close

Вход

Забыли?

вход по аккаунту

?

Effect of Fluctuation on Plane Front Propagation in Bistable Nonequilibrium Systems.

код для вставкиСкачать
Annalen der Physik. 7. Folge, Band 40, Heft 4/5, 1983, S. 277-286
J. A. Barth, Leipzig
Effect of Fluctuation on Plane Front Propagation
in Bistable Nonequilibrium Systems1)
By L. S C ~ A N S K Y - G E I EA.
R ,S. MIKHAILOV~)and W. EBELIN~
Sektion Physik, Bereich 04 der Humboldt-Universitiit Berlin/DDR
Abstract. A theory of front propagation in bistable stochastic media is developed. The effect
of fluctuations is twofold: 1)they lead to random variations of the front shape. 2) they result in
random motion of the front positions. Its general expression for the dispersion of the position for
arbitrary stochastic noises wit,h spatial correlation length I, and correlation time t, is given. For
rate functions with cubic non1,inearities the dispersion formula is further analysed.
Fluktuationseffektebei der Ausbreitung einer ebenen Front in bistsbilen
Nichtgleichgewichtssystemen
I n h a l t s u b e r s i c h t . Eine Theorie der Frontausbreitung in bistabilen stochastischen Medien
wird entwickelt. Der EinfluB der Fluktuationen fiihrt zu einer zufiilligen Veriinderung der Form der
Front, wie auch zu einer Zufallsbewegung der Frontposition. Ein allgemeiner Ausdruck fur die Dispersion der Frontposition fur beliebige stochastische Quellen mit einer riiumlichen KorrelationsliingeI,
und einer Korrelationszeit tcwird gegeben. Fur Ratenfunktionen mit kubischer Nichtlinearitat wird
die Dispersionsformel weiter analysiert.
1. Introduction
Bistable behaviour arises in many physical, chemical and biophysical situations
[l, 21. The investigation of systems whose final state depends on initial conditions is of
principal interest since here we have the simplest type of structure formation in equilibrium and nonequilibrium media. The first example we consider is bistability in chemical
reaction-diffusion systems. Its existence here is a typical nonequilibrium effect (dissipative structure); it is guaranteed by a steady supply of free energy into the system.
Bistable behaviour may be observed experimentally e.g. in enzymatic reactions with
second order inhibition [ 3 ] as well as in combustion phenomena [4].
Analytic investigations of chemical bistable systems are usually carried out using a
simple trimolecular model system introduced by SCHLOGL
[5]
A t a certain overcritical choice of a and b which are the densities of the speciesesA and
B the model guarantees bistable behaviour for the intermediate species X.This means
Dedicated to Prof. Yu. L. Klisnontovich, MOSCOW,
on the Occasion of the Goth Anniversary
of his Birthday.
2, On leave of absence from Department of Physics, Moscow State University, 117234 Moscow,
USSR.
L. SCEIMANSKY-GEIEB
et al.
278
that the chemical rate function
it = W ( n )= -kin3
+ ak,n2- k2n + kkb ,
(1.2)
where n(t)is the density of the species X , obtains three real roots 121, n2,n3 with W(ni)=O.
I n a bistable situation with two steady states inhomogeneous distributions are also
possible. A front wave of transition between the two steady states of the system can
propagate. I n one space dimension this front propagates with constant speed which
direction is defined by the parameters of the chemical reaction. Generally these parameters are fluctuating quantities. Their stochastic behaviour may be due t o several
factors as e.g. random light illumination as well as by random in- and output of raw
material. Another source of random effects could be the disordered spatial structure of
the material in which the reaction takes place e.g. in a disordered semiconductor. The
effect of such stochastic noises on the front propagation will be studied here. As a second
example let us consider disordered condensed media which may exist in two states (say
solid-liquid or excited-nonexcited) [S]. In such a case again fronts of transition may
be observed which propagate and are scattered by the random field in the disordered
material. We may think e.g. about the front between the solid and the liquid state originated by a powerful laser pulse (laser activated annealing [7]) where we have twodimensional plane fronts in a three dimensional medium.
