Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo
Development of the non-linear kinetic mean-field model suggested by George Martin in 1990 is discussed. Its steady-state limit is shown to coincide with Khachaturyan’s model. It is proved rigorously that Martin’s model and its 3DD version always provide decrease of free energy and are unable to mode...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
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irk-123456789-1518732019-05-26T01:25:15Z Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo Gusak, A. Zaporozhets, T. Фазовые превращения Development of the non-linear kinetic mean-field model suggested by George Martin in 1990 is discussed. Its steady-state limit is shown to coincide with Khachaturyan’s model. It is proved rigorously that Martin’s model and its 3DD version always provide decrease of free energy and are unable to model any overcoming of free-energy barrier, including nucleation. To enable nucleation processes within the mean-field models, the introduction of noise is necessary. Contrary to common way of noise introduction (noise of concentration), we introduce the noise of jump frequencies as a basic reason of fluctuations. The new method is called as Stochastic Kinetic Mean Field (SKMF). In this paper, we investigate and compare the dispersion and spatial correlations of concentration fluctuations by three methods—direct Monte Carlo simulation, numeric simulation by SKMF method, and analytic approximation within the scope of SKMF. Comparison confirms the correspondence of frequency noise to the averaging over finite number of Monte Carlo runs (over finite number of copies of the canonical ensemble). Предложено развитие нелинейной кинетической среднеполевой модели Жоржа Мартана 1990 года. Показано, что в приближении квазистационарности она соответствует модели Хачатуряна. Строго доказано, что модель Мартана и её 3DD-версия всегда обеспечивают уменьшение свободной энергии и не позволяют моделировать преодоление барьера свободной энергии вместе с зародышеобразованием. Для реализации процессов зародышеобразования в среднеполевых моделях необходимо вводить шум. В отличие от распространённого способа введения шума (как шума концентрации), мы вводим шум частоты обменов местами посредством скачков как основную причину флуктуаций. Новый метод называется SKMF (Stochastic Kinetic Mean Field). В этой работе исследуются и сравниваются дисперсия и пространственные корреляции флюктуаций концентрации, полученные с помощью трёх методов — прямого моделирования методом Монте-Карло, численного моделирования по методу SKMF, аналитического приближения в рамках SKMF. Сравнение этих методов подтверждает соответствие определённой амплитуды шума частоты усреднению по соответствующему конечному количеству Монте-Карло-запусков (по конечному числу копий канонического ансамбля). Запропоновано розвиток нелінійного кінетичного середньопольового моделю Жоржа Мартана 1990 року. Показано, що у наближенні квазистаціонарности він відповідає Хачатуряновому моделю. Строго доведено, що Мартанів модель та його 3DD-версія завжди забезпечують зменшення вільної енергії та не уможливлюють моделювати подолання бар’єру вільної енергії разом з зародкуванням. Для реалізації процесів зародкування в середньо-польових моделях необхідно вводити шум. На відміну від поширеного способу введення шуму (як шуму концентрації), ми вводимо шум частоти обмінів місцями через стрибки як основну причину флюктуацій. Нова метода називається SKMF (Stochastic Kinetic Mean Field). У цій роботі досліджуються та порівнюються дисперсія та просторові кореляції флюктуацій концентрації за допомогою трьох метод — прямого моделювання за методою Монте-Карло, чисельного моделювання за методою SKMF й аналітичного наближення в рамках SKMF. Порівняння цих метод підтверджує відповідність певної амплітуди шуму частот усередненню по відповідній скінченній кількості Монте-Карло-запусків (по скінченній кількості копій канонічного ансамблю). 2018 Article Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo / A. Gusak, T. Zaporozhets // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 11. — С. 1415-1435. — Бібліогр.: 26 назв. — англ. 1024-1809 DOI: 10.15407/mfint.40.11.1415 PACS: 05.40.Ca, 61.72.Bb, 64.60.Cn, 64.60.De, 66.30.Ny, 66.30.Pa, 81.30.Hd http://dspace.nbuv.gov.ua/handle/123456789/151873 en Металлофизика и новейшие технологии Інститут металофізики ім. Г.В. Курдюмова НАН України |
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Фазовые превращения Фазовые превращения |
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Фазовые превращения Фазовые превращения Gusak, A. Zaporozhets, T. Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo Металлофизика и новейшие технологии |
| description |
Development of the non-linear kinetic mean-field model suggested by George Martin in 1990 is discussed. Its steady-state limit is shown to coincide with Khachaturyan’s model. It is proved rigorously that Martin’s model and its 3DD version always provide decrease of free energy and are unable to model any overcoming of free-energy barrier, including nucleation. To enable nucleation processes within the mean-field models, the introduction of noise is necessary. Contrary to common way of noise introduction (noise of concentration), we introduce the noise of jump frequencies as a basic reason of fluctuations. The new method is called as Stochastic Kinetic Mean Field (SKMF). In this paper, we investigate and compare the dispersion and spatial correlations of concentration fluctuations by three methods—direct Monte Carlo simulation, numeric simulation by SKMF method, and analytic approximation within the scope of SKMF. Comparison confirms the correspondence of frequency noise to the averaging over finite number of Monte Carlo runs (over finite number of copies of the canonical ensemble). |
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Article |
| author |
Gusak, A. Zaporozhets, T. |
| author_facet |
Gusak, A. Zaporozhets, T. |
| author_sort |
Gusak, A. |
| title |
Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo |
| title_short |
Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo |
| title_full |
Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo |
| title_fullStr |
Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo |
| title_full_unstemmed |
Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo |
| title_sort |
martin’s kinetic mean-field model revisited—frequency noise approach versus monte carlo |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
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2018 |
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Фазовые превращения |
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http://dspace.nbuv.gov.ua/handle/123456789/151873 |
| citation_txt |
Martin’s Kinetic Mean-Field Model Revisited—Frequency Noise Approach versus Monte Carlo / A. Gusak, T. Zaporozhets // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 11. — С. 1415-1435. — Бібліогр.: 26 назв. — англ. |
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Металлофизика и новейшие технологии |
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AT gusaka martinskineticmeanfieldmodelrevisitedfrequencynoiseapproachversusmontecarlo AT zaporozhetst martinskineticmeanfieldmodelrevisitedfrequencynoiseapproachversusmontecarlo |
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2025-07-13T01:44:28Z |
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2025-07-13T01:44:28Z |
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| fulltext |
ФАЗОВЫЕ ПРЕВРАЩЕНИЯ
PACS numbers: 05.40.Ca, 61.72.Bb, 64.60.Cn, 64.60.De, 66.30.Ny, 66.30.Pa, 81.30.Hd
Martin’s Kinetic Mean-Field Model Revisited—Frequency
Noise Approach versus Monte Carlo
Andriy Gusak and Tetiana Zaporozhets
Bohdan Khmelnytsky National University of Cherkasy,
81 Shevchenko Blvd.,
18031 Cherkassy, Ukraine
Development of the non-linear kinetic mean-field model suggested by George
Martin in 1990 is discussed. Its steady-state limit is shown to coincide with
Khachaturyan’s model. It is proved rigorously that Martin’s model and its
3D version always provide decrease of free energy and are unable to model
any overcoming of free-energy barrier, including nucleation. To enable nu-
cleation processes within the mean-field models, the introduction of noise is
necessary. Contrary to common way of noise introduction (noise of concen-
tration), we introduce the noise of jump frequencies as a basic reason of fluc-
tuations. The new method is called as Stochastic Kinetic Mean Field (SKMF).
