Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice
Within the investigated model of a one-dimensional bi- and tri-molecular chemically reacted crystal, the asymptotic behaviour of the amplitude of a two-particle static structure factor (as a function of the crystal length) has been discovered. The nonlinear fluctuational scenario leads us to the...
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| Мова: | English |
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Інститут фізики конденсованих систем НАН України
1999
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice / O.I. Gerasimov, N.N. Khudyntsev // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 75-79. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1199212017-06-11T03:03:20Z Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice Gerasimov, O.I. Khudyntsev, N.N. Within the investigated model of a one-dimensional bi- and tri-molecular chemically reacted crystal, the asymptotic behaviour of the amplitude of a two-particle static structure factor (as a function of the crystal length) has been discovered. The nonlinear fluctuational scenario leads us to the conclusion as to the possibility of existence of an asymptotic metastable cluster fragmentation within initially homogeneous 1D systems. A connection between some possible effects and the properties of the fluctuations in reacting systems and reactive dynamics in a partially filled lattice is also shown. Запропонована модель одновимірного кристалу з дво- та тримолекулярними хімічними реакціями, для якої досліджена асимптотична поведінка амплітуди двочастинкового статичного структурного фактору (як функції довжини кристала). Аналіз нелінійного характеру флуктуацій свідчить про принципову можливість існування асимптотичних метастабільних станів у модельному одновимірному кристалі з однорідними початковими умовами. Також досліджено зв’язок між флуктуаційними властивостями та динамікою систем, що реагують, та деякими ефектами, що можуть виникати на цілком заповнених гратках. 1999 Article Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice / O.I. Gerasimov, N.N. Khudyntsev // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 75-79. — Бібліогр.: 9 назв. — англ. 1607-324X DOI:10.5488/CMP.2.1.75 PACS: 61.20.Gy, 82.40.-g. http://dspace.nbuv.gov.ua/handle/123456789/119921 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
Within the investigated model of a one-dimensional bi- and tri-molecular
chemically reacted crystal, the asymptotic behaviour of the amplitude of
a two-particle static structure factor (as a function of the crystal length)
has been discovered. The nonlinear fluctuational scenario leads us to the
conclusion as to the possibility of existence of an asymptotic metastable
cluster fragmentation within initially homogeneous 1D systems. A connection between some possible effects and the properties of the fluctuations
in reacting systems and reactive dynamics in a partially filled lattice is also
shown. |
| format |
Article |
| author |
Gerasimov, O.I. Khudyntsev, N.N. |
| spellingShingle |
Gerasimov, O.I. Khudyntsev, N.N. Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice Condensed Matter Physics |
| author_facet |
Gerasimov, O.I. Khudyntsev, N.N. |
| author_sort |
Gerasimov, O.I. |
| title |
Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice |
| title_short |
Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice |
| title_full |
Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice |
| title_fullStr |
Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice |
| title_full_unstemmed |
Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice |
| title_sort |
fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1d lattice |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119921 |
| citation_txt |
Fluctuational dynamics and the structure factor of nonlinear reactive systems on a 1D lattice / O.I. Gerasimov, N.N. Khudyntsev // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 75-79. — Бібліогр.: 9 назв. — англ. |
| series |
Condensed Matter Physics |
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AT gerasimovoi fluctuationaldynamicsandthestructurefactorofnonlinearreactivesystemsona1dlattice AT khudyntsevnn fluctuationaldynamicsandthestructurefactorofnonlinearreactivesystemsona1dlattice |
| first_indexed |
2025-07-08T16:54:49Z |
| last_indexed |
2025-07-08T16:54:49Z |
| _version_ |
1837098537862561792 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 1(17), p. 75–79
Fluctuational dynamics and
the structure factor of nonlinear
reactive systems on a 1D lattice
O.I.Gerasimov, N.N.Khudyntsev
Odesa State Hydrometheorological Institute,
100 Lvivska Str., 270011 Odesa, Ukraine
Received September 16, 1998
Within the investigated model of a one-dimensional bi- and tri-molecular
chemically reacted crystal, the asymptotic behaviour of the amplitude of
a two-particle static structure factor (as a function of the crystal length)
has been discovered. The nonlinear fluctuational scenario leads us to the
conclusion as to the possibility of existence of an asymptotic metastable
cluster fragmentation within initially homogeneous 1D systems. A connec-
tion between some possible effects and the properties of the fluctuations
in reacting systems and reactive dynamics in a partially filled lattice is also
shown.
