Crossover scaling in the two-dimensional three-state Potts model
We apply simulated tempering and magnetizing (STM) Monte Carlo simulations to the two-dimensional three-state Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the temperature-field plane at the Potts critical point and towards the Ising universalit...
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irk-123456789-1208032017-06-14T03:03:20Z Crossover scaling in the two-dimensional three-state Potts model Nagai, T. Okamoto, Y. Janke, W. We apply simulated tempering and magnetizing (STM) Monte Carlo simulations to the two-dimensional three-state Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the temperature-field plane at the Potts critical point and towards the Ising universality class for negative magnetic fields. Our data set has been generated by STM simulations of several square lattices with sizes up to 160x160 spins, supplemented by conventional canonical simulations of larger lattices at selected simulation points. We present careful scaling and finite-size scaling analyses of the crossover behaviour with respect to temperature, magnetic field and lattice size. Ми застосовуємо Монте Карло симуляцiї з симульованим темперуванням i намагнiченням (STM) до двовимiрної тристанової моделi Поттса у зовнiшньому магнiтному полi для того, щоб дослiдити кросоверну скейлiнгову поведiнку у площинi температура-поле при критичнiй точцi Поттса, а також клас унiверсальностi моделi Iзинга для негативних магнiтних полiв. Набiр наших даних був згенерований STM симуляцiями декiлькох квадратних ґраток розмiром до 160×160 спiнiв, доповненими звичайними канонiчними симуляцiями бiльших ґраток при вибраних симуляцiйних точках. Ми представляємо ретельний аналiз скейлiнгу i скiнченомiрного скейлiнгу кросоверної поведiнки по вiдношенню до температури, магнiтного поля i розмiру ґратки. 2013 Article Crossover scaling in the two-dimensional three-state Potts model / T. Nagai, Y. Okamoto, W. Janke // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23605:1-8. — Бібліогр.: 38 назв. — англ. 1607-324X PACS: 64.60.De, 75.30.Kz, 75.10.Hk, 05.10.Ln DOI:10.5488/CMP.16.23605 arXiv:1307.3865 http://dspace.nbuv.gov.ua/handle/123456789/120803 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We apply simulated tempering and magnetizing (STM) Monte Carlo simulations to the two-dimensional three-state Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the temperature-field plane at the Potts critical point and towards the Ising universality class for negative magnetic fields. Our data set has been generated by STM simulations of several square lattices with sizes up to 160x160 spins, supplemented by conventional canonical simulations of larger lattices at selected simulation points. We present careful scaling and finite-size scaling analyses of the crossover behaviour with respect to temperature, magnetic field and lattice size. |
| format |
Article |
| author |
Nagai, T. Okamoto, Y. Janke, W. |
| spellingShingle |
Nagai, T. Okamoto, Y. Janke, W. Crossover scaling in the two-dimensional three-state Potts model Condensed Matter Physics |
| author_facet |
Nagai, T. Okamoto, Y. Janke, W. |
| author_sort |
Nagai, T. |
| title |
Crossover scaling in the two-dimensional three-state Potts model |
| title_short |
Crossover scaling in the two-dimensional three-state Potts model |
| title_full |
Crossover scaling in the two-dimensional three-state Potts model |
| title_fullStr |
Crossover scaling in the two-dimensional three-state Potts model |
| title_full_unstemmed |
Crossover scaling in the two-dimensional three-state Potts model |
| title_sort |
crossover scaling in the two-dimensional three-state potts model |
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Інститут фізики конденсованих систем НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120803 |
| citation_txt |
Crossover scaling in the two-dimensional three-state Potts model / T. Nagai, Y. Okamoto, W. Janke // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23605:1-8. — Бібліогр.: 38 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT nagait crossoverscalinginthetwodimensionalthreestatepottsmodel AT okamotoy crossoverscalinginthetwodimensionalthreestatepottsmodel AT jankew crossoverscalinginthetwodimensionalthreestatepottsmodel |
| first_indexed |
2025-07-08T18:36:09Z |
| last_indexed |
2025-07-08T18:36:09Z |
| _version_ |
1837104911670575104 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 2, 23605: 1–8
DOI: 10.5488/CMP.16.23605