In this paper, we investigate and compare the dispersion and spatial correla-
tions of concentration fluctuations by three methods—direct Monte Carlo
simulation, numeric simulation by SKMF method, and analytic approxima-
tion within the scope of SKMF. Comparison confirms the correspondence of
frequency noise to the averaging over finite number of Monte Carlo runs
(over finite number of copies of the canonical ensemble).
Key words: kinetics, mean-field approximation, diffusion, noise, fluctua-
tion, correlation, probability.
Запропоновано розвиток нелінійного кінетичного середньопольового мо-
делю Жоржа Мартана 1990 року. Показано, що у наближенні квазистаці-
онарности він відповідає Хачатуряновому моделю. Строго доведено, що
Мартанів модель та його 3D-версія завжди забезпечують зменшення віль-
ної енергії та не уможливлюють моделювати подолання бар’єру вільної
енергії разом з зародкуванням. Для реалізації процесів зародкування в
Corresponding author: Tetiana Zaporozhets
E-mail: zaptet@ukr.net
Citation: Andriy Gusak and Tetiana Zaporozhets, Martin’s Kinetic Mean-Field Model
Revisited—Frequency Noise Approach versus Monte Carlo, Metallofiz. Noveishie
Tekhnol., 40, No. 11: 1415–1435 (2018), DOI: 10.15407/mfint.40.11.1415.
Ìåòàëëîôèç. íîâåéøèå òåõíîë. / Metallofiz. Noveishie Tekhnol.
2018, т. 40, № 11, сс. 1415–1435 / DOI: 10.15407/mfint.40.11.1415
Îттиски доступнû непосредственно от издателя
Ôотокопирование разрешено только
в соответствии с лицензией
2018 ÈМÔ (Èнститут металлоôизики
им. Ã. Â. Êурдюмова ÍÀÍ Óкраинû)
Íапечатано в Óкраине.
1415
mailto:zaptet@ukr.net
https://doi.org/10.15407/mfint.40.11.1415
https://doi.org/10.15407/mfint.40.11.1415
1416 Andriy GUSAK and Tetiana ZAPOROZHETS
середньо-польових моделях необхідно вводити шум. Íа відміну від поши-
реного способу введення шуму (як шуму концентрації), ми вводимо шум
частоти обмінів місцями через стрибки як основну причину ôлюктуацій.
Íова метода називається SKMF (Stochastic Kinetic Mean Field). Ó цій ро-
боті досліджуються та порівнюються дисперсія та просторові кореляції
ôлюктуацій концентрації за допомогою трьох метод — прямого моделю-
вання за методою Монте-Êарло, чисельного моделювання за методою
SKMF й аналітичного наближення в рамках SKMF. Порівняння цих ме-
тод підтверджує відповідність певної амплітуди шуму частот усереднен-
ню по відповідній скінченній кількості Монте-Êарло-запусків (по скін-
ченній кількості копій канонічного ансамблю).
Ключові слова: кінетика, середньопольове наближення, диôузія, шум,
коливання, ôлюктуація, кореляція, ймовірність.
Предложено развитие нелинейной кинетической среднеполевой модели
Жоржа Мартана 1990 года. Показано, что в приближении квазистацио-
нарности она соответствует модели Хачатуряна. Строго доказано, что мо-
дель Мартана и её 3D-версия всегда обеспечивают уменьшение свободной
энергии и не позволяют моделировать преодоление барьера свободной энер-
гии вместе с зародûшеобразованием. Для реализации процессов зародûше-
образования в среднеполевûх моделях необходимо вводить шум. Â отличие
от распространённого способа введения шума (как шума концентрации),
мû вводим шум частотû обменов местами посредством скачков как основ-
ную причину ôлуктуаций. Íовûй метод назûвается SKMF (Stochastic Ki-
netic Mean Field). Â этой работе исследуются и сравниваются дисперсия и
пространственнûе корреляции ôлюктуаций концентрации, полученнûе с
помощью трёх методов — прямого моделирования методом Монте-Êарло,
численного моделирования по методу SKMF, аналитического приближе-
ния в рамках SKMF. Сравнение этих методов подтверждает соответствие
определённой амплитудû шума частотû усреднению по соответствующе-
му конечному количеству Монте-Êарло-запусков (по конечному числу
копий канонического ансамбля).
Ключевые слова: кинетика, среднеполевое приближение, диôôузия,
шум, ôлуктуация, корреляция, вероятность.
