Linear Instabil i ty Mechanisms of Noise - Induced Phase Transit ions

Marta Ibafies 1, Jordi Garc ia-Ojalvo 2, Rafil Toral 3, and Jos@ M. Sancho 1

1 Departament d'Estructura i Constituents de la Matbria, Univ. de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Spain

2 Departament de Ffsica i Enginyeria Nuclear, Univ. Polit@cnica de Catalunya, Colom 11, E-08222 Terrassa, Spain

3 Departament de Fisica, Univ. de les Illes Balears, and Instituto Mediterr£neo de Estudios Avanzados (IMEDEA), E-07071 Palma de Mallorca, Spain

Abs t r ac t . We review the role of linear instabilities on phase transition processes induced by external spatiotemporal noise. In particular, we present a detailed linear stability analysis of a standard Ginzburg-Landau model with multiplicative noise. The results show the well-known constructive role of fluctuations in this case. The analysis is performed for both non-conserved and conserved dynamics, correspond- ing to order-disorder and phase separation transitions, respectively.

1 I n t r o d u c t i o n

Among all counterintuitive influence tha t external noise can exert on dynam- ical systems, spatial ordering has received a special a t tent ion in last years. Noise-induced patterns, for instance, have been observed in Swift-Hohenberg models in the presence of external fluctuations [1-3]. The mechanism through which the pat tern-forming instability arises is linear, and the role of noise (which as an external noise has to be interpreted in the Stratonovich sense) is to renormalize the coefficients of the corresponding dispersion relation [4, 5]. Subsequent investigations showed the existence of noise-induced phase t ran- sitions in field models 1 [6, 7], some of which were a t t r ibuted to a short- t ime instability tha t survives through observable t ime scales due to entrainment caused by spatial coupling [8]. However, it can also be shown in these cases tha t the mechanism through which noise destabilizes the disordered phase is again linear [9, 10]. In the following pages, we review in detail the influence of multiplicative noise in the linear destabilization of a homogeneous phase, which leads to the appearance of a noise-induced phase transition. This linear instability mechanism is not unique, since in some cases the destabilization is driven by a nonlinear mechanism [11]. Nevertheless, those situations are still a minority, and will not be considered in the following.

1 Throughout this paper, we use the term "phase transition" in the statistical- mechanics sense, to denote transitions between macroscopic states that display universal properties in the thermodynamic limit.

J .A. ~ ' e u n d a n d T. P6sche l (Eds.) : L N P 557, pp. 247 256, 2000. (~) Sp r inge r -Ve r l ag Ber l in He ide lbe rg 2000

248 M. Ibafies et al.

The stability analysis will be performed for two kinds of dynamics, the first of which is a standard relaxational model that evolves towards one of two phases (ordered or disordered) with no restriction. In the second case, the dynamics is restricted by a conservation law of the spatially averaged field, which leads in the ordered state to a process of phase separation. Whereas the linear stability analysis of the former, non-conserved systems is so far well known [4], that of the conserved case is introduced here in more detail.

2 Linear Stability Analysis

Transitions between two macroscopic phases in a given system occur due to the loss of stability of the initial state for certain values of the control pa- rameters. It is well known nowadays that some types of noise can modify the stability of a state, and thus change the parameter values at which the tran- sition takes place (i.e. the transition point). In many cases, the mechanism through which the destabilization arises is linear, and therefore the location of the corresponding transition point can be found by means of a linear sta- bility analysis. This analysis consists on studying the dynamical behavior of a perturbation applied initially to the state whose stability is being examined. In a linear approximation, valid only at short times, these perturbations ei- ther grow or decay exponentially in time. In the first case the initial state is unstable, in the second one it is stable.