http://www.icmp.lviv.ua/journal
Crossover scaling in the two-dimensional three-state
Potts model
T. Nagai1, Y. Okamoto1,2,3,4, W. Janke5,6
1 Department of Physics, Graduate School of Science, Nagoya University, Nagoya, Aichi 464–8602, Japan
2 Structural Biology Research Center, Graduate School of Science, Nagoya University, Nagoya,
Aichi 464–8602, Japan
3 Center for Computational Science, Graduate School of Engineering, Nagoya University, Nagoya, Aichi
464–8603, Japan
4 Information Technology Center, Nagoya University, Nagoya,
Aichi 464–8601, Japan
5 Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, 04009 Leipzig, Germany
6 Centre for Theoretical Sciences (NTZ), Universität Leipzig, Postfach 100 920, 04009 Leipzig, Germany
Received February 26, 2013, in final form April 19, 2013
We apply simulated tempering and magnetizing (STM) Monte Carlo simulations to the two-dimensional three-
state Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the
temperature-field plane at the Potts critical point and towards the Ising universality class for negative magnetic
fields. Our data set has been generated by STM simulations of several square lattices with sizes up to 160×160
spins, supplemented by conventional canonical simulations of larger lattices at selected simulation points. We
present careful scaling and finite-size scaling analyses of the crossover behaviour with respect to temperature,
magnetic field and lattice size.
Key words: three-state Potts model, phase transitions, critical phenomena, crossover scaling, Monte Carlo
(MC) simulations, simulated tempering and magnetizing (STM)
PACS: 64.60.De,75.30.Kz,75.10.Hk,05.10.Ln
1. Introduction
The two-dimensional three-state Potts model in an external magnetic field [1, 2] has several inter-
esting applications in condensed matter physics [2], and its three-dimensional counterpart serves as
an effective model for quantum chromodynamics [3–6]. When one of the three states per spin is dis-
favoured in an external (negative) magnetic field, the other two states exhibit Z2 symmetry and one ex-
pects a crossover from Potts to Ising critical behaviour. In the vicinity of the Potts critical point, another
crossover effect takes place when approaching the critical point along different paths in the temperature-
field plane.
To cover such a two-dimensional parameter space, generalized-ensemble Monte Carlo simulations are
a useful tool [7–10]. Well-known examples are the multicanonical (MUCA) algorithm [11, 12], the closely
related Wang-Landau method [13, 14], the replica-exchange method (REM) [15, 16] (see also [17, 18]), also
often referred to as parallel tempering, and simulated tempering (ST) [19, 20]. Inspired by recent multi-
dimensional generalizations of generalized-ensemble algorithms [21–23], the “Simulated Tempering and
Magnetizing” (STM) method has been proposed by two of us and tested for the classical Ising model
in an external magnetic field [24, 25]. Recently, we have extended this new simulation method to the
two-dimensional three-state Potts model and by this means generated accurate numerical data in the
temperature-field plane [26]. Herewe focus on a discussion of the two abovementioned crossover-scaling
© T. Nagai, Y. Okamoto, W. Janke , 2013 23605-1
http://dx.doi.org/10.5488/CMP.16.23605
http://www.icmp.lviv.ua/journal
T. Nagai, Y. Okamoto, W. Janke
scenarios that include (for the Potts-to-Ising crossover, in particular) the analysis of the specific heat
which provides the clearest signals.
The rest of this article is organized as follows. In section 2 we briefly discuss the model and review the
STMmethod. In section 3 we present the results of our crossover-scaling analyses at the phase transitions
with respect to temperature, magnetic field and lattice size. Finally, section 4 contains our conclusions
and an outlook to the future work.
2. Model and simulation method
The two-dimensional three-state Potts model in an external magnetic field is defined through the
Hamiltonian
H = E −hM , (2.1)
E =−
∑
〈i , j〉
δσi ,σ j
, (2.2)
M =
N
∑
i=1
δ0,σi
, (2.3)
where N = L2 denotes the total number of spins σi ∈ 0,1,2 arranged on the sites of a square L ×L lattice
with periodic boundary conditions, δ is the Kronecker delta function and h is the external magnetic
field. The sum in (2.2) runs over all nearest-neighbour pairs. Note that the magnetization M defined in
(2.3) takes on the value M = N for the ordered state in 0-direction, M = 0 for the ordered states in 1- or
2-direction, and M = N /3 in the disordered phase.