(Received June 17, 2018)
1. INTRODUCTION
Mean-field approximation is typically used for simplified analysis of
many equilibrium properties of gases, plasma, and condensed matter
[1–7]. Its main tricks are Eq. (1a) using unary probability distributions
for calculations of the potential energy like
3( )( ) (( ) ) ( )U K d r′ ′ ′= − r∫∫∫r r r r r (1a)
and Eq. (1b) simultaneously using Boltzmann distribution for this
MARTIN’S KINETIC MEAN-FIELD MODEL REVISITED 1417
unary probability with approximation (1a) for potential energy, mak-
ing this problem self-consistent one:
3
exp( ( ) )
( )
exp( ( ) )
U kT
U kT d r
−
r =
′ ′−∫∫∫
r
r
r
. (1b)
Interesting quasi-1D modification of the mean-field approach to ki-
netic problems of atomic transport in solid state was suggested by
George Martin in 1990 [8]. In this approach, the master equation for
probability Cp of finding atom A at the site belonging to plane number
‘p’, based on balance of local in- and out-fluxes for any site, self-
consistently uses the mean-field approximation for calculation of en-
ergy barriers in the jump frequencies:
1 , 1 1 1,
1 , 1 1 1,
(1 ) (1 )
(1 ) (1 ) .
p
v p p p p p p p p
p p p p p p p p
dC
Z C C C C
dt
C C C C
− − − −
+ + + +
= − − Γ − − Γ +
+ − Γ − − Γ
(2a)
Here, 2l vZ Z Z= + is a total number of nearest neighbours, Zl is a
number of nearest neighbours in the central plane ‘p’ perpendicular to
the concentration gradient, Zv—number of nearest neighbours in the
right ‘p + 1’ and in the left plane ‘p − 1’ (Fig. 1):
, 1
, 1 exp .p p
p p
E
kT
+
+
Γ = ν −
(2b)
, 1p p+Γ is a frequency (probability per unit time) of exchange between
atom A in plane ‘p’ and atom B in plane ‘p + 1’. , 1p pE + —the difference
between the saddle-point energy E
s
and interaction energy of jumping
atoms before energy.
In Ref. [9], we generalized Martin’s equations to the 3D-case with
the following kinetic equations for ‘concentration’ (probability) at site
‘i’ surrounded by nearest neighbours (the sites indicated by ‘in’):
( )
1
[ ]
[ ] [ ] [ ( ), ( )] [ ] [ ] [ ( ), ( )] .
Z
A
A B B A
in
dC i
C i C in i A in B C i C in in A i B
dt =
= − Γ + Γ∑ (3)
Fig. 1. Quasi-1D model of atomic migration (direction <111> for f.c.c. lattice).
1418 Andriy GUSAK and Tetiana ZAPOROZHETS
Here, the jumps are restricted (within the Martin’s model) only to the
first co-ordination shell (exchanges between nearest neighbours). Ex-
change frequencies between A and B in the neighbouring sites ‘i’ and
‘in’ are determined in [8, 9] via Arrhenius law like
( ),
0 0
[ ] [ ]
[ ( ), ( )] exp exp
s
A Bi in E E i E inQ
i A in B
kT kT
− +
Γ = ν − = ν −
(4)
with saddle-point assumed the same for all jumps, and with energies
before jump calculated taking into account interaction only with Z
nearest neighbours (VAA, VBB, VAB) and without any account of correla-
tions—within the mean-field approximation:
1
[ ] ( [ ] [ ] ),
Z
A A AA B AB
in
E i C in V C in V
=
= +∑
1
[ ] ( [ ] [ ] ).
Z
B A BA B BB
inn
E in C inn V C inn V
=
= +∑ (5)
Martin’s approach was later applied to strongly non-linear diffusion
in nanofilms with sharp gradients of the jump frequencies. Among
other effects, this approach predicted a possibility of the sharpening of
diffusion profiles (instead of traditional smoothening) and other non-
linear effects at the initial stages of interdiffusion [10, 11].
2. STEADY-STATE LIMIT OF 3D-GENERALIZATION
OF THE MARTIN’S MODEL
In his original paper [8], G. Martin considers in detail the steady-state
case of the kinetic equations in quasi-1D scheme. We will also start (in
our general 3D case) from analysis of the steady-state solutions of Eqs.
(3)–(5). Obviously, all time derivatives (for all sites ‘i’) in Eq. (3) are
equal to zero if the detailed balance is satisfied:
[ ] [ ] [ ( ), ( )] [ ] [ ] [ ( ), ( )]A B B AC i C in i A in B C i C in in A i BΓ = Γ (6a)
or
[ ] [ ] [ ] [ ] [ ] [ ]
exp exp .
[ ] [ ]
A A B A A B
B B
C i E i E i C in E in E in
C i kT C in kT
− − =
(6b)
Equation (6b) is a steady state condition, which can be interpreted as
the equalizing of chemical potentials throughout the system. Due to
the exchange mechanism of diffusion, it is sufficient to equalize the
reduced chemical potential (change of free energy due to substitution
of atom B by the atom A):
MARTIN’S KINETIC MEAN-FIELD MODEL REVISITED 1419
[ ] [ ] const.i inµ = µ ≡ µ = (7)
Here,
mix
1
[ ]
[ ] [ ] [ ] ln [ ] [ ]
[ ]
[ ]
ln 2 [ ] ( ),
[ ]
A
A B A B
B
Z
A
A AB BB
inB
C i
i i i kT E i E i
C i
C i
kT V C in Z V V
C i =
µ = µ − µ = + − =
= − + −∑
(8)
[ ] ln [ ] [ ] const,A A Ai kT C i E iµ = + + [ ] ln [ ] [ ] constB B Bi kT C i E iµ = + +
are chemical potentials within the mean-field approximation;
mix ( ) / 2AB AA BBV V V V= − + —mixing energy.
As we know, Martin’s kinetic equations always tend to steady state,
and steady state in closed system means equilibrium—stable or meta-
stable. Thus, Eq. (8) immediately leads to the self-consistent set of
non-linear algebraic equations
mix
1
[ ] 2 ( )
exp [ ] exp ( 1, , )
(1 [ ])
Z
A AB BB
A
inA
C i V Z V V
C in i N
C i kT kT=
µ − − = = … −
∑
(9)
with constraint of matter conservation:
1
[ ] .
