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

код для вставкиСкачатьAnnalen 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 ,

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