By mapping the integer valued spins σi to spin vectors ~si = [cos(2πσi /3),sin(2πσi /3)] one readily
sees that E = (2/3)(−
∑
〈i , j 〉~si~s j − N ) and M = (2/3)(M (x) + N /2), where M (x) is the component of the
magnetization vector ~M =
∑
i~si in field direction (assumed to be along the x-axis). In this equivalent
notation, it is fairly obvious that the Z3 symmetry for h = 0 is broken to Z2 for negative external magnetic
fields (see figure 1).
Figure 1. Schematic sketch illustrating the behaviour of the spins of the three-state Potts model in an ex-
ternal magnetic field h. For h > 0, the spin state 0 is favoured, whereas the states 1 and 2 are disfavoured.
For h = 0, all three states are equivalent. For h < 0, the spin state 0 is disfavoured and the states 1 and 2
are related to each other by Z2 symmetry.
Another frequently employed definition of the magnetization is the so-called “maximum definition”
Mmax = N mmax ≡
3
2
{
max
j=0,1,2
[
N
∑
i=1
δ j ,σi
]
−
N
3
}
, (2.4)
which yields the physically more intuitive value of 1 when the system is in one of the three ordered
phases and 0 is in the disordered phase, respectively.
Let us now turn to a brief description of the employed Monte Carlo simulation method. In the con-
ventional ST scheme [19, 20], the temperature is considered as an additional dynamical variable besides
the spin degrees of freedom. The STM method is a generalization to a two-dimensional parameter space
23605-2
Crossover scaling in the two-dimensional three-state Potts model
where the magnetic field is treated as the second additional dynamical variable similar to the tempera-
ture [24–26]. Here, one considers
e−(E−hM)/T+a(T,h) (2.5)
as a joint probability for (x,T,h) (∈ X ⊗ {T1,T2, · · · ,TNT }⊗ {h1,h2, · · · ,hNh
}), where a(T,h) is a parameter,
x denotes a (microscopic) state, and X is the sampling space. We have set Boltzmann’s constant to unity.
Note that the temperature and external field are discretized into NT and Nh values, respectively.
A suitable candidate for a(Ti ,h j ) can be obtained from the (empirical) probability of occupying each
set of parameter values,
P (Ti ,h j ) = e− f (Tk ,hl )+a(Tk ,hl ) , (2.6)
where e− f (Tk ,hl ) =
∫
dx e−(E−h j M)/Ti . This shows that the dimensionless free energy f (Ti ,h j ) is the proper
choice for a(Ti ,h j ) in order to generate a uniform distribution of the number of samples according to T
and h. This implies a random-walk-like evolution of T and h in STM simulations as it is demonstrated
in figure 2 for a 80×80 lattice. The block structures reflect the first-order phase transition line at h = 0
in the Potts model and the second-order phase transition at the effective Ising transition temperature
Tc ≈ 1.1346 for negative magnetic field.
0.5
1
1.5
2
0 0.25 0.5 0.75 1
T
x10
6
MC sweeps
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.25 0.5 0.75 1
h
x10
6
MC sweeps
Figure 2. Time history of temperature T and magnetic field h in STM simulations for the linear lattice
size L = 80.
3. Results
Our STM simulations were performed for lattice sizes L = 5,10,20,40,80, and 160 with the total num-
ber of sweeps varying between about 160×106 and 500×106 , where a sweep consisting of N single-spin
updates with the heat-bath algorithm followed by an update of either the temperature T or the field h. We
used the Mersenne Twister [27] as quasi-random-number generator. Statistical error bars are estimated
using the jackknife blocking method [28–31].