N
A A
i
C i NC
=
=∑ (10)
Equation equivalent to Eq. (9) was suggested by A. Khachaturyan
[7] with arguments of Fermi–Dirac-type equation. He used it in his
method of concentration waves describing ordered binary structures
(see also [12–14]). These equations may be rather effectively applied
for the construction of equilibrium phase diagrams of binary and mul-
ticomponent alloys, but with important restriction of the universal
rigid lattice. Thus, limiting (steady-state) case of Martin’s kinetic ap-
proach provides self-consistent mean-field thermodynamics. Now let
us consider the relaxation processes in Martin’s approach.
3. TIME EVOLUTION OF FREE ENERGY IN MARTIN’S MODEL
AND ITS 3D GENERALIZATION
To analyse the time evolution of the free energy, we will represent the
kinetic equation (3) in terms of individual pairs of sites ‘i’, ‘in’ that
automatically provides the matter conservation:
( , )
1
[ ]
,
i inZ
A A
in
dC i dC
dt dt=
= ∑
1420 Andriy GUSAK and Tetiana ZAPOROZHETS
( , )
[ ] [ ] [ ( ), ( )] [ ] [ ] [ ( ), ( )
i in
A
A B B A
dC
C i C in i A in B C i C in in A i B
dt
= − Γ + Γ (11)
Here,
( , )i in
AdC dt can be called the ‘partial’ time derivative showing
change of concentration in site ‘i’ (and opposite change of concentra-
tion in neighbouring site ‘in’) due to exchange only between these two
sites. Then
( )
( )
( , )/2
( , )
/2
( , )
[ ] [ ]
[ ] [ ]
ln [ ] [ ] ln [ ] [ ]
[ ] [ ]
[ ] [ ] [ ( ), ( )] [ ] [ ] [ ( ), ( )]
i inNZ
A
AB AB
i in
A ANZ
A B A B
B B
i in
A B B A
dCdF
i in
dt dt
C i C in
kT E i E i kT E in E in
C i C in
C i C in i A in B C i C in in A i B
= µ − µ =
+ − − + − × =
× − Γ + Γ
∑
∑
Further,
0
/2
( , )
exp
[ ] [ ]
[ ] [ ] exp
[ ] [ ] [ ] [ ] [ ] [ ]
ln exp ln exp
[ ] [ ]
[ ] [ ] [ ]
exp
[ ]
s
B B
B B
NZ
i in A A B A A B
B B
A A B
B
dF E
kT
dt kT
E i E in
C i C in
kT
C i E i E i C in E in E in
C i kT C in kT
C i E i E i
C i kT
= −ν − ×
+ ×
×
− − × − ×
− ×
∑
[ ] [ ] [ ]
exp .
[ ]
A A B
B
C in E in E in
C in kT
− −
In notations of the reduced chemical potentials, it gives:
0
/2
( , )
exp
[ ] [ ]
[ ] [ ] exp ( [ ] [ ])
[ ] [ ]
exp exp .
s
NZ
B B
B B AB AB
i in
AB AB
dF E
dt kT
E i E in
C i C in i in
kT
i in
kT kT
= −ν − ×
+ × µ − µ ×
µ µ × −
∑
(12)
It is evident that the expression ( 1 2)(exp( 1) exp( 2))f f f f− − is always
positive except case f1 = f2 when it is equal to zero. Therefore, expres-
sion
[ ] [ ] [ ] [ ]
exp expAB AB AB ABi in i in
kT kT kT kT
µ µ µ µ − −
in Eq. (12) is al-
MARTIN’S KINETIC MEAN-FIELD MODEL REVISITED 1421
ways positive except case of equal chemical potentials.
In this product (which differs from typical expressions in the entro-
py production or the free energy release rate), the first factor (differ-
ence of reduced chemical potential between neighbouring sites) is the
driving force of exchange, and second factor (difference between ex-
ponents of reduced chemical potentials divided by kT) corresponds to
general nonlinear expression for the flux of exchanging atom. This is a
main reason why Martin’s scheme is better adjusted to the early stages
of diffusion at very sharp concentration gradients and is able to pre-
dict such non-trivial behaviour as the possibility of concentration pro-
file sharpening instead of smoothening [10, 11]. (Exponential form of
the driving force was suggested by M. Ivanov et al. [15].)
Thus, taking also into account the common ‘minus’ before the sum
in Eq. (12), the time derivative of the free energy is always negative
except steady-state (absolute or metastable minima), when it is zero.
So, we just proved that the Martin’s equation and its 3D generaliza-
tion might describe only evolution with minimization of the free energy.
Nucleation process, or any other process related to overcoming the
free energy barrier, cannot be described by KMF.
To model the evolution from metastable state to the stable state by
overcoming of the nucleation barrier, it is necessary to introduce addi-
tional noise—noise of initial conditions [9] or, better, the dynamic
noise during the evolution.
4. STOCHASTIC GENERALIZATION OF KMF
As just mentioned, noise should be introduced into the mean-field
scheme, to describe the first-order phase transformations. In linear
theories of atomic transport, the noise of concentration and of order
parameter was introduced into the Onsager scheme following the fluc-
tuation-dissipation theorem, for example, by Khachaturyan et al. [16,
17]. Yet, this method had some drawbacks:
(1) Onsager scheme for atomic local fluxes was linear and not self-
consistent, Onsager coefficients and their activation energies were not
interrelated with local composition and its energetics (contrary to
Martin’s approach);
(2) order parameter fluctuations were introduced independently of
concentration fluctuation; on the contrary, in Martin’s approach the
order is not something independent, instead, it is determined by the
oscillations of local concentration (unary probabilities at the sites) be-
tween sublattices;
(3) it seems more natural to introduce the fluctuations of the jump fre-
quencies as a true reason of the noise.