Key words: chemical reactions, structure factor, 1D lattice.
PACS: 61.20.Gy, 82.40.-g.
1. Introduction
One-dimensional models are known to be an effective tool for solving a variety
of problems in statistical mechanics. In the past decades much attention in such
areas as chemical reactions, random walks and aggregation problems has been paid
to the role of dimensionality [1]. In particular, work carried out in Brussels by
Nicolis, Provata, Prakash, Tretyakov and Turner has showed that restricting space
to low dimension can cause deviations from the mean field behaviour, depending on
the type of the nonlinearity involved. For instance, while the bimolecular reaction
A+X ←→ 2X shows the mean field behaviour on a 1D completely filled lattice, the
trimolecular reaction A+2X ←→ 3X stabilizes in such a lattice in a nonequilibrium
locally frozen asymptotic state in which the ratio of the average number A to X
particles is a constant quite different from the mean-field value.
The work carried out within the framework of the IUAP project during our
stay in Brussels focused on two topics: the properties of the fluctuations in the
reacting systems and the study of reactive dynamics in a partially filled lattice. A
c© O.I.Gerasimov, N.N.Khudyntsev 75
O.I.Gerasimov, N.N.Khudyntsev
manuscript summarizing the results is currently being prepared for publication in
Europhysics Letters. We hereafter summarize the prinsipal steps and conclusions.
2. Fluctuations in a completely filled 1D lattice
Consider the bimolecular reaction 2X ←→ A + X on an ideal totally filled
1D lattice of M sites, bearing in mind that a given species can only react with
its nearest neighbours and that particles cannot overlap. We adopt as an initial
condition a uniform configuration containing only X particles. Since at equilibrium
all the positive configurations of A of X particles, except the one of a lattice filled
completely by A, can be generated from this initial condition, the probability
distribution gM (NA) is given by
gM (NA) =
(
M
NA
)
/ M−1
∑
NA=0
(
M
NA
)
= 21−M
(
M
NA
)
(1)
with NA +NX = M.
From (1) one recovers asymptotically (M →∞) the previous result of Nicolis
et al. r = 〈NA〉 / 〈NX〉 = 1. Furthermore, one can compute a covariant matrix of
the fluctuations around the mean particle numbers
〈
δN2
A
〉
=
〈
N2
A
〉
− 〈NA〉2 = −
M
4
,
〈δNAδNX〉 = −
〈
δN2
A
〉
=
M
4
, (2)
etc.
Coming now to the trimolecular model and adopting the same initial condition,
one can easily see that because of the geometric constrains involved, for each given
NA, at least NA − 1 sites cannot be occupied by A particles. The probability
distribution replacing (1) is thus
gM (NA) =
(
M − 1−NA
NA
)
M−1
∑
NA=0
(
M − 1−NA
NA
) =
2M
√
5
(
1 +
√
5
)M −
(
1−
√
5
)M
(
M − 1−NA
NA
)
.
(3)
Again, the expression reproduces the value 〈NA〉 / 〈NX〉 ≈ 0.38 obtained pre-
viously by Nicolis et al. However, one is now also in the position to compute the
properties of the fluctuations. For instance, one finds
1
M
〈δNXδNA〉 =
2
5
(
1 +
√
5
)
2√
5
(
1−
√
5
) − 1
5
. (4)
76
Fluctuational dynamics and structure factor
3. Partially filled lattice
Next we allow for vacancies in the lattice, starting with an initial configuration
in which NX particles (NX < M) are distributed randomly among the M sites
available. We also introduce three auxiliary variables:
– the number of nearest neighbours pairs occupied simultaneously by X par-
ticles – NXX ,
– the number of nearest neighbours pairs of which only one is occupied by X
particles – N0X ,
– the number of nearest neighbours pairs both of which are empty – N00.
The following relations between these variables are easily established
2NXX +N0X = 2NX ,
2N00 +N0X = 2 (M −NX) , (5)
showing that of these three variables only one can be chosen independently, say
NXX . The number of different configurations of X particles with only NXX pairs
is then
GM (NX , NXX) =
(
NX
NX −NXX
)(
M −NX
NX −NXX
)
. (6)
Since A particles can only be generated from the configurations involving contin-
uous X particles, the conditional probability to find NA particles in the system is
(
NXX
NA
)
(up to factor), and the equilibrium distribution of the lattice is, for the
bimolecular model,
gM (NA, NX) =
NX−1
∑
NXX=0
2−NXX
(
NX
NX −NXX
)(
M −NX
NX −NXX
)(
NXX
NA
)
. (7)
This expression can be reduced to a form exhibiting a confluent hypergeometric
function and used to calculate the fluctuations of the number of particles. The
principal result of this analysis is that now the fluctuations behave anomalously
〈δNAδNX〉 ≈ λ2M2, (8)
where λ = N
(0)
X /M is an initial filling fraction.