As a third example population fronts in real spaces or on high-dimensional phenotype spaces [B] may by considered. We may think e.g. about fronts in hypercyclic systems [91. Stochastic behaviour of the reaction parameters necessarily occurs e.g. due
to the inhomogeneous distribution of building stones of the phenotypes or hypercircles
along the direction of propagation.
The general feature of all these situations is that they are described by a nonlinear
partial differential equation of the type
a
-n(r,
at
t ) = W(n(r,t ) ; k&r, t ) , ... , k l ( r ,t ) )
a
+ -.ara D- . -n(r,
ar
t),
(1.3)
where r is the state vector in the corresponding space and D is a diffusion-like tensor.
W ( n ;k,,
kl) are assumed to be S-shaped (see Fig. 1)with three zeroes a t n = %, n,, n,
for constant kitwo of which (121, n3)correspond to states that are stable with respect to
small fluctuations of the density.
...
Fig. 1. The bistable rate function W(n)with two stable (nl, n3)and one unstable (aa)steady states
Effect of Fluctuation on Front Propagation in Bistable Systems
279
Further we assume that ki(r, t ) are fluctuating fields
k , ( r , t ) = kp
+ dki(r, t ) ,
where 6ki are small stochastic noises wit,h the properties
(dki(r, t ) ) = 0
and
( d k i ( r , t ) Gki(r,t ’ ) ) = diiSi(lr- r’l, I t - t ’ l ) .
We consider the situation where the pair correlation function factorizes into a product
of space and time dependent parts. Such assumption is not necessary in the derivation
of a general formula of the front propagation dispersion. However, it simplifies the
final analysis of the problem.
The characteristic spatial correlation lengths are 1: and the characteristic correlation
times of stochastic functions dki(r, t ) are 7:.
If z, is infinite we are dealing with stationary random media. Also purely time dependent cases if 1,. is large may be considered. Analytical results we obtain by comparison of these parameters with the width of the interface of the front 1, or the time l&
resp. where u is the speed of the front.
In section 2 we discuss the deterministic theory of front propagation in bistable
media. General effects of fluctuations on the front are investigated in section 3. The
aection 4 is devoted to applications of the theory for nonlinear rate functions with a
polynom of third degree like in the Schlogl model.
2. Front Propagation in Bistable Medis
We consider here the case when the diffusion tensor D is constant and the density
n(r,t ) changes only in one space direction. Then the one-dimensional bistable medium
is described by the nonlinear partial differential equation :
where W(n)is by assumption a S-shaped function with three zeroes at % < n, < 1s.
If we introduce the inertial coordinate:
t=r-vt
.solution of the ordinary differential equation
dno
dad
W(no) v - D -= 0
dt
dEa
correspond to a steady density profile moving with the speed v in the direction of r
n = no(r - vt) = nO(5).
(2.3)
+
+
Further we assume boundary conditions
n(-
00,
t ) = n,; n(00, t ) = n,; t < 00
and vanishing derivatives dnldl at infinity. Then the speed of the front [S]
(2.4)
L.SCEIMANSKY-GEIER
et al.
280
is mainly defined by the value of the integral standing in the nominator. The supersaturation
711
is a measure of the stability of the steady states n, and n,. For example x > 0 means
that n, is stable and n, is meatastble. Its sign defines the direction of the front motion.
We obtain
z > 0: n(r, t ) + n ( r , o o ) = n,;
x < 0: n ( r , t )
n(r, 00) = n,;
< co,
r>-
r
00.
(2.7)
The front is a t rest for x = 0 and therefore x = 0 is a kind of Maxwell-condition.
Problems of front propagation may also be considered in the spherically symmetric
case [l,6, 9, 101. Front propagation means here the increase or decrease of a droplet-like
configurat,ion of a new steady state a t the origin of coordinates imbedded in the alternative stable state. If R(t) is the radius of the droplet instead of the constant speed the
front moves with the R-dependent velocity [6, 101
v(R) = R = ~ (-lRLIR).