Due to its random-walk-like nature, the STM method, combined with reweighting techniques such
as WHAM [32–34] or MBAR [35], yields the density of states n(E , M) (up to an overall constant) in a
wide range of the two-dimensional parameter space. Using these data it is straightforward to compute a
two-dimensional map of any thermodynamic quantity that can be expressed in terms of E and M . As an
example, figure 3 shows the specific heatC = (〈E 2〉−〈E〉2)/T 2 and susceptibility χ= (〈M2
max〉−〈Mmax〉2)/T
per spin as functions of T and h for L = 80. We see a line of phase transitions starting at the Potts critical
point at h = 0, T Potts
c = 1/ln(1+
p
3) = 0.9950 which, for strong negative magnetic fields, approaches the
Ising model limit with the critical point at h → −∞, T
Ising
c = 1/ln(1+
p
2) = 1.1346. For all h < 0, the
Z3 symmetry of the 3-state Potts model in zero field is broken to Z2 symmetry (recall figure 1) and by
universality the critical behaviour is expected to be Ising-like.
23605-3
T. Nagai, Y. Okamoto, W. Janke
Figure 3. (Color online) (a) Specific heat per site C/L2 and (b) magnetic susceptibility per site χ/L2 as
functions of T and h for L = 80. The solid vertical line corresponds to T = 1.1346, which is the critical
temperature of the Ising model (in 2-state Potts model normalization).
For positive magnetic fields, the phase transition disappears altogether. However, for finite lattices
and small h > 0, the singular behaviour persists to some extent due to finite-size effects. More precisely,
the peaks of, e.g., the specific heat shown in figure 4, growwith an increasing lattice size L until L is larger
than the (finite) correlation length of the system. This can be interpreted as a crossover in the dependence
of field h and lattice size L.
Figure 4. (Color online) Specific heat per site C/L2 as a function of T . With increasing system size L, the
peaks become more pronounced [L = 5 (dashed red), 10 (dotted green), 20 (solid black), 40 (dash dotted
purple), and 80 (solid blue)]. (a) h = 0.0, (b) h = 0.005, (c) h = 0.01, (d) h = 0.02.
To study, in the vicinity of the Potts critical point, the crossover-scaling behaviour in the T −h plane,
we calculated the magnetization m = M/L2 by reweighting. Its scaling form is given by [36]
m(T,h,L) = L−β/ν
Ψ(tLyt ,hLyh ) , (3.1)
where yt = 1/ν and yh = (β+γ)/ν are the usual scaling dimensions which can be expressed in terms of
standard critical exponents. For easier reference, we have collected the exactly known critical exponents
of the two-dimensional Ising and Potts models in table 1. The actually observed exponent depends on
the precise path in which the critical point is approached in the T −h plane. According to the crossover
23605-4
Crossover scaling in the two-dimensional three-state Potts model
Table 1. Critical exponents for the two-dimensional Ising and three-state Potts models [yt = 1/ν, yh =
(β+γ)/ν] [2].
Model yt yh α β γ δ ν
Ising 1 15/8 0 (log) 1/8 7/4 15 1
Potts 6/5 28/15 1/3 1/9 13/9 14 5/6
scaling formalism [36] in the limit of an infinite lattice, if t−yh /yt h (in the Potts model t−14/9h) is small
enough, then the magnetization obeys m ∼ tβ (= t 1/9), and otherwise it scales as m ∼ h1/δ (= h1/14),
where t = (Tc −T )/Tc. Figure 5 (a) shows that as long as finite-size effects are negligible (L6/5t ≫ 0.1)
and t ≫ (h/6)9/14 (i.e., t−14/9h is small), then the critical behaviour is m ∼ t 1/9. Figure 5 (b) shows that if
finite-size effects are negligible (L28/15h ≫ 0.1) and t ≪ (h/6)9/14 (i.e., t−14/9h is large), then the critical
behaviour is m ∼ h1/14. Thus, figure 5 clearly shows that the line h = 6t 14/9 gives the boundary of the two
scaling regimes.
Figure 5. (Color online) The difference between magnetization and its expected scaling behaviours
around the critical point for L = 80. Shown are (a) |mL2/15 −1.2(L6/5t)1/9| where the amplitude 1.2 was
obtained by fitting the magnetization data to t1/9 and (b) |mL2/15−(L28/15h)1/14|. In both plots, the solid
line corresponds to h = 6t14/9.