4. Structure factor
To get an idea of the type of spatial inhomogeneities locally created in the
lattice we evaluated the structure factor of the system, a quantity the additional
interest of which is in its experimental accessibility.
77
O.I.Gerasimov, N.N.Khudyntsev
Let g be a wave number associated with the inhomogeneities. The structure
factor is then
SAX (g) =
∞
∑
l=−∞
∞
∑
l′=−∞
eig(l−l′)
〈
∑
n∈NA
δkrnl
∑
m∈NX
δkrml′
〉
, (9)
where l and l′ denote the lattice sites. Performing the summations over l and l′ we
obtain
SAX (g) =
1
2 (1− cos g)
〈(
1− eigNA
) (
1− eigNX
)〉
. (10)
In the long wavelength (“hydrodynamic”) limit g → 0 this expression reduces
to
SAX (g) ∼ 〈δNAδNX〉 − Γg2. (11)
The g-dependence of this function reflects the existence of a spatial variability.
However, since the extremum of SAX is at g = 0, no preferred length scale emerges.
The situation is likely to change in the trimolecular model which is currently under
investigation.
5. Acknowledgements
The authors would like to thank J.W.Turner for useful discussions. O.I.G.
thanks the Belgian Federal Office for Scientific, Technical and Cultural Affairs for
the financial support of his research under the Poles d’Attraction Interuniversi-
taire Program. He expresses gratitude to the Center for Nonlinear Phenomena
and Complex Systems at Universite Libre de Bruxelles, Belgium, for the financial
support, hospitality and fruitful cooperation.
References
1. Nicolis G., Prigogine I. Self-organization in Nonequilibrium Systems. New York, Wiley,
1977.
2. Provata A., Turner J.W., Nicolis G. Nonlinear chemical dynamics in low dimensions:
an exactly soluble model. // Journ. Stat. Phys., 1993, vol. 70, p. 1195–1213.
3. Prakash S., Nicolis G. Dynamics of fluctuations in a reactive system of low spatial
dimension. // Journ. Stat. Phys., 1996, vol. 82, p. 297–322.
4. Tretyakov A., Provata A., Nicolis G. Nonlinear chemical dynamics in low-dimensional
lattices and fractal sets. // Journ. Phys. Chem., 1995, vol. 99, p. 2770–2776.
5. Monthus C., Hilhorst H.J. The pair correlation function in a randomly sequentially
filled 1D lattice. // Phisica A, 1991, vol. 175, p. 263–274.
6. Hill T.L. Statistical Thermodynamics. London, Addison Wesley, 1960.
7. Feller W. An Introduction to Probability Theory and Its Applications. New York,
Wiley, 1971.
8. Prudnicov A.P., Brychkov Yu.A., Marichev O.J. Integrals and Series (vol. 1–3). New
York, Gordon and Breach Sci. Pub., 1990.
78
Fluctuational dynamics and structure factor
9. Gradstein J.C., Ryzhik J.M. Tables of Integrals, Sums, Series and Products. Moscow,
Nauka Pub., 1963 (in Russian).
Флуктуаційна динаміка та структурний фактор
нелінійної системи з реакціями на одновимірній
гратці
Герасимов О.І., Худинцев М.М.
Одеський гідрометеорологічний інститут,
270011 Одеса, вул. Львівська, 100
Отримано 16 вересня 1998 р.
Запропонована модель одновимірного кристалу з дво- та тримоле-
кулярними хімічними реакціями, для якої досліджена асимптотич-
на поведінка амплітуди двочастинкового статичного структурного
фактору (як функції довжини кристала). Аналіз нелінійного характе-
ру флуктуацій свідчить про принципову можливість існування асим-
птотичних метастабільних станів у модельному одновимірному кри-
сталі з однорідними початковими умовами. Також досліджено зв’я-
зок між флуктуаційними властивостями та динамікою систем, що ре-
агують, та деякими ефектами, що можуть виникати на цілком запов-
нених гратках.
Ключові слова: хімічні реакції, структурний фактор, одновимірна
гратка.
PACS: 61.20.Gy, 82.40.-g.
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