Dynamics of a stochastic linear-growth equation with spatiotemporal correlated noise
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Dynamics of a stochastic linear-growth equation with spatiotemporal correlated noise Ning-Ning Pang Department of Physics, National Taiwan University, Taipei, Taiwan, Republic of China ~Received 10 July 1997! The Edwards-Wilkinson ~EW! equation @Proc. R. Soc. London, Ser. A 381, 17 ~1982!# with long-range spatiotemporal noise correlation of the power-law form for arbitrary substrate dimensionality is analyzed in detail. We obtain in closed form not only the interfacial width but also various correlation functions charac- terizing the system. This information can be employed for the comparison with experiments directly, permit- ting determination of various parameters. Furthermore, we demonstrate explicitly that the interface described by the EW equation with correlated noise for arbitrary substrate dimensionality is still a self-affine fractal governed by dynamic scaling. @S1063-651X~97!07112-2# PACS number~s!: 05.40.1j, 81.10.Aj, 02.50.2r The kinetic roughening phenomenon of growing inter- faces in random media has brought about much interest for its widespread applications in nature @1#. Mathematically, the partial differential equations with stochastic noise compo- nents are often used to describe the continuum limit of the evolution of the growing interfaces @1#. One of the best- known examples is the Edwards-Wilkinson ~EW! equation @2#, consisting of two terms: the Laplacian component and the white-noise component. The EW equation has been widely accepted as a standard model appropriate to kinetic roughening phenomena in which the competing effects are surface tension relaxation and the incessant peppering of sto- chastic noise. Recently, it was pointed out @3,4# that the spa- tiotemporal white-noise assumption for the EW equation is invalid if the substrate dimensionality is higher than two. Moreover, it shows @3,4# that the noise correlation must be less singular than the d function in order to obtain the physi- cally reasonable solution. This motivates us to study the EW equation with long-range noise correlation in the higher ~than two! substrate dimensionality in order to see how the interfacial behavior varies with the different noise correla- tion. In addition, it has also been conjectured recently @5# that the EW equation with temporal correlated noise is suit- able for describing the interface roughening in driven mag- netic systems with quenched disorder, of which the experi- mental realization is a diluted antiferromagnet in an applied magnetic field. Jost and Usadel @5# conjectured that the ran- dom fields at the moving interface play the role of an effec- tive noise. The correlations between the random fields at the interface have been measured numerically @5# to be tempo- rally power-law correlated, u t 2t 8 u 20.8 . Therefore, a rigorous study of the EW equation with correlated noise is the very next step necessary for establishing the whole theoretical foundation related to the EW equation. In this paper we give a comprehensive analysis of the EW equation with long-range spatiotemporal correlated noise of the power-law form for arbitrary substrate dimensionality. We specifically choose the noise correlation to be long-range correlated of the power-law form for two reasons. First, it is pointed out in Ref. @8# that the power-law correlated noise may represent the effect of removing fast degrees of free- dom. Second, the results of this paper then can be used to clarify whether the conjecture @5#, related to the domain wall roughening driven through a medium with random fields, is correct. The calculations in this work generalize the previous results in Ref. @6#, which mainly deals with the EW equation with spatial correlated noise in 1 11 dimensions. The outline of this paper is as follows. We first review the usual tech- nique of solving the stochastic diffusion-type equations and point out its limitations. Then we study extensively the in- terfacial width and the various correlation functions. We ob- tain not only the values of the critical exponents but also the exact expressions of the scaling functions for arbitrary sub- strate dimensionality and noise correlation of the power-law form. Finally, we discuss the implications of the results ob- tained and their applications related to experiments. Here we consider the interfacial profile h ( x, t ) in d 11 dimensions, governed by the EW equation with spatiotempo- ral correlated noise: ] t h ~ x, t ! 5n „ 2 h ~ x, t ! 1h ~ x, t ! , ~1! where h ( x, t ) is the interface height, x is a coordinate in the d -dimensional substrate, and h ( x, t ) represents the noise. The noise h ( x, t ) has zero mean and its correlated variance is given as ^ h ~ x, t ! h ~ x8 , t 8 ! & 52 Du x2x8 u 2 r 2d u t 2t 8 u 2 u 21 , ~2! with 0 ,r ,d /2 and 0 ,u , 1 2 to account for the decaying long-distance noise correlation. For the self-containment of the paper, we first briefly review the usual technique of deal- ing with the stochastic diffusion-type equations, which has been mentioned in the previous work of Ref. @6#. As usual, by applying a Fourier transformation to k space, we get ] t h ˜ ~ k, t ! 52n k 2 h ˜ ~ k, t ! 1h ˜ ~ k, t ! . ~3! The solution for Eq. ~3!, under the assumption of the flat initial condition, is easily obtained h ˜ ~ k, t ! 5e 2n k 2 t E 0 t e n k 2 t h ˜ ~ k, t ! d t . ~4! Here the correlated variance between noises in k space is PHYSICAL REVIEW E DECEMBER 1997 VOLUME 56, NUMBER 6 56 1063-651X/97/56~6!/6676~4!/$10.00 6676 © 1997 The American Physical Society
Transcript of Dynamics of a stochastic linear-growth equation with spatiotemporal correlated noise
PHYSICAL REVIEW E DECEMBER 1997VOLUME 56, NUMBER 6
Dynamics of a stochastic linear-growth equation with spatiotemporal correlated noise
Ning-Ning PangDepartment of Physics, National Taiwan University, Taipei, Taiwan, Republic of China
~Received 10 July 1997!