In 2016, we (jointly with Debrecen team) introduced a new simula-
tion method called SKMF (Stochastic Kinetic Mean Field), based on in-
1422 Andriy GUSAK and Tetiana ZAPOROZHETS
troduction of the jump frequencies noise [18]:
Lang Lang
, , , ,
1
(1 )( ) (1 ) ( ) ,
Z
i
i j i j i j i j j i j i
j
dC
C C C C
dt =
= − − Γ + dΓ − − Γ + dΓ ∑ (13)
Lang 3(2random 1).n
i,k
A
=
dt
dΓ − (14)
In analytical form, Langevin noise of frequencies satisfies the fol-
lowing condition:
Lang Lang 2( ) ( ) ( ).i,j k,m n ik jmt t A t t′ ′< dΓ dΓ >= d d d − (15)
Among other results, we found simple regularities for modelling
fluctuations in an ideal solid solution by SKMF.
a) Composition deviation (mean-squared fluctuation of concentra-
tion at one site) is proportional to the frequency noise amplitude An:
2
0
(1 )
( ) .n
C C
C = A
−
d
Γ
(16)
b) Using of certain frequency noise amplitude in SKMF is equivalent
to using of M
runs
runs of Monte Carlo simulation, with
runs 0
2
.
(1 ) n
M
C C A
Γ
=
−
(17)
In other words, M
runs
is a finite number of copies in the canonical en-
semble over which the averaging is done. Zero noise is equivalent to the
infinite number of copies in the canonical ensemble, and it is mean-
field.
Both equations (16) and (17) were discussed, proven analytically and
checked numerically for the case of ideal solution in [18].
Here, we consider in details the fluctuations in non-ideal solutions
with positive as well as with negative mixing energies V
mix. One should
distinguish two cases: (1) homophase fluctuations at ‘high’ tempera-
ture and (2) heterophase fluctuations (nucleation) at ‘low’ tempera-
ture. For example, under positive mixing energy, ‘high’ temperature
means
mix2
(1 )
ZV
T C C
k
> − , when the regular solution demonstrates
only homophase fluctuations without decomposition. In case 2 of ‘low’
temperature, the system (within certain composition interval) demon-
strates decomposition (V
mix
> 0) or ordering (for example, f.c.c. alloy
with V
mix
< 0, C = 1/4 or C = 1/2) or ordering with decomposition (for
example, f.c.c. alloy with V
mix
< 0, C = 1/8). Below in this paper, all ex-
MARTIN’S KINETIC MEAN-FIELD MODEL REVISITED 1423
amples will be calculated for the case of f.c.c. lattice. Case 2 will be
considered elsewhere.
5. FLUCTUATIONS OF CONCENTRATION IN F.C.C. SOLID
SOLUTION WITH NON-ZERO MIXING ENERGY
Here, we consider a homogeneous (except local fluctuations) binary
f.c.c. solid solution with equal average probability (concentration) of A
atom being found at any site: i jC C C< >=< >= . Local (at site ‘i’) con-
centration is fluctuating:
.i iC C C= + d
As before, we are, first of all, interested in concentration dispersion
(which is of course positive and the same for each site of globally ho-
mogeneous system:
2
0 .i ix C C C≡< d d >=< d > (18)
We also will discuss two spatial correlations:
1 ,i inx C C≡< d d > 2 .i innx C C≡< d d > (19)
Here, ‘in’ is any nearest neighbour site of the site ‘i’—(site at the first
co-ordination shell around ‘i’), ‘inn’ is the next nearest neighbour site
of the site ‘i’—(site at the second co-ordination shell around ‘i’). De-
spite initial neglect of correlation in the basic equations of SKMF, it is
physically evident that in case of positive mixing energy, when the al-
loy has a tendency to decomposition (which becomes successful at low
temperature), the neighbouring sites should demonstrate the tendency
to the same sign of fluctuation, so that the spatial correlation
1 i inx C C≡< d d > is expected to be not zero, but positive. On the contra-
ry, in case of negative mixing energy, when the alloy has a tendency to
ordering, 1 i inx C C≡< d d > is expected to be not zero, but negative. Im-
mediate reason of correlations is a dependence of jump frequencies on
the local concentration fluctuations:
,
, 0 0
0
( ( ) ( ))
( ( ) ( )) exp exp
( ) ( ) ( ) ( ) ( )
exp exp 1 .
s
i j A B
i j
s
A B A B A B
Q E E i E j
A i B j
kT kT
E E E E i E j E i E j
kT kT kT
− +
Γ = ν − = ν − =
− + d + d d + d = ν − ≈ Γ +
�
(20)
Thus, in case of non-ideal alloy, the variations of frequencies are
1424 Andriy GUSAK and Tetiana ZAPOROZHETS
caused not only directly by Langevin noise of frequencies, but as well
by the local variations of concentrations:
Lang conc
, , , ,i j i j i jdΓ = dΓ + dΓ
where
conc
,i jdΓ is a variation of frequency generated by local deviation of
composition influencing the activation energy:
conc
,
conc
,
( ) ( )
( ( ) ( )) ,
( ) ( )
( ( ) ( )) .
A B
i j
B A
i j
E i E j
A i B j
kT
E i E j
B i A j
kT
d + d
dΓ ≈ Γ
d + d
dΓ ≈ Γ
�
�
(21)
Thus,
Lang Lang
, ,
1 1
1
(1 )
( ) ( ) ( ) ( )
(1 ) .
Z Z
i
i j i j j i
j j
Z
A B B A
j
d C
Z C C C C
dt
E i E j E i E j
C C
kT kT
= =
=
d = − Γd + Γ d − − dΓ − dΓ −
d + d d + d − − Γ −
∑ ∑
∑
(22)
Variations of basic energies are:
mix
1
mix
1
( ) ( ) 2 ( ( )),
( ) ( ) 2 ( ( )).
Z
A B A
in
Z
B A A
jn
E i E i V C in
E j E j V C jn
=
=
d − d = − d
d − d = d
∑
∑
(23)
Substitution of Eq. (23) into Eq. (22) gives:
Lang Lang
, ,
1 1
( )
( ) ( ) (1 )
Z Z
A
A A i j j i
j j
d C i
Z C i C j C C
dt = =
d = − Γd + Γ d − − dΓ − dΓ − ∑ ∑
( ) ( )
mix
1 1 1
2
(1 ) ( ) ( ) .
Z Z Z
A A
j in jn
V
C C C in C jn
kT = = =
− − Γ − d + d
∑ ∑ ∑ (24)
Multiply this equation by dCi and make an averaging for steady state:
2
2
1
mix
1 1 1
Lang Lang
, ,
1
( )
0 ( ) ( ) ( )
2
2
(1 ) ( ) ( ) ( ) ( )
(1 ) .