In the case of stochastic systems, the linear stability analysis needs to be performed on a statistical moment of the pertubed state. Contrarily to homogeneous (zero-dimensional) systems, the onset of instability for spa- tial[y extended systems is the same for all statistical moments (at least when moded-coupling contributions are discarded) [5]. It is especially interesting in this case to perform the analysis on the structure ]unction, since this quantity (which is the Fourier transform of the second statistical moment) is proportional to the intensity of scattered light in X-ray and neutron diffrac- tion experiments. Moreover, for conserved systems (whose first moment is constant in time) we are forced to study the second statistical moment (or a higher-order one).

We will now perform the linear stability analysis of the structure function for the particular case of models A and B (using the terminology of criti- cal dynamics [12]) with spatiotemporal multiplicative noise. Model A is the prototype of a non-conserved system, model B of a conserved one.

2.1 M o d e l A

This nonconserved model is defined by

0¢(r,t) Ot

- - - a ¢ - ¢ 3 + D V 2 ¢ + ¢ ~ ( r , t ) + r l ( r , t ) , (i)

Linear instability mechanisms of noise-induced phase transitions 249

where both additive and multiplicative noises are Gaussian with zero mean and correlations

(,l(r, t) ~](r', t')} = 2 ~ 5(r - r') ~(t - t') (2a)

(~(r, t) ~(r', t') ) = 2 a 2 c(]r - r '[) 5(t - t') , (2b)

and e and a 2 are the additive and multiplicative noise intensities, respectively. The function c(]r - rrl) is the spatial correlation function of the external noise, which becomes 5(r - r r) in the limit of zero correlation length. This external multiplicative noise can be understood as fluctuations in the control parameter a.

The linear stability analysis is performed in a discrete version of the model. In a d-dimensional discrete lattice of mesh size Ax, (1) takes the form:

d¢i - a C i - C 3 + D E D i j C j + T l i ( t ) +¢ i~ i ( t ) , (3) dt

J where ¢i - ¢(r i ) , r i = A x i , i C [0, L - 1] d and L is the number of cells on each side of the regular lattice. The sum runs over the whole lattice, and only one index is used to label all cells, independently of the dimension of the sys tem./gi j accounts for the discretized Laplacian operator

V2 ~ E D i j - 1 Ax 2 E (Snn(i)J- 2dS,j) , (4) J J

where nn(i) represents the set of all sites which are nearest neighbors of cell i. The discrete noises ~li(t) and ~i(t) are still Gaussian with zero mean and correlations

5ij (~?i(t) ,lj(t')) = 2e ~ 5(t - t') (5a)

(~,(t) ~j(t')} = 2 a 2 cli-jl 5(t - t ' ) , (5b)

where Cli_Jl is a convenient discretization of the function c(Ir - r t l ) , which in the limit of zero correlation length becomes 5ij /Ax d.

In order to study the stability of the homogeneous state (¢i(t) = 0 Vi), we linearize (3) and look for the dynamical equation of the two point correlation function {¢i¢j ):

d (¢ iC j} = - 2 a {¢i¢j} + D E (Dis{¢sCj)+/9 j8{¢8¢i ) ) + $

+ + + (6)

The last four terms of this equation can be calculated with the help of Novikov's theorem [13], which in our case takes the forms:

(~i(t)¢i(t)¢j(t)} = ~ f dt'{~i(t)~(t')) (5(¢i(t)Oj(t)) 5 8(t,) } (Ta)

250 M. Ibafies et al.

(,i(t)¢j(t)) = ~., / dt'(~i(t)~s(t')) \ (fCj(t) ) .

Now, by formally integrating the linear terms of (3),

(7b)

) ¢~(t) = ¢~(0) + dt' - a ¢,(t') + D ~ 15~ ¢~(t') + ,~( t ' ) + ¢~(t') ~(t ' ) 3

(8) we can calculate the response functions at equal times

5¢~(t) t,=t (~¢~(t) t,=t = 5 ~ . (9) ~ ( t ' ) = ¢~(t) ~ , 5,~(t')

Making use of these expressions, (Ta) and (7b) become

(~i(t)Oi(t)¢j(t)) = E cr 2 cli_~ I ((fi~(¢~¢j) + 5j~(¢~0i)) 8

= a 2 (¢i¢j) (Co + cli-jl ) (10a)

( r ] i ( t ) ¢ j ( t ) ) = E ~ S (lOb) -A~x d5i~5~ = Ax~ 5i~ • 8

For the second statistical moment we thus have

8

E (11) +2~ 2 <¢~¢~) (cl,_j , + co) + 2 Z-Tz~ 5~.

