We study a numerical closure approach for systems of chemically reacting systems on lattices with low-dimensional support, for which a mean-field approximation is insufficiently accurate because of lateral interaction on the lattice. We introduce a hierarchy of macroscopic state variables, taking particle clusters into account, whose time evolution is obtained via microscopic (kinetic Monte Carlo) simulation. The macroscopic state variables are chosen such that they can be straightforwardly conserved during reconstruction of a microscopic configuration (the so-called lifting step). We present and compare the effects of different alternatives to initialize the remaining degrees of freedom. We illustrate the strong interplay between the number of macroscopic state variables and the specifics of the lifting and that, for a given lifting operator, accuracy of the macroscopic dynamics does not necessarily improve monotonically when adding macroscopic state variables.

1.
G.
Nicolis
,
Faraday Discuss.
120
,
1
10
(
2002
).
2.
A. V.
Shabunin
,
F.
Baras
, and
A.
Provata
,
Phys. Rev. E
66
(
3
),
036219
(
2002
).
3.
V. P.
Zhdanov
,
Elementary Physicochemical Processes on Solid Surfaces
,
Fundamental and Applied Catalysis
(
Plenum
,
1991
).
4.
D. T.
Gillespie
,
J. Comp. Phys.
22
(
4
),
403
434
(
1976
).
5.
A.
Chatterjee
and
D.
Vlachos
,
J. Comput.-Aided Mater. Des.
14
,
253
308
(
2007
).
6.
M.
Neurock
,
S. A.
Wasileski
, and
D.
Mei
,
Chem. Eng. Sci.
59
,
4703
4714
(
2004
).
7.
D.-J.
Liu
and
J. W.
Evans
,
J. Chem. Phys.
124
,
154705
(
2006
).
8.
Y. G.
Kevrekidis
,
C. W.
Gear
,
J. M.
Hyman
,
P. G.
Kevrekidis
,
O.
Runborg
, and
C.
Theodoropoulos
,
Commun. Math. Sci.
1
(
4
),
715
762
(
2003
).
9.
Y. G.
Kevrekidis
and
G.
Samaey
,
Annu. Rev. Phys. Chem.
60
,
321
344
(
2009
).
10.
W.
E
and
B.
Engquist
,
Commun. Math. Sci.
1
(
1
),
87
132
(
2003
).
11.
W.
E
,
B.
Engquist
,
X.
Li
,
W.
Ren
, and
E.
Vanden-Eijnden
,
Comm. Comp. Phys.
2
(
3
),
367
450
(
2007
).
12.
A. G.
Makeev
,
D.
Maroudas
,
A. Z.
Panagiotopoulos
, and
Y. G.
Kevrekidis
,
J. Chem. Phys.
117
(
18
),
8229
8240
(
2002
).
13.
A. G.
Makeev
and
Y. G.
Kevrekidis
,
Surf. Sci.
603
(
10–12
),
1696
1705
(
2009
).
14.
C. W.
Gear
,
T.
Kaper
,
Y. G.
Kevrekidis
, and
A.
Zagaris
,
SIAM J. Appl. Dyn. Syst.
4
(
3
),
711
732
(
2005
).
15.
A.
Zagaris
,
C. W.
Gear
,
T. J.
Kaper
, and
Y. G.
Kevrekidis
,
ESAIM: Math. Modell. Number. Anal.
43
,
757
784
(
2009
).
16.
R. J.
Glauber
,
J. Math. Phys.
4
(
2
),
294
307
(
1963
).
17.
A.
Tretyakov
,
A.
Provata
, and
G.
Nicolis
,
J. Phys. Chem.
99
(
9
),
2770
2776
(
1995
).
18.
S.
Prakash
and
G.
Nicolis
,
J. Stat. Phys.
82
,
297
322
(
1996
).
19.
A. G.
Makeev
,
D.
Maroudas
, and
Y. G.
Kevrekidis
,
J. Chem. Phys.
116
(
23
),
10083
10091
(
2002
).
20.
X.
Guo
,
Y.
De Decker
, and
J. W.
Evans
,
Phys. Rev. E
82
,
021121
(
2010
).
21.
D. R.
de Souza
and
T.
Tomé
,
Physica A
389
(
5
),
1142
1150
(
2010
).
22.
A.
Provata
,
J. W.
Turner
, and
G.
Nicolis
,
J. Stat. Phys.
70
,
1195
1213
(
1993
).
23.
R. M.
Ziff
,
E.
Gulari
, and
Y.
Barshad
,
Phys. Rev. Lett.
56
,
2553
2556
(
1986
).
24.
S.
Prakash
and
G.
Nicolis
,
J. Stat. Phys.
86
,
1289
1311
(
1997
).
25.
E.
Abad
,
P.
Grosfils
, and
G.
Nicolis
,
Phys. Rev. E
63
,
041102
(
2001
).
26.
J. W.
Evans
,
D. K.
Hoffman
, and
H.
Pak
,
Surf. Sci.
192
,
475
450
(
1987
).
27.
G.
Samaey
,
T.
Lelièvre
, and
V.
Legat
,
Comput. Fluids
43
(
1
),
119
133
(
2011
).
28.
B.
Nadler
,
S.
Lafon
,
R. R.
Coifman
, and
Y. G.
Kevrekidis
,
Appl. Comput. Harmon. Anal.
21
(
1
),
113
127
(
2006
).
29.
M.
Belkin
and
P.
Niyogi
,
Neural Comput.
6
(
15
),
1373
1396
(
2003
).
You do not currently have access to this content.