(2.8)
Rkplays the role of a critical radius like in nucleation theory. This case will be considered
in a separate paper. Generally for a n arbitrary #-shaped, function W ( n ) eq. (2.2) is
difficult to solve. On the other hand, the case where W ( n ) is a polynom of the third
power is quite well investigated and an analytic solution of eq. (2.2) is given by MONTROLL
[ll],see also [lo, 121:
nO(E)= TL,
+
(fi3
- nl) [1
+ exp (B5)1-1
(2.9)
with
The speed of the front can be calculated from (2.9) as [ l a ]
v =
vz
2 (n, + n, - 2
4 .
(2.10)
There exist two characteristic lengths in the deterministic problem. Looking for the
width of the interface of the profile we obtain
-
I, = 4
f?
(n3- nl)-l.
(2.11)
The second length is found from the speed of the front
(2.12)
and corresponds t o the critical radius in nucleat,ion theory. It is of interest to point out
that the ratio of these two lengths does not depend on the diffusion coefficients and is
defined only by the supersaturation of the system :
(2.13)
Effect of Fluctuation on Front Propagation in Bistable Systems
281
We introduce here the relative supersaturation
(2.14)
which is one of the main values determining the behaviour of the system.
3. Stochastic Motion of tho Propagating Front
Following (1.4) we now assume that the reaction-constants are fluctuating values
[13]. If the variations 8ko(x,t ) . .. Sk&, t ) from the mean values k: may be supposed
small the rate function W can be expanded in terms of Ski (i = 0, .. . , 2) and it will be
sufficient to keep only the linear term
W ( n ;k, ..., k,) = W ( n ;k:, ..., k!)
2
These variations would induce fluctuations of the local density 72 near the steady profile
n?(Q of the propagating front which due to the smallness of 6ki also would be supposed
small :
4 2 , t ) = no(E)
6n(z,t ) ,
(3.2)
Then their evolution would be described by the linear version of eq. (1.3)
+
or if we change to the inertial system
-asn
_ - i.sn
at
here
+ 5 (5, t ) ,
3 is the stationary linear operator
g(5, t ) is the fluctuating random field given by the expression
(3.5)
The linear operator 3 is not hermitian but it possesses a complete set of eigenfunctions
@#) (see [14]) and therefore 6n and 8 may be decomposed as3)
WE,t ) = 2;.cj(t)@ j ( E )
x 5 , t ) = z;.&(t) @j(O
(3.6)
(3-7)
*
If l i are the eigenvalues of the operator
I;
mi= -lpj
(3.8)
the decomposition coefficients Ci(t)obey the stochastic differential equations
C,(t) = --liCj
+ &(t).
(3.9)
3) Eigenfunctions dii and eigenvaliies li of the linear operator r can be found in an analytical
way for a very important case where W ( n )is given by a cubic polynomial, i.e. for the Schliigl model.
The analytical expressions for Gjand di are obtained in ref. [15, 161.
L. SCHIW4NSKY-GEIERet
282
81.
It is known [ 141 that eigenfunctions of nonhermitian operators cannot be expected to
be orthogonal. However there exists a biorthogonality relationship between the sets
of eigenfunctions of hermitially adjointed operators. In our case the hermitially adjointed
operator
(3.10)
has eigenvalues and eigenfunctions @ ( f )
f++Qi
(3.11).
= AiQi
with properties
2 = 17;p i , Gi)
-
(3.12)
dii,
where dii is the Kronekker-Symbol. It can also directly be verified that due t o the special
form of the operator r^ there is the following algebraic relation between the two sets of
eigenfunctions
(3.13)
W t )= exp ( ~ 5 1 0@@).
)
With (3.11). .. (3.12) and (3.6) we find the decomposition coefficients of the fluctuating
field
(3.14)
Invariance under transformations of the front which do not modify its profile implies
that among the eigenvalues of the operator I
' there should be present the value Lo = 0
which corresponds to the eigenfunction
(3.15)
It will be useful t o distinguish in the decomposition series (3.6) the part with j = 0
(3.16)
Since
we can rewrite (3.16) as well in the form
4 5 ,t ) = nO(E+ Co(t))+ dn,(5,t).