The Edwards-Wilkinson~EW! equation@Proc. R. Soc. London, Ser. A381, 17 ~1982!# with long-rangespatiotemporal noise correlation of the power-law form for arbitrary substrate dimensionality is analyzed indetail. We obtain in closed form not only the interfacial width but also various correlation functions charac-terizing the system. This information can be employed for the comparison with experiments directly, permit-ting determination of various parameters. Furthermore, we demonstrate explicitly that the interface describedby the EW equation with correlated noise for arbitrary substrate dimensionality is still a self-affine fractalgoverned by dynamic scaling.@S1063-651X~97!07112-2#
PACS number~s!: 05.40.1j, 81.10.Aj, 02.50.2r
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The kinetic roughening phenomenon of growing intefaces in random media has brought about much interesits widespread applications in nature@1#. Mathematically, thepartial differential equations with stochastic noise compnents are often used to describe the continuum limit ofevolution of the growing interfaces@1#. One of the best-known examples is the Edwards-Wilkinson~EW! equation@2#, consisting of two terms: the Laplacian component athe white-noise component. The EW equation has bwidely accepted as a standard model appropriate to kinroughening phenomena in which the competing effectssurface tension relaxation and the incessant peppering ofchastic noise. Recently, it was pointed out@3,4# that the spa-tiotemporal white-noise assumption for the EW equationinvalid if the substrate dimensionality is higher than twMoreover, it shows@3,4# that the noise correlation must bless singular than thed function in order to obtain the physically reasonable solution. This motivates us to study theequation with long-range noise correlation in the high~than two! substrate dimensionality in order to see how tinterfacial behavior varies with the different noise corretion. In addition, it has also been conjectured recently@5#that the EW equation with temporal correlated noise is sable for describing the interface roughening in driven mnetic systems with quenched disorder, of which the expmental realization is a diluted antiferromagnet in an applmagnetic field. Jost and Usadel@5# conjectured that the random fields at the moving interface play the role of an effetive noise. The correlations between the random fields atinterface have been measured numerically@5# to be tempo-rally power-law correlated,ut2t8u20.8. Therefore, a rigorousstudy of the EW equation with correlated noise is the vnext step necessary for establishing the whole theorefoundation related to the EW equation.
In this paper we give a comprehensive analysis of theequation with long-range spatiotemporal correlated noisethe power-law form for arbitrary substrate dimensionaliWe specifically choose the noise correlation to be long-racorrelated of the power-law form for two reasons. First, itpointed out in Ref.@8# that the power-law correlated noismay represent the effect of removing fast degrees of frdom. Second, the results of this paper then can be useclarify whether the conjecture@5#, related to the domain wal
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roughening driven through a medium with random fields,correct. The calculations in this work generalize the previoresults in Ref.@6#, which mainly deals with the EW equatiowith spatial correlated noise in 111 dimensions. The outlineof this paper is as follows. We first review the usual tecnique of solving the stochastic diffusion-type equations apoint out its limitations. Then we study extensively the iterfacial width and the various correlation functions. We otain not only the values of the critical exponents but alsoexact expressions of the scaling functions for arbitrary sstrate dimensionality and noise correlation of the power-lform. Finally, we discuss the implications of the results otained and their applications related to experiments.
Here we consider the interfacial profileh(x,t) in d11dimensions, governed by the EW equation with spatiotemral correlated noise:
] th~x,t !5n¹2h~x,t !1h~x,t !, ~1!
whereh(x,t) is the interface height,x is a coordinate in thed-dimensional substrate, andh(x,t) represents the noiseThe noiseh(x,t) has zero mean and its correlated variancegiven as
^h~x,t !h~x8,t8!&52Dux2x8u2r2dut2t8u2u21, ~2!
with 0,r,d/2 and 0,u, 12 to account for the decaying
long-distance noise correlation. For the self-containmenthe paper, we first briefly review the usual technique of deing with the stochastic diffusion-type equations, which hbeen mentioned in the previous work of Ref.@6#. As usual,by applying a Fourier transformation tok space, we get
] th~k,t !52nk2h~k,t !1h~k,t !. ~3!