The structure function can be defined as

1 ((~t, (t)(~_t, ( t)) (12) S . ( t ) - (LAx)~

where Ct, (t) - ¢ (k . , t) is the Fourier transform of ¢i (t)

1 X-" ei"~k. ~ (13) ¢,(t) = A x d E e - i ' " k " ¢ i , ¢,(t) - (LAx)d A.~ ~ ' ' i tt

with k t, 2~ [0, L - = A~Ltt, and tt E 1] d. From definitions (13) the following relations can be easily verified

E e~k"'("'-"~)= LdSij E e- i (k"-k~)" ' = Lds"'" (14) t~ i

By using definition (13) we can write the dynamical equation for the structure function as

dt E eiU.'(~ -~') (¢i¢j) , (15) i , j

Linear instability mechanisms of noise-induced phase transitions 251

and substituing (11)

a s . _ 2 a S . ( t ) + D Ee k.'('J -'') + dt

i , j , s

+2a2 coS,(t) + 2a 2 - E eik,'(rJ-r')cli_jl(¢iCj) + 2e. (16) i , j

Now we have to rewrite the Laplacian terms. A Fourier-transformed Lapla- cian operator does not couple variables with different moment. In fact, it can be seen that the Laplacian term leads to

where the following relation has been taken into account

E D~8(¢~¢j}- (L2--x)2d Ee i t ' v "Je ik ' " 'Dp(¢ ,¢ . } , (18) 8 V~p

D~ : ~ (7],,1:1 cos (k , , r i ) - 1) can be understood as and the Fourier

transform of the discrete Laplacian. On the other hand, using definition (13) and relations (14), the last contribution of the multiplicative noise in (16) can be written as

z 20-2 -L- eikz.(rj-rl)cii_jl(¢iCj) = 0 - 2 E "d~, S._~(t) , (19)

Therefore, taking into account (17) and (19), the equation for the structure function becomes finally

1 dS~(t)dt -- 2w~S~(t) + 2z + 2a2 (LAx) - - - - - - J E ~s~-~( t ) , (20)

v

with the additional dispersion relation w, = a - a 2 Co - D/)~ . In the con- tinuous and thermodynamic limit (Ax --+ 0 and L --+ c~), the dynamical equation for the structure function is

0 -2 f OS(k,t_____~) _ -2w(k) S(k,t) + 2e + 2 ~ - ~ j dk' c(Ik - k'l) S(k ' , t ) (21)

Ot

with w(k) = a - a 2 c(O) + D k S (22)

Looking at this dispersion relation, it is readily seen that perturbations grow when w(k) < 0 for some interval of k values. Hence, the homogeneous state

252 M. Ibafies et al.

¢ ( r , t) = 0 is stable if w(k) > 0 V k. This is satisfied for a - a 2 c(0) > 0 and thus we can define an effective control parameter

a e f f ---- a - - 6 2 c ( 0 ) (23)

such tha t for aeff > 0, the homogeneous state ¢ ( r , t ) = 0 is stable. The transit ion point is then

at = a 2 c(O) , (24)

which in discrete space is written as at = a2co. This transit ion point in- creases with noise intensity and decreases with noise correlation length be- cause c(0) ~ A -d. In the deterministic case (a 2 = 0 and c = 0) at = 0 and thus, for a > 0 the disordered homogeneous s tate ¢ ( r , t) = 0 is stable, whereas for a < 0 this disordered state is unstable. In the presence of multi- plicative noise at = 0 .2 c ( 0 ) > 0 , and hence the homogeneous disordered state is stable for a smal ler region of the control parameter than in the determin- istic case. Hence, fluctuations in the control paramete r induce order in the system.

