltzmann distribution, and the function
〈δXδX〉R=R′ is defined through fluctuations of the distribution function.
At T < Tc the size of coherent region, i.e., the size of “natural” domain for
polydomain state is defined by the correlation radius of “slow” fluctuations, i.e., r c ∼
413
Yu.L.Klimontovich
√
1/δ for T < Tc. Thus, the correlation radius of “slow” fluctuations is macroscopic
for all temperatures T < Tc. It monotonously increases. For this reason, it ensures
the existence of the coherence inside the domains.
It is natural, that collective properties exhibit themselves only in the region of
temperatures at which the minimal value of the correlation radius is larger than the
physically infinitesimal length lph. The beginning of the phase transition takes place
for the temperature for which the inequality begins to be true:
rc =
√
〈X2〉 > lph. (94)
The quantities rc, 〈X2〉 are defined by the formulas (56), (58) and (79). The inequal-
ity takes place only for a sufficiently small value of the nonlinearity parameter b (or
for a sufficiently small value of the dimensionless parameter X 2
T b).
The stated results are valid only for large systems, since there were used the
solutions of the kinetic equation received without accounting for the finiteness of
the system. For small systems, the additional dependence of characteristics of phase
transition on the sizes of all systems appears.
9.3. Stochastic resonance in the case of the second order pha se transition
It is possible to represent the bistable element of ferroelectrics as a small over-
damped particle moving (at T < Tc ) in a double-well potential under random
Langevin forces resulting from thermal fluctuations. Suppose that this element is
exposed to an additive periodic external force. Let this force be smaller than the
barrier height.
The stochastic resonance arises when the frequency of the external force is close
to the Kramers rate r0 ≡ 1/τtr, which defines the mean switching rate in the non-
modulated system [17].
For nonsymmetrical phase of ferroelectrics (for temperatures T < Tc ) the Kram-
ers rate is given by the formula
r0 ≡
1
τtr
=
Γ
2π
(|U ′′(Xmax)|U ′′(Xmin))
1/2
exp
(
−∆U
X2
T
)
. (95)
Here ∆U = U(Xmax)− U(Xmin) is the nonmodulated barrier height.
From the expression for the function U(X) for the nonsymmetrical phase of
ferroelectrics it follows that:
(|U ′′(Xmax)|U ′′(Xmin))
1/2
=
√
2
Tc − T
Tc
for Tc > T (96)
and
∆U = U(Xmax)− U(Xmin) =
1
4b
(
Tc − T
Tc
)2
. (97)
Taking into account these results, we can represent the expression for the Kramers
rate for nonsymmetrical phase of ferroelectrics in the following form:
r0 ≡
1
τtr
=
Γ√
2π
Tc − T
Tc
exp
(
− 1
4X2
T b
(
Tc − T
Tc
)2
)
. (98)
414
Description of the second order phase transitions
To obtain the “standard” Kramers rate for the bistable element, it is necessary
to make the following substitution in the latter formulas
Tc − T
Tc
→ 1. (99)
Now the Kramers rate is defined by the formula
r0 ≡
1
τtr
=
Γ√
2π
exp
(
− 1
4X2
T b
)
, X2
T =
kT
mω2
0
. (100)
We see that in the ferroelectrics in the critical region there is the alternative possi-
bility to control the stochastic resonance. The increase of temperature (Tc − T > 0)
decreases the height of the barrier.
In the paper [18] the new phenomenon of the mean switching frequency locking
between the states of a bistable system driven by a periodic force was found. Now
a new possibility has appeared to observe this effect in the electric circuit with the
ferroelectrics elements.
References
1. Landau L.D. To the theory of phase transitions. // JETP, 1937, vol. 7, p. 19–35.
2. Landau L.D., Lifshits E.M. Statistical Physics. Moscow, Nauka (in Russian).
3. Stanley H.E. Introduction to Phase Transition and Critical Phenomena. Oxford,
Clarendon Press, 1971.