The solution for Eq.~3!, under the assumption of the flainitial condition, is easily obtained
h~k,t !5e2nk2tE0
t
enk2th~k,t!dt. ~4!
Here the correlated variance between noises ink space is
6676 © 1997 The American Physical Society
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56 6677DYNAMICS OF A STOCHASTIC LINEAR-GROWTH . . .
^h~k,t !h~k8,t8!&52Druku22rdd~k1k8!ut2t8u2u21.~5!
For d51, the prefactorDr is simply
Dr5D
p E0
`
du u2r21 cosu5D
pG~2r!cos~rp!. ~6!
For d.1, the prefactor
Dr5DSd21
~2p!d E0
`
du u2r21E0
p
du sind22u exp~ iu cosu!
5D
ApSd21GS d21
2 D~2p!d E
0
`
du u2r21S 2
uD ~d22!/2
3J~d22!/2~u!
5DAp22r21Sd21
~2p!d
GS d21
2 DG~r!
GS d
2212r D , ~7!
with Sd21 , G, and J(d22)/2 denoting the surface area of(d21)-dimensional unit sphere, the Gamma function, athe Bessel function of the first kind@7#, respectively. Notethat 0,r,(d11)/4 is required for the convergence ofDr .Consequently, the application of the Fourier method isstricted to the range in which the value of the noise spacorrelation exponentr is between 0 and (d11)/4. To theauthor’s knowledge, the limitation for applying the Fouritechnique has not been discussed in the literature. The aprocedure is a direct generalization of the previous results@6#to arbitrary substrate dimensionality.
The interfacial widthW(L,t) is defined as
W2~L,t ![L2dK E0
L
ddx@h~x,t !2hL~ t !#2L , ~8!
with L denoting the lateral segment of the surface,hL(t)[L2d*0
Lh(x,t)ddx, and^ & the statistical average. The nottions for the interfacial width are in accordance with tnotations in Ref.@6#. Substituting Eqs.~4! and ~5! into Eq.~8!, we obtain
W2~L,t !5L2r14u122df ~nt/L2!. ~9!
Consequently, the steady-state roughness exponentx5r12u1(22d)/2, revealing a spatiotemporal symmetthrough the combinationr12u, while the dynamical expo-nentz52, independent of the values of the noise correlatexponents~r and u!. In contrast, the Kardar-Parisi-Zhan~KPZ! equation with correlated noise@8,9# does not share theabove symmetries, due to the presence of the nonlinear(“h)2 in the KPZ equation.
In Eq. ~9!, the scaling function
d
-l
ve
n
rm
f ~nt/L2!5Dr
un2u11 E ddq
q2r14u12
3S 12)i 51
d222 cosqi
qi2 D g~q2nt/L2!, ~10!
where the notations are defined asq[(q1 ,q2 ,...,qd) andq[uqu and the functiong(y5q2nt/L2) has the form
g~y!5E0
y
du u2ue2u1e22yE0
y
du u2ueu
5g~2u11;y!1~21!2u11e22yg~2u11;2y!,
~11!
with g(2u11;6y) denoting the incomplete gamma funtion @7#. When t→`, g(q2nt/L2) approaches a constanG(2u11). Thus we find the scaling function
f ~nt/L2!u t→`5Dr
un2u11 G~2u11!E ddq
q2r14u12
3S 12)i 51
d222 cosqi
qi2 D , ~12!
which gives us the amplitude prefactor for the steady-sinterfacial width. Note that the restriction (d22)/2,r12u,d/2 is necessary to avoid the divergence of Eq.~12!. Theabove restriction on the noise correlation in fact implies tthe interfacial roughness exponentx @5r12u1(22d)/2#is positive and less than 1, namely,Wsat/L vanishes in thethermodynamic limit. Thus the kinetically roughened intefacial profile, described by the Edwards-Wilkinson equatiwith long-range spatiotemporal correlated noise, is ssmooth on macroscopic length scales. In contrast,Mullins-Wolf-Villain equation with correlated noise@10#
] th~x,t !5n¹4h~x,t !1h~x,t ! ~13!
behaves anomalously in the sense that the roughness enentx is larger than 1 for 111 and 211 dimensions. Typi-cally, anomalously scaling interfaces suggest multiscalbehaviors of the system@11#. However, a rigorous theoreticatreatment is still lacking at the present time.
In the following, we calculate in detail the various corrlation functions. The notations for correlation functions ain accordance with the notations in Ref.@10#. We are inter-ested in the height difference correlation function, whichdefined as
Cg~x,t,t0
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6678 56NING-NING PANG
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56 6679DYNAMICS OF A STOCHASTIC LINEAR-GROWTH . . .
