Macroscopic Effects of the Perturbation of Particle Velocity
Distribution in Chemical Wave Fronts
A. Lemarchand
Universite Pierre et Marie Curie, CNRS UMR 7600, Laboratoire de Physique Theorique des Liquides,
4, place Jussieu, case courrier 121, 75252 Paris cedex 05, France
Abstract. Nonequilibrium effects induced by a fast chemical reaction are shown to modify the speed and the shape of
wave fronts propagating in a reactive medium. For a trigger wave, the analytical results deduced from Boltzmann
equation agree well with the microscopic simulation results obtained with the Direct Simulation Monte Carlo method.
For a front propagating into an unstable state, the simulation results depend on the mean number^ of particles in a cell.
INTRODUCTION
The motivation of this work is to weave some links between the microscopic description and the macroscopic
properties of spatio-temporal structures observed in inhomogeneous chemical systems. A way to better understand
these links is to identify generic situations whffe the macroscopic deterministic dynamics is sensitive to even weak
perturbations and thus, in particular, to the details of the underlying microscopic dynamics. A weHoiown example
of such a sensitive situation is the vicinity of a bifurcation where theamplitude of fluctuations diverges. But this
critical behavior only occurs for specific values of the parameters. We are also interested here in another class of
generic situations where internal fluctuations can be intuitively suspected of playing a rolat an observable scale: It
is the case of dynamical systems admitting a continuum of simultaneously stable solutions, easy to be visited by
small fluctuations. The propagation of chemical wave fronts offers simple as well as rich examples of these two
kinds of situations.
TWO TYPES OF WAVE FRONTS
At a macroscopic level, a wave fTontA(x-Ut) is a uniformly translating solution of reactiondiffusion equations,
moving in directionjc with constant speed U and replacing the stationary state, A = A$ , by the stable stationary state,
A = Al. Depending on whether the stationary state A = A$ is stable or not, the properties of the wave front are
different. In a bistable system, a trigger wave propagatiig between two stable stationary states has a uniquely
defined speed C/0 . In the case of the propagation into an unstable state, there exists a continuum of stable wave front
solutions propagating with any speed U9 greater than a marginal value t/min ; the problem of selection of the
propagation speed has been extensively studied since the pioneering works of Fisher [1] and of Kolmogorov,
Petrovsky, and Piskunov [2] in 1937. In each case, the macroscopic descrifion leads to expressions of the
propagation speed as function of chemical rate constants and diffusion coefficient. We suspect that the sensitivity of
a trigger wave to small perturbations of microscopic origin will be in general weak but could be enhanofe in the
vicinity of a bifurcation. On the contrary, a front propagating into an unstable state should exhibit a high sensitivity
whatever the parameters.
Kinetic theory studies based on Boltzmann equation have revealed that the perturbation of particle/elocity
distribution induced by a chemical reaction may significantly influence the dynamics of gaseous chemical systems
[3-6]. The existence of a departure from the equilibrium distribution comes from the dependence of molecule
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
609
reactivity on energy, so that a chemical process affects more strongly the populations of reactant molecules
belonging to a certain range of velocities. The basic consequence of this nonequilibrium effect at a macroscopic
level is a modification of the rate constants and transportcoefficients appearing in the macroscopic equations for the
concentrations of chemical species. The point here is thus to determine if these perturbations are sufficient to modify
mean wave front properties such as its speed.
We choose the following modified version of Schlogl model [7]
2A + B
3A
(1)
A -> B
(2)
to study a trigger wave, and Fisher model [1,2]
A + B -> 2A
(3)
for the wave front propagating into an unstable state. In eafa case, the total concentration of species A and B is
constant and we derive from the perturbation solution of Boltzmann equation [810] the modified reaction-diffusion
equation for the concentration of species A which include the corrections due to the noequilibrium effects. The
speed and shape of the perturbed front is calculated using a numerical solution of this modified reactiowliffusion
equation. These theoretical predictions are compared with the results of microscopic simulations using the Direct
Simulation Monte Carlo (DSMC) method [11].
TRIGGER WAVES
In the case of Schlogl model [12-14] and provided the reaction is not extremely fast, the agreement between the
results of the two approaches is good even for such a sensitive quantity like the fouit order cumulant (or kurtosis)
of the particle velocity distribution (see Fig. 1).
0.05 -
200
400
600
-0.05
300
800
cell number
400
500
cell number
600
FIGURE 1. (a) Front profiles in the moving frame for Schlogl model. The dotted line corresponds to the
macroscopic prediction without nonequilibrium corrections, (b) Spatial variations in the moving frame of the
kurtosis Kyz of the particle A velocity distribution and of the relative deviation of its second moment from its
equilibrium value. The solid lines correspond to microscopic simulation results using DSMC method, the dashed
lines to analytical results based on Boltzmann equation. The activation energy of forward reaction (1) is equal to
thermal energy.
The nonequilibrium effects for the trigger wave are the strongest when the activation energy of the reaction is
comparable to thermal energy. The largest effects are observed on the speed of the front, for which the relative
correction reaches 65% for parameter values close to that correponding to a stationary interface between the two
stable stationary states. The plateau height and the front width are also affected by the nonequilibrium effects, but
their relative corrections do not exceed 10%. We expect stronger corrections in the vioiity of the bifurcation
610
associated with the coalescence of the unstable stationary state with the largest stable stationary state. The previous
study [14] of a homogeneous bistable system had revealed the existence of diverging corrections to the largesttable
stationary state value, ,4 = 4, near this bifurcation. However, the conditions allowing us to avoid the problem of
nucleation, prevent us from studying here the vicinity of this bifurcation.