(3.17)
We argue here that there are two principal effects induced by stochastic variations of
the parameter of the medium. A t first they lead to stochastic deviations of the front
shape which are described by the term dn,(t, t ) . Secondly they excite a random motion
of the position of the front.
This position at the moment t is given by
(3.18)
x ( t ) = xo
vt - Co(t)
+
and C, obeys the stochastic differential equation
co = So(4.
(3.19)
Co itselfes gives the random contribution to the propagation speed v of the front.
Effect of Fluctuation on Front Propagation in Bistable Systems
283
Due to (3.5) and (1.5) we have <30(t))
= 0 and therefore there would beno average
shift4) of the front in comparison with the deterministic position xo vt. However, the
propagating front would perform a stochastic motion of the diffusional type. The mean
dispersion of the front shifts A s = x(t) - xo - vt at the moment t is
+
t
(Ax2),=
t
J / (50(4)30@2)dtl dt2
0 0
so that in the long time limit t
ly with time
( A d ) , = 9* t .
(3.20)
min {-re, lJv>(comp. (1.6)) the dispersion increases linear(3.21)
The effective diffusion constant SD of the stochastic wandering of the propagating front
is determined from eqs. (3.20) and (3.14) [13]
(3.22)
This formula is very general since in its derivation no assumptions about the statistical
character of the random fields were made and therefore eq. (3.22) is valid for Gaussian,
Poissonian and other cases.
4. Application
We now intend to apply the theory t o the very important case where W ( n )is a polynoin of the third power. The advantage of this problem is that no(t)is explicitly known.
There are many problems that lead in first approximation t o polynoms of the third
power near a critical point:
8
W ( n )=
2 ktuini.
i=O
We have sketched the deterministic theory of front propagation in media with (4.1) a t
the end of section 2.
We assume further the correlation function of the following kind:
Expression (4.2) is written in such a form that for a small correlation time z, or for a
small correlation length 1, the exponential functions reduce t o Dirac-&functions with
weights Ci
lim Si= Ci6(t - t ' ) 6(r - d).
(4.3)
I , +o
Tc +o
4)
Here we average over a statistical ensemble of propagating fronts with the same initial position.
L.SCHIMANSKY-GEIEB
et al.
384
The calculation of the denominator of eq. (3.22) with
dno
_ -_--. n3 - n1
d&
1,
1
cosh2 (2E/Z0) '
where 1, is due to (2.11) can be easily peformed. We obtain
which converges for arbitrary a.
Next the time integration with (4.2) in (3.22) may be carried out:
Here
x =
vz: + 7?z:
appears as an effective correlation length. During the time z, the correlation is transported
by the front with velocity v which leads t o the new correlation length vz,..
Further analysis of (4.5) can be done analytically for the two limit cases distinguished
by the value of the ratio %/lo.
1. x / l o + 0: I n this case action of fluctuations is localized and therefore the first
part of the exponential function in (4.5) may be written as a &function. The analysis
leads t o a n integral which can be calculated. We obtain
91,
sin2 xa
9=
(n3- n1)2'( n o r ) 2 (1 - a 2 ) 2
(4.7)
The expression (4.7) is always finite excluding limit situations of marginal stability
101 I = 1. I n these situations the product of r-functions which contains expressions of the
form norisin no1 diverges.
2. x / l , -+00 :I n this case the fronts are narrow in comparison with fluctuation ranges.
It is sufficient to approximate the first part of the exponential function by one and we
obtain:
Again in limit situations I o1 I --f 1 the diffusion coefficient of the stochastic wandering
of the front becomes infinitely large. On the other hand the effect of fluctuations in
situations differing from (a1 = 1 is small due to the proportionality of 9 to ldx.