4. Patashinskii A.Z., Pokrovskii V.L. Fluctuation Theory of Phase Transitions. Moscow,
Nauka, 1982 (in Russian).
5. Ma Sh. Modern Theory of Critical Phenomena. London, Benjamin; Moscow, Mir,
1980.
6. Anisimov M.A. Critical Phenomena in Liquids and Liquid Crystals. Moscow, Nauka,
1987 (in Russian).
7. Balescu R. Equilibrium and Nonequilibrium Statistical Mechanics. New York, John
Wiley and Sons, 1975.
8. Grossberg A.Ju., Khokhlov A.R. Statistical Physics of Macromolecules. Moscow, Nau-
ka, 1989 (in Russian).
9. Dotsenko V.S. Physics of spin glass state. // Uspekhi Fiz. Nauk, 1993, vol. 163, p. 1–
37.
10. Klimontovich Yu.L. Statistical Physics. Moscow, Nauka, 1982; New York, Harwood
Academic Publishers, 1986.
11. Klimontovich Yu.L. Statistical Theory of Open Systems. Moscow, Yanus; Dordrecht,
Kluwer Academic publishers, 1995.
12. Ginsburg V.L. Theory of ferroelectrics phenomena. // Uspechi Fiz. Nauk, 1949, vol. 38,
p. 490–525.
13. Strukov B.A., Levanuk A.P. Physical fundamentals of ferroelectrics phenomena in
crystals. Moscow, Nauka-Fizmatlit, 1995 (in Russian).
14. Lifshits E.M., Pitaevskii L.P. Physical Kinetics. Moscow, Nauka, 1979.
415
Yu.L.Klimontovich
15. Klimontovich Yu.L. Two alternative descriptions of second order phase transitions:
Landau theory and the self-consistent approach. // Physics Letters A, 1996, vol. 210,
p. 65–70.
16. Klimontovich Yu.L. From the Hamiltonian mechanics to a continuous media. Dissipa-
tive structures. Criteria of self-organization. // TMP, 1993, vol. 96, p. 385–416.
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Missouri, 1992.
18. Shulgin B., Neiman A., Anishchenko V. Mean switching frequency locking in stochastic
bistable systems driven by a periodic force. // Phys. Rev. Lett., 1995, vol. 23, p. 4157–
4160.
Термодинамічний та кінетичний опис фазових
переходів другого роду
Ю.Л.Клімонтович
Фізичний факультет
Московського державного університету ім. М.В.Ломоносова,
Воробйовиє гори, 119899 Москва, Росія
Отримано 19 листопада 1999 р.
Проведено опис термодинаміки і кінетики фазових переходів моделі
сегнетоелектриків на основі кінетичного рівняння для функції роз-
поділу, залежної від “параметра порядку”, координат і часу.
Для однодоменних сегнетоелектриків застосовано самоузгоджений
підхід до розрахунку першого момента. Кінетичне рівняння зводить-
ся до релаксаційного рівняння Гінзбурга-Ландау. Сприйнятливість
описується законом Кюрі, а теплоємність має стрибок.
Розрахунки проведено для однодоменних та полідоменних сегне-
тоелектриків. У першому випадку використовується самоузгоджене
наближення для першого моменту. У другому випадку самоузгодже-
не наближення робиться для другого моменту. В цьому випадку від-
значається стрибок сприйнятливості, а теплоємність описується за-
коном Кюрі.
Показано, що формула Орнштейна-Церніке справедлива не для про-
сторового корелятора флуктуацій, а лише для спектральної густини
цієї кореляційної функції при нульовій частоті.
В кінетичній теориї фазових переходів усі фізичні величини мають
скінчені значення в критичній точці. Таким чином, немає проблеми
“розбіжностей”.
Ключові слова: фазові переходи, самоузгоджені наближення,
проблема “розбіжностей”, когерентні флуктуації, термодинамічні
функції, критична точка
PACS: 64.60.-i, 64.60.Fr, 64.60.My, 65.50.tm
416
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