Since the three-state Potts model in a negative magnetic field is expected to behave like the Ising
model, we also investigated the crossover behaviour between these two models using finite-size scaling
techniques. For the susceptibility maximum χmax ∝ Lγ/ν, the finite-size scaling exponent of the Potts
and Ising model is given by γ/ν = 26/15 = 1.7333. . . and 7/4 = 1.75, respectively. Figure 6 shows that
the exponents are so similar that we can hardly distinguish the difference, despite the accuracy of the
measurements. The difference is much more pronounced for the maxima of the specific heat which are
expected to scale with the system size L with an exponent α/ν = 2/5 for the Potts and α/ν = 0, i.e.,
logarithmically, for the Ising model. We also measured different quantities, which are the maximum
values of
dln〈mmax〉
dβ ,
dln〈m2
max〉
dβ ,
dln〈U2〉
dβ ,
dln〈U4〉
dβ , and
d〈mmax〉
dβ . Here, U2 = 1− 〈m2
max〉
3〈mmax〉2 and U4 = 1− 〈m4
max〉
3〈m2
max〉2
are the Binder cumulants [37]. The derivatives were obtained by using [38]
dln〈mk
max〉
dβ
= 〈E〉−
〈mk
maxE〉
〈mk
max〉
, (3.2)
dln〈U2k〉
dβ
=
〈m2k
max〉
3〈mk
max〉
2
{
〈E〉−2
〈mk
maxE〉
〈mk
max〉
+
〈m2k
maxE〉
〈m2k
max〉
}
, (3.3)
d〈mmax〉
dβ
= 〈mmax〉〈E〉−〈mmaxE〉 . (3.4)
Figure 6 shows our results. Note that
dln〈mmax〉
dβ |max,
dln〈m2
max〉
dβ |max,
dln〈U2〉
dβ |max,
dln〈U4〉
dβ |max, and
23605-5
T. Nagai, Y. Okamoto, W. Janke
d〈mmax〉
dβ |max are expected to behave asymptotically as L1/ν, L1/ν, L1/ν, L1/ν, and L(1−β)/ν , respectively, as
the lattice size L increases [31]. These critical exponents are presented for the Potts model by 1/ν = 6/5
and (1−β)/ν = 16/15, and for the Ising model by 1/ν = 1 and (1−β)/ν = 7/8 (see table 1). We observe
that all quantities for h = 0 (red curve with filled circles) follow the Potts case and that those for negative
external field (green curve with filled up triangles and blue curve with filled down triangles) follow the
Ising case in the limit of large L. In fact, the two curves for h = −0.5 and h = −1.0 converge into almost
the same line as L increases. On the other hand, the (green) curve for h =−0.5 exhibits greater deviation
from the scaling behaviour for small L. This can also be understood as another crossover effect governed
by h and L.
Figure 6. (Color online) Finite-size scaling behaviour of χmax, Cmax,
dln〈U4〉
dβ
|max, and
d〈mmax〉
dβ
|max for
three characteristic h values.
4. Conclusions
In this work, we reported the scaling and finite-size scaling analyses of the two-dimensional three-
state Potts model in a magnetic field based on the data generated using the Simulated Tempering and
Magnetizing (STM) method [24, 25]. In such simulations, the random walk in temperature and magnetic
field covers a wide range of these parameters so that STM simulations enable one to study crossover
phenomena with a single simulation run [26].
By this means we calculated the magnetization, susceptibility, energy, specific heat and related quan-
tities as functions of temperature, magnetic field, and lattice size around the critical point using reweight-
ing techniques. These data allowed us to extract the crossover behaviours of phase transitions. First, at
the Potts critical point for h = 0, we observed a clear crossover of the scaling behaviours of the magneti-
zation with respect to temperature and magnetic field. Second, from an analysis of the specific heat and
23605-6
Crossover scaling in the two-dimensional three-state Potts model
other quantities, a crossover in the scaling laws with respect to (negative) magnetic field and lattice size
was identified, thereby verifying the expected crossover from 3-state Potts to Ising critical behaviour.
The data of the present work yield the two-dimensional density of states n(E , M) (up to an overall
constant) which determines the weight factor for two-dimensional multicanonical simulations. We can
also perform two-dimensional multicanonical simulations, which will be an interesting future task.