I n a general case of an arbitrary ratio x/lo we find only the first term in a series with
i=O:
Effect of Fluctuation on Front Propagation in Bistable Systems
285
with
(4.10)
Results of computing (4.10) are-shown in Fig. 2. We again obtain the divergence for
strong supersaturation I oc I +-1. For a sharp front situation 5B becomes small.
Higher terms of the series may be computed as well. But they depend on the average
reaction rates kp too and we will restrict our investigation here to limit situations (4.7)
and (4.8).
Fig. 2. Results of computing J,,
0; b ) -1:= l ;
4x2
~ ) 20"- = 2
4x2
The divergence of 5B at marginal points of stability I a I = 1 is of course of unphysical
manner. Necessarily here nonlinear effects must be included and investigations based
on linear operators fail. Near to the marginal points of stability effects of fluctuations
also lead to large stochastic variations of the front shape. This is due to the existence of
slowly vanishing modes of the spectrum of eigenvalues of the T-operator (comp. [15, 161
for the case, where W ( n )is a polynom of the third power).
References
EBELINQ,
W. ; FEISTEL,R. : Physik der Selbstorganisation und Evolution. Berlin: AkademieVerlag 1982.
I. : Selforganization in Nonequilibrium Systems. New York: WileyNICOLIS,G. ; PRIQOQINE,
Interscience 1977.
G.; EBELINQ,
W.: J. Nonequ. Thermodyn. 2 (1977) 1.
CZAJKOWSKI,
F R A N K - ~ E N E TD.
ZK
A.Y
:Diffusion
,
and Heat Transfer in Chemical Kinetics. Moscow: Nauke
1967.
SCHLOOL,
F. : Z. Phys. 263 (1972) 147.
PATASCHINSKY,
A. S.; SCHWMILO,
V. I.: Zh. Eksp. Teor. Fiz. 77 (1979) 1417.
286
L. SCJ~IMAMSKY-GEIER
et al.
[7] SURKO,
C. M.; SIMONS,
A. L.; AUSTON,
D. M.; GOLOVCHENKO,
J. -4.; SLUSHER,
R. E. ; VENKATENSA, T.N. c.: Appl. Phys. Lett. 34 (1979) 635.
[8] FEISTEL,
R.; EBELINQ,W.: Biosystems 16 (1982) 291.
[9] EBELINQ,W. ; SONNTAQ,
I. ;SCHIMANSKY-GEIER,
L. : Studia Biophys. 84 (1981) 87.
[lo] SCEIMANSKY-GEIER,
L.; EBELINQ,
W.: Ann. Physik (Leipzig) 40 (1983) 10.
[ll]MONTROIL,
E. W.: In: Statistical Mechanics. RICE,S. A.; FREED,
K. F.; LIQHT,J. C. eds.
Chicago: University of Chicago 1972.
[12] NITZAN,A.; ORTOLEVA,P.; Ross, J.: Proc. Faraday Symp. of the Chem. SOC. 9 (1974) 241.
1131 MTnrrArr.ov, A. S.; SCHIMANSKY-GEIER,
L.; EBELINQ,
W.: Phys. Lett. 96 A (1983) 463.
[14] Lax,M.: Fluctuations and Coherent Phenomena, in: Statistical Physics, Phase Transitions, and
Superfluidity. CHRETIEN
et al. eds. London: Gordon and Breach 1968.
[15] ~M~QYARI,
E.: J. Phys. A 16 (1982) L139.
[16] SCHLOGL,
F.; ESCHER,
C.; BERRY,R. S.: Phys. Rev. A 27 (1983) 2698.
Bei der Redaktion eingegangen am 1. September 1983.
Anschrift d. Verf.: Dr. L. SCEIMANSKY-GEIER
Humboldt-UniversitLt zu Berlin
Sektion Physik, Bereich 04
DDR-1040 Berlin, Invalidenstr. 42
,
Документ
Категория
Без категории
Просмотров
0
Размер файла
458 Кб
Теги
front, effect, bistable, propagation, system, fluctuations, plan, nonequilibrium
1/--страниц
Пожаловаться на содержимое документа