Z
i
i A A
j
Z Z Z
A A A A
j in jn
Z
i i j i j i
j
d C d C
C Z C C i C j
dt dt
V
C C C i C in C i C jn
kT
C C C C
=
= = =
=
d < d >
< d >= = = − Γ < d > +Γ < d d > −
− − Γ − < d d > + < d d > −
− − < d dΓ > − < d dΓ >
∑
∑ ∑ ∑
∑
(25)
MARTIN’S KINETIC MEAN-FIELD MODEL REVISITED 1425
Here, ‘j’ correspond to nearest neighbours of i, ‘in’ are the neighbours
of i, ‘jn’—neighbours of j, 4 sites are simultaneously the neighbours
for i and j (see Fig. 2).
Here, in ‘first approximation’, we consider concentrations fluctua-
tions in sites as statistically correlated only for nearest neighbours
(neglecting, so far, the correlation in the second co-ordination shell):
2
0
1
,
( ( ))( ( )) ( ) ( ) , is a neighbour of
0, inall other cases.
A A A A
C x k m
C k C m C i C in x m k
< d >= =
< d d >= < d d >=
(Let us remind that, in ‘zeroth approximation’, the correlation be-
tween nearest neighbours is taken as just zero). Then, in our ‘first ap-
proximation’, 1
1
( ) ( )
Z
A A
in
C i C in Zx
=
< d d > =∑ . In the sum
1
( ) ( )
Z
A A
jn
C i C jn
=
< d d >∑
over the nearest neighbours of site j, which in turn is a neighbour of
site ‘i’, one term survives as x0, since i is one of the neighbours of j; also
four terms survive as x1 (one can geometrically check that each of
nearest neighbours of ‘i’ has other 4 nearest neighbours of ‘i’ in its
first co-ordination shell—see Fig. 2).
Therefore,
0 1
1
( ) ( ) 1 4 .
Z
A A
jn
C i C jn x x
=
< d d > = +∑ (26)
So,
1 1 1
1 0 1 0 1
( ) ( ) ( ) ( )
( 4 ) ( 4) .
Z Z Z
A A A A
j in jn
C i C in C i C jn
Z Zx x x Zx Z Z x
= = =
− < d d > + < d d > =
− + + = − −
∑ ∑ ∑
Thus, Eq. (25) is reduced to
Fig. 2. Neighbouring sites i and j have four common neighbours.
1426 Andriy GUSAK and Tetiana ZAPOROZHETS
mix mix
0 1
Lang Lang
, ,
1
2 2
1 (1 ) 1 (1 ) ( 4)
(1 ) .
Z
i j i i i j
j
V V
Z x C C Z x C C Z
kT kT
C C C C
=
Γ + − − Γ + − − =
= − < d dΓ > − < d dΓ > ∑
(27)
In full analogy with Appendix in [18], one can show that for any i,
Lang Lang 2
, ,
1
(1 ) .
Z
i j i i i j n
j
C C C C ZA
=
< d dΓ > − < d dΓ > = − ∑ (28)
Thus,
mix mix
2 2 2
0 1
2 16
1 (1 ) 1 (1 ) (1 ) .n
V V
x C C x C C C C A
kT kT
+ − − + − = − Γ
(29)
One can prove (using additional long algebra) that the account of
non-zero correlations in the second co-ordination shell converts Eq.
(29) into the following equation:
mix mix mix
0 1 2
2 2 2
(1 )2 (1 )16 (1 )4
1 1
(1 ) .n
C C V C C V C C V
x x x
kT kT kT
C C A
− − −
+ − + + =
= − Γ
(30)
We have now three unknowns— 2
0 ( ) ,x C i= < d > 1 ( ) ( 1) ,x C i C in=< d d >
and 2 ( ) ( 2)x C i C in= < d d > (‘in1’ and ‘in2’ correspond to the first and
second co-ordination shells, so that we need two more equations. For
this, we take once more the kinetic equation (16) and multiply it in oth-
er way—first, by the fluctuation of concentration in one of the sites
‘in1’, neighbouring to ‘i’, and, second, by the fluctuation in the ‘in2’
in the second co-ordination shell. At that, due to equivalence of all
sites in homogeneous alloy,
1
1
1 1
1
1
0
2 2
i in
in i
i in in i
in i
d C d C
C C
dt dt
d C d C d C C
C C
dt dt dt
d d
< d > = < d > =
d d < d d > = < d > + < d > = > =
(31)
2
2
2 2
2
1
0.
2 2
i in
in i
i in in i
in i
d C d C
C C
dt dt
d C d C d C C
C C
dt dt dt
d d
< d > = < d > =
d d < d d > = < d > + < d > = > =
(32)
Substituting Eq. (24) for id C dtd into Eqs. (31), (32), and making ra-
MARTIN’S KINETIC MEAN-FIELD MODEL REVISITED 1427
ther long and tiresome algebra, one gets, together with Eq. (29) with
notations
mix
(1 ) ,
V
C C
kT
υ ≡ −
2
2 2(1 ) nA
I C C≡ −
Γ
, (33)
the set of three algebraic equations for three unknowns x0, x1, x2:
0 1 2
0 1 2
0 1 2
(1 2 ) ( 1 16 ) 4 ,
(1 16 ) ( 8 6 ) (2 24 ) ,
2 ( 1 12 ) (3 8 ) 0.
x x x I
x x x I
x x x
+ υ + − − υ + υ =
+ υ + − + υ + + υ =
υ + − + υ + + υ =
(34)
Solution is following:
2
analyt analyt
0 02 3
19 54 416
( ) ,
19 54 1204 1136
x I k I
− υ − υ
= = υ
− υ − υ − υ
(35a)
analyt analyt
1 12 3
2(19 36 )
( ) ,
19 54 1204 1136
x I k I
+ υ
= υ = υ
− υ − υ − υ
(35b)
analyt 2 analyt
2 22 3
212
( ) .
