Advanced experimental techniques in chemistry and physics provide increasing access to detailed deterministic mass action models for chemical reaction kinetics. Especially in complex technical or biochemical systems the huge amount of species and reaction pathways involved in a detailed modeling approach call for efficient methods of model reduction. These should be automatic and based on a firm mathematical analysis of the ordinary differential equations underlying the chemical kinetics in deterministic models. A main purpose of model reduction is to enable accurate numerical simulations of even high dimensional and spatially extended reaction systems. The latter include physical transport mechanisms and are modeled by partial differential equations. Their numerical solution for hundreds or thousands of species within a reasonable time will exceed computer capacities available now and in a foreseeable future. The central idea of model reduction is to replace the high dimensional dynamics by a low dimensional approximation with an appropriate degree of accuracy. Here I present a global approach to model reduction based on the concept of minimal entropy production and its numerical implementation. For given values of a single species concentration in a chemical system all other species concentrations are computed under the assumption that the system is as close as possible to its attractor, the thermodynamic equilibrium, in the sense that all modes of thermodynamic forces are maximally relaxed except the one, which drives the remaining system dynamics. This relaxation is expressed in terms of minimal entropy production for single reaction steps along phase space trajectories.

1.
M. J. Pilling and P. W. Seakins, Reaction Kinetics (Oxford University Press, Oxford, 1995).
2.
J. Warnatz, in 24th Symposium on Combustion (Combustion Institute, Pittsburgh, 1992), p. 553.
3.
D. E.
Koshland
, Jr.
,
Science
280
,
852
(
1998
).
4.
M. S.
Okino
and
M. L.
Mavrovouniotis
,
Chem. Rev.
98
,
391
(
1998
).
5.
M. J.
Davis
and
R. T.
Skodje
,
J. Chem. Phys.
111
,
859
(
1999
).
6.
S. H.
Lam
and
D. A.
Goussis
,
Int. J. Chem. Kinet.
26
,
461
(
1994
).
7.
J. Warnatz, U. Maas, and R. W. Dibble, Combustion (Springer, Berlin, 1996).
8.
K.-C.
Chou
,
J. Biol. Chem.
264
,
12074
(
1989
).
9.
D. E.
Koshland
, Jr.
,
J. Phys. Chem.
60
,
1375
(
1956
).
10.
D. J. M.
Park
,
J. Theor. Biol.
46
,
31
(
1974
).
11.
R. H.
Snow
,
J. Phys. Chem.
70
,
2780
(
1966
).
12.
T.
Turányi
,
A. S.
Tomlin
, and
M. J.
Pilling
,
J. Phys. Chem.
97
,
163
(
1993
).
13.
G.
Li
,
H.
Rabitz
, and
H.
Toth
,
Chem. Eng. Sci.
46
,
95
(
1991
).
14.
G.
Li
,
H.
Rabitz
, and
H.
Toth
,
Chem. Eng. Sci.
49
,
343
(
1994
).
15.
H.
Rabitz
,
M.
Kramer
, and
D.
Dacol
,
Annu. Rev. Phys. Chem.
34
,
419
(
1983
).
16.
T.
Turányi
,
J. Math. Chem.
5
,
203
(
1990
).
17.
U.
Maas
,
Appl. Math. (Germany)
3
,
249
(
1995
).
18.
U.
Maas
,
Comput. Visual. Sci.
1
,
69
(
1998
).
19.
U.
Maas
and
S. B.
Pope
,
Combust. Flame
88
,
239
(
1992
).
20.
M.
Hadjinicolaou
and
D. A.
Goussis
,
SIAM J. Sci. Comput. (USA)
20
,
781
(
1999
).
21.
A. N.
Yannacopoulos
,
A. S.
Tomlin
,
J.
Brindley
,
J. H.
Merkin
, and
M. J.
Pilling
,
Physica D
83
,
421
(
1995
).
22.
P. Deuflhard and J. Heroth, in Scientific Computing in Chemical Engineering (Springer, New York, 1998), p. 29.
23.
J. Zobeley, D. Lebiedz, A. Ishmurzin, and U. Kummer, Nonlinear Dynam. (submitted).
24.
L.
Petzold
and
W.
Zhu
,
AIChE J.