There are other techniques that allow us to find the onset of instability. One of them is the study of the s ta t ionary state of the s t ructure function equation given by the linear stability analysis [see (21)]. In contrast with the above discussion, this method takes into account the mode-coupling te rm which depends also on the multiplicative noise. For the part icular case of multiplicative noise white in space [~(Ik - k'l) = 1], the steady s tate for the s tructure function can be obtained from (21)

1 [~ + 2 ast(O) ] (25) s s t ( k ) =

where Gst(r ) = (27r)-d f dke iU'rSs t (k ) is the correlation function. By inte- grating again the above equation, we find tha t the value Gst (0) is

~ ' 1 f dk (26) G s t ( 0 ) - l - a27 ' 7 - (27r) d w ( k ) "

Hence, the resulting s tat ionary structure function is

6" ~ C - , ( 2 7 ) Sst(k) - w(k) ' 1 - a27

where e' is a renormalized additive noise intensity. In the subcritical region, where nonlinear terms are supposed to be negligible, it is expected tha t this linear result agrees satisfactorily with the behavior of the full nonlinear model. Nevertheless this s ta t ionary solution will diverge for 7or 2 = 1, which indicates tha t at tha t point the result is not valid anymore. Hence, the value of the control paramete r at which the s ta t ionary structure flmction diverges is the transit ion point a = at. The condition 7 a 2 = 1 explicitly reads,

a 2 f dk

1 - (2~r) d j ~ d a t - a 2 + D k 2 ( 2 8 )

Linear instability mechanisms of noise-induced phase transitions 253

which is the same as the one found by Becker and Kramer [4]. It is possible to see that the contribution of the mode-coupling term [last term in (21)] to the transition point is quite small (of order ~ -~ in d = 1). Other methods such as mean field approximations also give corrections to the transition point of order D -n.

2.2 M o d e l B

The conserved model corresponding to the one studied in the previous Section is defined by

O¢(r,t) _ V2 [a¢ + ¢3 _ DV2¢+¢~(r,t) ] +Tl(r,t) ' (29) Ot where ¢ ( r , t ) represents, for instance, the local difference of concentrations of each phase in the case of a binary alloy. Multiplicative noise represents fluctuations in the control parameter a. Both additive and multiplicative noises are Gaussian, with zero mean and a correlation for the additive noise given by

(~(r, t) ~(,", t ')) = - 2 e V 2 6(r - r ' ) 5(t - t') (30)

and (2b) for multiplicative noise. The discrete version of the model in a d- dimensional lattice of mesh size Ax is:

d ¢ i - ~ bis [aCs+¢3-DE DsJCJ +¢~(t) J +~i(t), (31)

where ¢i --- ¢(r i ) , as before. The discrete noises ~?~(t) and ~i(t) are still Gaussian, with zero mean and correlations given by

= D ,j - t ' ) ( 3 2 )

and (5b). As done in the previous Section, we will study the stability of¢i ( t ) = 0 Vi,

taking only into account the linear terms in (31). In this case, the dynamical equation for the two-point correlation function (¢i¢j) is

[ s m

+ E Dis -a{¢iCs) -- n E 5 s r n ( ¢ m ¢ i ) -'~ ( ¢ i C s ~ s ) 2r- (?~j~)i)* (33) 8 ~7l

Novikov's theorem allows us to calculate all noise terms in the previous equa- tion. The procedure is the same as before [see (7a)-(10b)] and the results

254 M. Ibafies et al.

are

(¢j¢,~,) = 0.2 E DsmCl,_ml(¢mCj} + 0.2 E/~j ,~Cl,_ml(¢m¢~) (343) r n m

(~?iCj}- eDij (34b) z~x d

Hence, for the second statistical moment we have

+ a 2 ~ DjmCl*--ml{¢mCs}]+ Es L),8 - a ( ¢ i ¢ , ) - D Em Dsm(¢.~¢i)