19 54 1204 1136
x I k I= υ = υ
− υ − υ − υ
(35c)
Thus, our analytic approximation of SKMF method predicts propor-
tionality to I (actually, to the squared noise amplitude multiplied by
2 2(1 )C C− ). To check this prediction, we found x0, x1, x2 by direct nu-
meric simulation of fluctuations according to numeric solution of the
Eqs. (13), (14). Results of simulation are shown in Fig. 3. Indeed, all of
the above-mentioned three characteristics are proportional to the re-
duced noise parameter
2 2 2(1 ) nI C C A≡ − Γ as predicted:
numeric numeric
0 0 ( ) ,x k I= υ
numeric numeric
1 1 ( ) ,x k I= υ
numeric numeric
2 2 ( ) .x k I= υ
So far, we calculated dispersion per one site. This is not convenient
for comparison with Monte Carlo and phenomenological thermody-
namics. Therefore, we also calculated the dispersion for the cluster
containing n sites.
In Figure 4, we compare theoretical predictions and numeric results
for the dependence of factors k0, k1, k2 on the renormalized mixing en-
ergy
mix(1 )C C V kTυ ≡ − for the case when dispersion and correlations
are calculated for ‘cluster’ containing one site. In Figure 4, we can see
that the difference between analytic approximation and numeric simu-
lation is not more than few percent; it is almost ideal for zero and nega-
tive mixing energies and increases for large positive mixing energies.
To compare with Monte Carlo, we should consider concentrations for
cluster containing at least n = 1 + 12 = 13 sites. Therefore, we recalcu-
lated the results for dispersion for the case of larger clusters. Composi-
tion fluctuation in clusters containing central atom and some part of
the neighbourhood, can be found for various cluster definitions, but
1428 Andriy GUSAK and Tetiana ZAPOROZHETS
we choose a cluster containing 13 equivalent sites with the same
‘weight’ in calculating the average:
2
12
analyt 0 1 2
0
1
13 48 61
( 13) ( ) ( ) .
13 169in
x x x
x n C i C in
=
+ +
= = d + d =
∑ (36)
In Figure 5, we compare analytic approximation and numeric results
of SKMF modelling for dispersion x0(n = 13) calculated for concentra-
tions averaged over cluster of 1 + 12 sites as a function of reduced noise
I, and for corresponding coefficient k0(n = 13) = x0(n = 13)/I as a func-
tion of renormalized mixing energy v.
Now, let us compare SKMF simulation results with Monte Carlo
simulation. Standard Metropolis algorithm for exchange mechanism
was applied.
Concentration for every site at each step was calculated as an aver-
age over cluster containing 13 = 1 + 12 sites (central atom plus first co-
ordination shell of the f.c.c. lattice) and 19 = 1 + 12 + 6 (central site
Fig. 3. Dependences of dispersion x0 (a) and of correlations in the first and sec-
ond co-ordination shells x1 (b), x2 (c) on
2 2 2(1 ) nI C C A≡ − Γ for positive 0.04,
negative −0.04 and zero mixing energy
mix(1 ) .C C V kTυ ≡ − So far, all charac-
teristics were calculated for single site (n = 1) by numeric SKMF modelling.
MARTIN’S KINETIC MEAN-FIELD MODEL REVISITED 1429
plus two co-ordination shells). Monte Carlo results for these two choic-
es of averaging clusters are shown in Fig. 6.
To compare the kinetic Monte Carlo (KMC) simulation results with
SKMF, it could present some problem, since SKMF model contains
noise amplitude An. Luckily, as mentioned above, at least for the case
of ideal solution we managed to solve this problem [18]. Namely, we
relate the noise amplitude to the number of runs of Monte Carlo simu-
lation, in case when the concentration at each site at each moment is
calculated as an average over the M
runs
copies of the system. At that,
dispersion of concentration (for an ideal solution) is equal to
2
2 2(1 ) nA
I C C≡ −
Γ in SKMF approach and to runs
(1 )C C
M
−
in Monte Carlo
approach with averaging over M
runs
runs. To make these two expres-
sions for dispersion in two approaches coinciding, we got interrelation
(17)— runs 0
2
.
(1 ) n
M
C C A
Γ
=
−
Now, we will try to derive the analogue of interrelation for the case
Fig. 4. Dependences of factors k0 (a), k1 (b), k2 (c) on
mix(1 )C C V kTυ ≡ − for
analytic SKMF (solid line) approximation and for numeric SKMF (dots) mod-
elling. All calculations made for ‘cluster’ containing one site (n = 1).
1430 Andriy GUSAK and Tetiana ZAPOROZHETS
of regular solid solution.
As for Monte Carlo approach, we made the calculations with averag-
ing over M
runs
runs (simultaneous copies within canonical ensemble).
We specially checked that all above-mentioned characteristics are in-
versely proportional to Mruns
(see Fig. 7). For example, at
mix(1 ) 0.02C C V kTυ ≡ − = ,
KMC runs
0 runs
0.0251
( 13, 0.02, ) .x n M
M
= υ = � (37)
As for analytic approximation of the SKMF approach, we passed to
Fig. 5. Comparison between analytic approximation (solid lines) and numeri-
cal results of SKMF modelling (dots) for dispersion x0(n = 13) calculated for
concentrations averaged over cluster of 1 + 12 sites as a function of reduced
noise
2 2 2(1 ) nI C C A≡ − Γ (a) and for corresponding coefficient
0 0( 13) ( 13)k n x n I= = = as a function of renormalized mixing energy
mix(1 )C C V kTυ ≡ − (b).
Fig. 6. Dependences of dispersion
KMC
0x and of correlations in the first and
second co-ordination shells
KMC
1 ,x
KMC
2x versus
mix(1 )C C V kTυ ≡ − calculated
for concentrations averaged over clusters containing n = 13 (filled dots) and
n = 19 (unfilled dots) by numeric SKMF modelling.
MARTIN’S KINETIC MEAN-FIELD MODEL REVISITED 1431
the clusters containing many sites, n >> 1. Then, analytic approxima-
tion of the short-range order in SKMF model predicts that
2
0 1 1 2 2
0 1 1 2 22
1
1 1
( ) ( ),
K
i
nx nZ x nZ x
C i x Z x Z x
n nn=
+ +
< d > ≈ = + +
∑ (38)
where co-ordination numbers for the first and second shells of f.c.c.
lattice are Z1 = 12, Z2 = 6.