WAVE FRONTS PROPAGATING INTO AN UNSTABLE STATIONARY STATE
In the case of Fisher model [12,16-24], the unusual sensitivity of the dynamics even far from any bifurcation
makes the analysis difficult and the results deduced from microscopic simulations using DSMC method appear as a
superposition of different effects (see table I). For example, the front speed varies not only with the frequency of the
reactive collisions but also with the mean numberJV of particles simulated in a spatial cell, at least in the range
20 < N < 104 . Note that choosingN=l
is sufficient to recover the macroscopic limit in a nonsensitive situation.
The comparison of several approaches at different scales [23,24] has been necessary to discriminate between the
different sources of disagreement with the prediction of the deterministic theory, and in particular to isolate the
nonequilibrium effects from finite-size effects due to internal fluctuations and discretization of the variables.
TABLE I. Comparison of two types of wave fronts: corrections to their propagation speed induced by different kinds of
perturbations with respect to the deterministic macroscopic description. The results are deduced from the comparison of
several methods of description at different scales. The presence of an X in a cell mans that the method used takes into
account the mentioned perturbation
Discretization of the
Fluctuation of the
Departure from
__________________________variables____________variables__________local equilibrium___
Langevin equations
Master equation
X
X
X
Boltzmann equation
X
DSMC
X
X
X
Consequences on
mean front speed
for Schlogl front
„ _ /^ ^ ^-3/2
" \ /
Ref. [13]
/^v _
^-3/2
\ /
»
Refs. [12]
(u)=U'Q
n rn>n
Ref. [14]
Consequences on
mean front speed
for Fisher front
-rjm n _/r/\ n (\r\\~2
* \ /
JV » 108
Ref. [20]
ITJ\ — TJm n
A/™^
\ /
*
N < 104
Refs. [12,16,23]
(u}=u'mm
RefS [22 23
'
> ^
CONCLUSION
The effects of the discretization of the variables, of the internal fluctuations, and of a departure from local
equilibrium on the mean propagation speed of the two types of wave fronts are summarized in Table I. Beyond the
quantitative characterization of macroscopic consequences of such microscopic effects, we give with Bher front an
611
example of sensitive dynamical system, such that the microscopic description using DSMC does not lead to its
macroscopic properties for typical values of the mean number of particles in a cell. These effects are not artefacts
but actual finite-size effects: They are found in very good agreement with the results deduced at a mesoscopic level
from stochastic descriptions using a master equation or Langevin equations. They should be detectable in
experimental systems involving a small number of pirticles.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
R. A. Fisher, Ann. Eugenics 7, 335 (1937).
A. Kolmogorov, I. Petrovsky, and N. Piskunov, Bull. Univ. Moscow Ser. Int. Sec. A 1, 1 (1937).
I. Prigogine and E. Xhrouet, Physica 15, 913 (1949).
J. Ross and P. Mazur, J. Chem. Phys. 35, 19 (1961).
B. Shizgal and M. Karplus, J. Chem. Phys. 52, 4262 (1970); id. 54, 4345 (1971); ibid. 54, 4357 (1971).
Cf. a number of papers in Far-from-equilibrium Dynamics of Chemical Systems, edited by J. Gorecki et a/., World
Scientific, Singapore, 1994.
Z. Schlogl, Z. Phys. 253, 147 (1972).
S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases, Cambridge University Press, Cambridge,
1970.
B. Nowakowski and A. Lemarchand, J. Chem. Phys. 106, 3965 (1997).
B. Nowakowski, J. Chem. Phys. 109,3443 (1998).
G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon, Oxford, 1994.
M. Karzazi, A. Lemarchand, and M. Mareschal, Phys. Rev. E 54, 4888 (1996).
D. A. Kessler, Z. Ner, and L. M. Sander, Phys. Rev. E 58, 107 (1998)
A. Lemarchand and B. Nowakowski, Phys. Rev. E, to appear (2000).
A. Lemarchand and B. Nowakowski, Physica A 271, 87 (1999).
A. Lemarchand, A. Lesne, and M. Mareschal, Phys. Rev. E 51, 4457 (1995).
C. Dellago and H. A. Posch, Physica A 240, 68 (1997).
R. van Zon, H. van Beijeren, and C. Dellago, Phys.Rev. Lett. 80, 2035 (1998).
R. van Zon, Ph. D. Thesis, Chaos in dilute hard sphere gases in and out of equilibrium, Shaker Publishing, Maastricht, 2000.
E. Brunet and B. Derrida, Phys. Rev. E 56, 2597 (1997).
A. Lemarchand and B. Nowakowski, Europhys. Lett. 41, 455 (1998).
A. Lemarchand and B. Nowakowski, J. Chem. Phys. 109, 7028 (1998).
A. Lemarchand and B. Nowakowski, J. Chem. Phys. 111,6190(1999).
A. Lemarchand, J. Stat. Phys., 101, 579 (2000).
612