/)ij (35) + 0 . 2 + - •

m m

We now look for the equation of the structure function. Substituing (35) into (15) and considering relations (14) and (18), we finally find

dSu(t)dt - 2D, [a- Du (D-0.2Cl) +0.2~m DomCm Su(t)

+20. 2 / ) , (AxL)-d Z ~,_,S~ (t) - 2 ¢/9, (36) z]

In the continuous and thermodynamic limit, we have

OS(k, t) k2 ~2 f Ot - 2 w(k)S(k , t )+2ek2-2k2~-~ dk"d(Ik-k'])S(k',t )

with the dispersion relation

w(k) = a + 0.2 [~2c(]rD] r=0

In discrete space, this relation reads

(37)

+ (D - 0.2 c(0)) k 2 . (38)

w u = a + 2d0.2(cl - co) - (D - 32 cl)Du. (39)

The dispersion relation indicates that for w(k) > 0 Vk, the homogeneous null state is stable. This occurs for a + 0.2 IV 2 c([rD] r=o > 0 so that, as in the previous Section, we can define an effective control parameter

aeff = a + 0 .2 [V 2 c(l'PD] r=O ' ( 4 0 )

such that the homogeneous null state is stable for positive values of aeff. Hence, the onset of stability is now given by

at = - a 2 [~ 72 c(Ir]) ] r=0 ' (41)

Linear instability mechanisms of noise-induced phase transitions 255

which in a discrete space is written as

at : 2d a 2 (co - c1) • (42)

As in model A, the transition point in the presence of multiplicative noise is p o s i t i v e and therefore, fluctuations in the control parameter i n d u c e or-

d e r in the system. However, the effective control parameter is different be- tween the two models. In model A, the dependence of the transition point on the spatial correlation of the noise is merely due to a natural "soften- ing" effect of noise correlation [when the noise is spatially correlated, its effective intensity is ~r 2 c(0) ~ a 2 A-d]. In model B, the Laplace operator introduces a more complicated dependence on.the spatial correlation of the noise, at ~ a2A -(2+d). Moreover, the expression of w ( k ) for model B (38) has a noise dependence term that can be considered as a modification of the spa- tial coupling parameter D. We can thus define an effective spatial coupling parameter Deft = D - a 2 c(0). These results axe consistent with those coming from a mean-field approximation in the limit of infinite coupling [14].

As it has been done in the previous Section, we can look for the stat ionary state of the structure function and find a condition for the onset of instability which takes into account, in contrast with the above discussion, the coupling term between Fourier modes. For the case of multiplicative noise white both in time and space (c([k - k'[) = 1), the stat ionary structure function is

Sst(k) = (43)

with ¢1 given by expressions (27) and (26). This solution differs from that corresponding to model A in the expression of w ( k ) [compare (22) and (38)]. The transition point is given by 7or 2 = 1, for which the stat ionary structure function diverges, and is equal at first order to the result given in (41).

For the particular case of a spatial correlation c(]r[) of gaussian type

1 (44) c ( I r [ ) - (Av/~) d exp - 2 A 2 ] ,

whose width A characterizes the correlation length of the noise, and which becomes a delta function for A -+ 0, the transition points for models A and B are, respectively, [see (24) and (41)]

a 2 d a 2

aA -- (27r)d/2 Ad' aB - - ( 2 7 1 - ) 4 / 2 ) ~ d + 2 " (45)

3 C o n c l u s i o n s

Noise-induced phase transitions for which linear destabilization is the domi- nant mechanism have been examined in detail by means of a linear stability

256 M. Ibafies et al.

analysis. The study is performed on both non-conserved and conserved mod- els. In the two cases, noise is seen to have an ordering effect in the system, although the influence is seen to be different in each situation. The role of spatial correlation of the noise is somewhat simple in the non-conserved case, but clearly non-trivial in the conserved model.

Acknowledgments

This work has been financially supported by the Direcci6n General de Ense- fianza Superior, under projects PB94-1167, PB96-0241, PB97-0141-C02-01, and PB98-0935.

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