Taking into account the analytical solution (35), one gets
2
0 1 2
1
1 1
( ) ( 12 6 )
n
i
C i x x x
n n=
< d > ≈ + + =
∑
2 2
2 3
1 (19 54 416 ) 12 (38 72 ) 1272
.
19 54 1204 1136
I
n
− υ − υ + υ + υ + υ
=
− υ − υ − υ
Further, here, we will limit ourselves by the linear approximation in
terms of renormalized mixing energy. It gives
2 22 2 mix
1
1 (1 24 ) (1 )
( ) 1 24 (1 ) .
n
n
i
AC C V
C i I C C
n n n kT=
+ υ −
< d > ≈ = + − Γ
∑ (39)
On the other hand, according to thermodynamic theory of fluctua-
tion, in regular solutions, the composition fluctuation in the cluster
containing n sites, is equal to
2
mix2
mix
2
1 1 1 (1 )
.
22 1 (1 )
(1 )
n
kT kT C C
C
kTn n n ZVg ZV C C
C C kTC
−
< d > = = =
∂ − − − −∂
Fig. 7. Dispersion
KMC
0 ( 13)x n = versus M
runs
(a) and logarithm x
KMC(n, M
runs)
for n = 13 (filled dots) and n = 19 (unfilled dots) versus lnMruns
(b) at
mix(1 ) 0.02.C C V kTυ ≡ − =
1432 Andriy GUSAK and Tetiana ZAPOROZHETS
If one averages the results over M
runs
copies of the canonical ensem-
ble, then the dispersion will be M
runs
times less:
runs
2
runs mix,
1 1 (1 )
.
1 2 (1 ) ( )n M
C C
C
M n ZV C C kT
−
< d > =
− −
(40)
In the same approximation of small values of
mix(1 ) ( ),C C V kTυ = −
Eq. (40) transforms into
runs
mix
2
runs,
1 1
(1 ) 1 24 (1 ) .
n M
V
C C C C C
n kTM
< d > ≈ − + −
(41)
Equalizing Eqs. (41) and (39) for regular solution within the cluster of
N sites gives:
2 mix
2 2
mix
runs
1
(1 ) 1 24 (1 )
1 1
(1 ) 1 24 (1 ) ;
nA V
C C C C
n kT
V
C C C C
M n kT
− + − = Γ
= − + −
(42)
so,
runs 0
2(1 ) n
M
C C A
Γ
=
− that totally coincides with Eq. (8) for ideal so-
Fig. 8. Comparison of Monte Carlo dispersion under averaging over fixed
number of MC runs for the clusters of size n = 13 (diamond)
KMC
0 ( 13)x n = with
analytic (solid lines)
analyt runs
0(1 ) ( 13)C C k n M− = and numeric (square)
numeric runs
0(1 ) ( 13)C C k n M− = SKMF results as a function of renormalized
mixing energy
mix(1 ) .C C V kTυ ≡ − Data are given for M
runs
= 1, but practi-
cally the same is valid for any M
runs.
MARTIN’S KINETIC MEAN-FIELD MODEL REVISITED 1433
lution and M
runs
runs of Monte Carlo.
Thus, for the regular solution, at least for linear approximation
over renormalized mixing energy, our interpretation of noise ampli-
tude remains the same as for ideal solution—inverse squared noise am-
plitude is equivalent to averaging over finite number M
runs
(Eq. (8)) of
the copies of canonical ensemble. In other words, we predict that in
regular solution
runs
SKMF
2 KMC 0
runs,
( ) (1 )
.
n M
k n C C
C
M
−
< d > ≈ (43)
To check a validity of this analytic prediction, we compare
KMC runs
0 ( 13, )x n M= and
SKMF runs
0(1 ) ( 13) .C C k n M− = Numeric experi-
ment confirms the above-described analytic theory, at least, for small
positive and for any reasonable negative mixing energy (see Fig. 8).
6. CONCLUSIONS
1. G. Martin’s approach to kinetic mean-field and its generalization to
3D contain mean-field thermodynamics of Khachaturyan [7] as a
steady-state limit.
2. Martin’s kinetic is nonlinear in respect to fluxes, and therefore
should be more appropriate for simulation of atomic migration and
phase transformations in the sharp concentration gradients.
3. 3D-generalization of Martin’s KMF equations is proved (for the
first time) to provide only negative or zero time derivative. Therefore,
KMF cannot provide the first-order transformations with overcoming
the nucleation barrier.
4. We introduce frequency noise instead of concentration noise as a
basic reason of stochastic behaviour leading to concentration fluctua-
tions and to overcoming the nucleation barriers.
5. Introduction of frequency noise leads to characteristics of fluctua-
tions coinciding with Monte Carlo approach under the following inter-
relation between noise amplitude and the number of ensemble copies
(MC runs) over which the averaging is done:
runs 0
2
.
(1 ) n
M
C C A
Γ
=
−
6. Dependences of dispersion and correlations on the mixing energies
are reasonably well described by analytic approximation (35), especial-
ly for negative and for small positive mixing energies.
7. SKMF is much faster than MC and gives analogous results for fluctu-
ations. Therefore, it looks reasonable to use SKMF for modelling of nu-
cleation behaviour, keeping in mind the interrelation (8). It might be a
good tool to check and distinguish the alternative theories of nucleation
[22–24], especially in the sharp concentration gradients [25, 26].
1434 Andriy GUSAK and Tetiana ZAPOROZHETS
ACKNOWLEDGEMENTS
This work was supported by both the Marie Curie International Re-
search Staff Exchange Scheme Fellowship within the 7th
European
Community Framework Programme under Grant 612552 and the Min-
istry of Education and Science of Ukraine.
Authors are grateful to George Martin, Zoltan Erdélyi, Rafal Ko-
zubski, and Helen Zapolsky for fruitful discussions.
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
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(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
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/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
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/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
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/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
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/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|