As a final remark we should like to stress that the present method is useful not only for spin systems
as considered here but also for other complex systems with many degrees of freedom. Since our method
does not require any change of the frequently rather intricate energy calculations, it should be highly
compatible with the available program packages.
Acknowledgements
We thank the Information Technology Center, Nagoya University, the Research Center for Computa-
tional Science, Institute for Molecular Science, and the Supercomputer Center, Institute for Solid State
Physics, University of Tokyo, for computing time on their supercomputers. This work was supported, in
part, by JSPS Institutional Program for Young Researcher Overseas Visit (to T.N.) and by Grants-in-Aid
for Scientific Research on Innovative Areas (“Fluctuations and Biological Functions") and for the Com-
putational Materials Science Initiative from the Ministry of Education, Culture, Sports, Science and Tech-
nology, Japan (MEXT). W.J. gratefully acknowledges support by DFG Sonderforschungsbereich SFB/TRR
102 (Project B04) and the Deutsch-Französische Hochschule (DFH-UFA) under Grant No. CDFA–02–07. T.N.
also thanks the Nagoya University Program for “Leading Graduate Schools: Integrative Graduate Educa-
tion and Research Program in Green Natural Sciences” for support of his extended stay in Leipzig.
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Скейлiнґ кросоверу у двовимiрнiй тристановiй моделi
Поттса
Т. Нагаi1, Ю. Окамото1,2,3,4, В. Янке5,6
1 Вiддiл фiзики, унiверситет м. Нагоя, Нагоя, Айчi 464–8602, Японiя
2 Центр дослiджень структурної бiологiї, унiверситет м. Нагоя, Нагоя, Айчi 464–8602, Японiя
3 Центр комп’ютерних наук, унiверситет м. Нагоя, Нагоя, Айчi 464–8602, Японiя
4 Центр iнформацiйних технологiй, унiверситет м. Нагоя, Нагоя, Айчi 464–8602, Японiя
5 Iнститут теоретичної фiзики, унiверситет Ляйпцiгу, 04009 м. Ляйпцiг, Нiмеччина
6 Центр теоретичних природничих наук (NTZ), унiверситет Ляйпцiгу, 04009 Ляйпцiг, Нiмеччина
Ми застосовуємо Монте Карло симуляцiї з симульованим темперуванням i намагнiченням (STM) до дво-
вимiрної тристанової моделi Поттса у зовнiшньому магнiтному полi для того, щоб дослiдити кросоверну
скейлiнгову поведiнку у площинi температура-поле при критичнiй точцi Поттса, а також клас унiверсаль-
ностi моделi Iзинга для негативних магнiтних полiв. Набiр наших даних був згенерований STM симуляцi-
ями декiлькох квадратних ґраток розмiром до 160×160 спiнiв, доповненими звичайними канонiчними
симуляцiями бiльших ґраток при вибраних симуляцiйних точках. Ми представляємо ретельний аналiз
скейлiнгу i скiнченомiрного скейлiнгу кросоверної поведiнки по вiдношенню до температури, магнiтного
поля i розмiру ґратки.
Ключовi слова: тристанова модель Поттса, фазовi переходи, критичнi явища, скейлiнг кросоверу,
симуляцiї Монте Карло, симульоване темперування i намагнiчення (STM)
23605-8
http://dx.doi.org/10.1103/PhysRevE.86.056705
http://dx.doi.org/10.1016/j.phpro.2012.05.016
http://dx.doi.org/10.1088/1742-5468/2013/02/P02039
http://dx.doi.org/10.1145/272991.272995
http://dx.doi.org/10.1093/biomet/61.1.1
http://dx.doi.org/10.1007/978-3-540-74686-7_4
http://dx.doi.org/10.1103/PhysRevLett.63.1195
http://dx.doi.org/10.1002/jcc.540130812
http://dx.doi.org/10.1002/jcc.540161104
http://dx.doi.org/10.1063/1.2978177
http://dx.doi.org/10.1103/RevModPhys.46.597
http://dx.doi.org/10.1007/BF01293604
Introduction
Model and simulation method
Results
Conclusions
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