45
,
869
(
1999
).
25.
S. J.
Fraser
,
J. Chem. Phys.
88
,
4732
(
1988
).
26.
A. H.
Nguyen
and
S. J.
Fraser
,
J. Chem. Phys.
91
,
186
(
1989
).
27.
M. R.
Roussel
and
S. J.
Fraser
,
J. Chem. Phys.
93
,
1072
(
1990
).
28.
M. R.
Roussel
and
S. J.
Fraser
,
J. Phys. Chem.
95
,
8762
(
1991
).
29.
R. T.
Skodje
and
M. J.
Davis
,
J. Phys. Chem. A
105
,
10356
(
2001
).
30.
A. N.
Gorban
,
I. V.
Karlin
,
V. B.
Zmievskii
, and
S. V.
Dymova
,
Physica A
275
,
361
(
2000
).
31.
N. Peters and B. Rogg, Reduced Kinetic Mechanisms for Applications in Combustion Systems (Springer, Berlin, 1993).
32.
C. Chevalier, W. J. Pitz, J. Warnatz, C. K. Westbrook, and H. Melenk, in 24th Symposium on Combustion (Combustion Institute, Pittsburgh, 1992).
33.
A. T.
Norris
and
S. B.
Pope
,
Combust. Flame
100
,
211
(
1995
).
34.
D. A. Goussis and S. Lam, in 24th Symposium on Combustion (Combustion Institute, Pittsburgh, 1992), p. 113.
35.
A. C.
Heard
,
M. J.
Pilling
, and
A. S.
Tomlin
,
Atmos. Environ.
32
,
1059
(
1998
).
36.
U. Maas and S. B. Pope, in 25th Symposium on Combustion (Combustion Institute, Pittsburgh, 1994), p. 1349.
37.
C.
Correa
,
H.
Niemann
,
B.
Schramm
, and
J.
Warnatz
,
Proc. Comb. Inst.
28
,
1607
(
2000
).
38.
J.
Nafe
and
U.
Maas
,
Combust. Theory Modell.
6
,
697
(
2002
).
39.
M. R.
Rousse
and
S. J.
Fraser
,
Chaos
11
,
196
(
2001
).
40.
A. N. Gorban, I. V. Karlin, and A. Y. Zinovyev, Preprint: http://ihes.fr/PREPRINTS/M03/M03-50.pdf (2003).
41.
A. N.
Gorban
and
I. V.
Karlin
,
Chem. Eng. Sci.
58
,
4751
(
2003
).
42.
H. G.
Kaper
and
T. J.
Kaper
,
Physica D
165
,
66
(
2002
).
43.
P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability, and Fluctuations (Wiley, New York, 1971).
44.
T. D. Donder, Affinity (Stanford University Press, Menlo Park, 1936).
45.
D. Kondepudi and I. Prigogine, Modern Thermodynamics (Wiley, England, 1936).
46.
L.
Onsager
,
Phys. Rev.
37
,
405
(
1931
).
47.
V. I.
Bykov
,
G. S.
Yablonskii
, and
T. A.
Akramov
,
Dokl. Akad. Nauk SSSR
234
,
621
(
1977
).
48.
M.
Grmela
,
I. V.
Karlin
, and
V. B.
Zmievski
,
Phys. Rev. E
66
,
011201
(
2002
).
49.
H.
Struchtrup
and
W.
Weiss
,
Phys. Rev. Lett.
80
,
5048
(
1998
).
50.
H. G. Bock and K. J. Plitt, in Proceedings of the Ninth IFAC World Congress, Budapest (Pergamon, Oxford, 1984).
51.
D. B.
Leineweber
,
I.
Bauer
,
H. G.
Bock
, and
J. P.
Schlöder
,
Comput. Chem. Eng.
27
,
157
(
2003
).
52.
D. B.
Leineweber
,
A.
Schläfer
,
H. G.
Bock
, and
J. P.
Schlöder
,
Comput. Chem. Eng.
27
,
167
(
2003
).
53.
M.
Diehl
,
H. G.
Bock
,
J. P.
Schlöder
,
R.
Findeisen
,
Z.
Nagy
, and
F.
Allgöwer
,
J. Process Contr.
12
,
577
(
2002
).
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