Through the studies ofG(x,t0) and Cs(x,t), we explicitlydemonstrate that the spatiotemporal correlations of surfluctuations are invariant under the rescalingx→bx, t→bzt,andh→bxh, regardless of whether the system is in the trasient regime or in the steady state. So the system descrby the EW equation with long-range spatiotemporal corlated noise is indeed a perfect example affirming thenamic scaling hypothesis.
Furthermore, motivated by the practical interest in desition techniques such as molecular-beam epitaxy~MBE!,various stochastic differential equations and cellulartomata@1# have been proposed recently to describe surfgrowth via MBE in different environments. Our results rgarding correlation functions then can be used to extvarious phenomenological parameters, e.g.,r, u, andD/n2u11, through comparisons with discrete growth modor experiments such as scanning tunneling microsc~STM! data or diffuse x-ray reflectivity measurements. Fexample, the noise correlation exponents~r andu! either aregiven in discrete growth models or can be obtained byperimental measurement of disorder correlation. Thenmeasurement of the equal-time height difference correlafunctions of self-affine surfaces in the steady state provius the information to determine the value ofD/n2u11, pro-vided the noise correlation exponents are known.
On the other hand, it is conjectured@5# that the EW equa-tion with temporal correlated noise is suitable for describthe interface roughening, above the depinning transitiondriven magnetic systems with quenched random fieldswhich the experimental realization is diluted antiferromanets in applied magnetic fields. The correlations between
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random fields at the interface, playing the role of an effectnoise, have been measured numerically@5# to be temporallypower-law correlatedut2t8u20.8. Thus our results are alsuseful to prove or disprove the above conjecture througcomparison with data generated from numerical models, ea comparison of the asymptotic functional forms of the crelation functions. The details will be discussed elsewh@12#.
In conclusion, we analyzed in detail the EdwardWilkinson equation with long-range spatiotemporal nocorrelation of the power-law form for arbitrary substrate dmensionality. We obtained in closed form not only the intefacial width but also various correlation functions charactizing the system. This information can be employed for tcomparison with experiments directly, permitting determintion of various parameters. Furthermore, we demonstraexplicitly that the interface described by the EW equatiwith correlated noise for arbitrary substrate dimensionalitystill a self-affine fractal governed by dynamic scaling andreveals a spatiotemporal symmetry through the combinar12u. Definitely, the most valuable problem is to obtain tfull probability functional for the interfacial profile. How-ever, the derivation of the temporal evolution of the fuprobability functional for the interfacial profile is much mortechnically challenging since the noise here is long-ranspatiotemporal correlated. Work on this problem is plannfor the near future.
This work was supported in part by the National ScienCouncil of Republic of China under Grant No. NSC 82112-M-002-019.
.
A
.,
@1# See, for example, A.-L. Baraba´si and H. E. Stanley,FractalConcepts in Surface Growth~Cambridge University PressCambridge, 1995!; T. Halpin-Healy and Y.-C. Zhang, PhysRep.254, 215 ~1995!; J. Krug, Adv. Phys.46, 139 ~1997!.
@2# S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London, SA 381, 17 ~1982!.
@3# K. Honda, Phys. Rev. E55, R1235~1997!.@4# K. Honda, Fractals4, 331 ~1996!.@5# M. Jost and K. D. Usadel, Phys. Rev. B54, 9314~1996!.@6# Y.-K. Yu, N.-N. Pang, and T. Halpin-Healy, Phys. Rev. E50,
5111 ~1994!.@7# See, for example,An Atlas of Functions, edited by J. Spanie
and K. B. Oldham~Springer-Verlag, Berlin, 1987!; Integrals
r.
and Series, edited by A. P. Prudnikov, Y. A. Brychkov, and OI. Marichev ~Gordon and Breach, New York, 1986!.
@8# E. Medina, T. Hwa, M. Kardar, and Y.-C. Zhang, Phys. Rev.39, 3053~1989!.
@9# N.-N. Pang, Y.-K. Yu, and T. Halpin-Healy, Phys. Rev. E52,3224 ~1995!; J. G. Amar, P.-M. Lam, and F. Family, PhysRev. A 43, 4548~1991!; C.-K. Peng, S. Havlin, M. Schwartzand H. Stanley,ibid. 44, R2239~1991!.
@10# S. Majaniemi, T. Ala-Nissila, and J. Krug, Phys. Rev. B53,8071 ~1996!.
@11# S. Das Sarma, S. V. Ghaisas, and J. M. Kim, Phys. Rev. E49,122 ~1994!.
@12# N.-N. Pang~unpublished!.
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