Stochastic dynamics of surfaces and interfaces

Transcription

1 Department of Physics Seminar I - 1st year, II. cycle Stochastic dynamics of surfaces and interfaces Author: Timotej Lemut Advisor: assoc. prof. Marko Žnidarič Ljubljana, 2018 Abstract In the seminar, we are interested in the behaviour of driven stochastic surfaces. We introduce roughness as standard deviation of height of the surface averaged over noise and study its large scale behaviour. From a systematical treatment of a general Langevin equation we arrive at two equations for height, namely Edward-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) equation. They both represent each its own universality class. We state some properties of both equations. We derive critical exponents of the EW class in any dimension and get an explicit equation for roughness. For KPZ we are able to find out the critical exponents only in dimension d = 1.

2 Contents 1 Introduction 1 2 Introduction of roughness and its critical exponents 2 3 Universality classes of random deposition 3 4 Edward-Wilkinson equation 4 5 Kardar-Parisi-Zhang equation 6 6 Discussion 8 1 Introduction We will be interested in behaviour of surfaces in the presence of noise, that are also driven in some way. Surface is described with a height function and is usually realised as some sort of boundary between two phases. Examples could be purely mathematical models or some table-top experiments, for example burning or wetting front of a paper. Other experimental realisations include boundaries between solid and liquid/gaseous phases of matter, domain walls, and boundaries between different phases in liquid crystals. For example, in the case of a boundary between solid and gaseous phase, driving is represented by deposition or evaporation of particles, while noise would be represented by thermal fluctuations. One of the simplest models of random interfaces are deposition models. Below we describe some of those, to which we will reference later in the seminar and which will help to illustrate some notions. They belong to different universality classes, meaning, have different large scale behaviour. Figure 1: Schemas of two models of random interfaces: A - Random deposition model with relaxation, B - Ballistic deposition [1]. The most straightforward is called the random deposition model (RD). It is defined over integers and its evolution rule is: pick a random site and deposit a particle on it. Height is the number of particles already deposited on each site. Adding a rule, that after every deposition on a randomly chosen site, we look at the neighbouring sites of the newly deposited particle and, if possible, we move the particle to one of the sites that are lower than the site on which the particle has been deposited, we get to the model called random deposition with relaxation model (RDR) and one realisation with several possible depositions of a new particle is depicted in figure 1.A. Simple variation of RD would be not to relax a particle after deposition, but apriori restrict the difference in height of each neighbours, so when a deposition of a particle on a chosen site would violate a restriction of maximum height difference, a particle would not be deposited. Such model is called restricted solid on solid model (RSOS) and actually belongs to a different universality class than RDR. In figure 1.B a picture of falling particles in the ballistic deposition model (BD) is displayed, where the particle is again deposited to a randomly chosen site, but sticks to the particles of the neighbouring sites. As we can see, voids and overhangs can form. We also include a picture (Figure 2) of some realisations of all the described models. In the second chapter we introduce roughness of the surface as a standard deviation of height of the surface averaged over noise. We guess that it diverges in the limit of infinite samples and infinite time, so we introduce its scaling form and define the critical exponents describing its divergence. In the next 1

3 Figure 2: Realisations of models: A - RD, B - RDR, C - BD, D - RSOS [2]. We notice a much rougher surface in the case of RD, where the sites are completely independent of each other. Voids and overhangs are visible in the case of BD model. chapter we derive simplest equations that could govern stochastic boundaries. We only use arguments of symmetry and power counting. In the next two chapters, we examine two of such equations. The first one is called Edward-Wilkinson (EW, [3]) equation and is actually just diffusion equation with additive noise, while the other one, called the Kardar-Parisi-Zhang (KPZ, [4]) equation, is nonlinear. We are able to get some intuition about its solutions and even determine critical exponents in dimension d = 1. For EW we are able to do that in arbitrary dimension and even get to the explicit expression for roughness. 2 Introduction of roughness and its critical exponents We imagine a boundary as a surface (for easier analysis without overhangs) defined over volume V in arbitrary dimension d, which we will represent with height function h(x, t), where x is d-dimensional vector and t is time. We define roughness of the surface W (L, t) as a function of its linear size L = V 1/d and time t as W 2 (L, t) = h 2 (x, t) h(x, t) 2, (1) where the overline denotes the spatial average and the average over noise (averaging over realisations). Looking back to the introduced models, for RD, we can already guess how its roughness will behave with respect to L and t. We notice that there is no coupling between different sites, which means that the size and the spatial dimension of the system L are irrelevant. RD is just a collection of independent random walks, where W does not depend on L at all, while dependence of W on time is the usual W (L, t) t. For roughness of finite sample not to diverge with time, there has to be some sort of smoothing mechanism, some interaction between neighbouring points on the interface, so that the finite surface cannot get arbitrarily rough. All other above models have some kind of coupling between sites, so roughness saturates after some critical time and also depends on L, lim W (L, t) = W (L, t ). (2) t For those cases, we expect that by taking an infinite sample, we will nevertheless encounter arbitrarily large deviations in height, lim W (L, t) =. (3) L,t By making that observation, we go on the same route as that of equilibrium phase transitions, by having free energy density replaced with roughness. On the grounds of the above considerations, one 2

4 uses the scaling ( ) { t W (L, t) = L α u β, u 1 w L z, where w(u) const., u 1, (4) and α is called the roughness exponent, β is the growth exponent and z = α/β the dynamic exponent. By using above ansatz we also calculate what kind of scaling transformation produces statistically the same profile. We are checking whether the interface we are describing is possibly self-similar (height profile scales with the same factor as space), or something similar. If we rescale space by a factor of b, time must be scaled by b z, so that the argument of the function w in equation (4) stays the same. Also from the same equation, we have that roughness will scale with b α and the same holds for height also, since it is linear in roughness. Scaling transformations are therefore: x bx t b z t (5) h b α h. Interface which scales in such a way is called self-affine. In the introduction, we also consider dependence of critical exponents on dimension. There exist lower and upper critical dimension, d l c and d u c, which are defined as: α(d l c) = 1 and α(d u c ) = β(d u c ) = 0. (6) Meaning, for d > d u c, fluctuations do not roughen the surface on large time and length scales, while for d < d l c, fluctuations make the surface super-rough. In that case not only does the difference of height diverge but the slope also diverges. 3 Universality classes of random deposition We examine a general Langevin-type equation for a moving boundary + 2 t h(x, t) + t h(x, t) = A[h] + η(x, t), (7) where A is some functional of h and η is white noise, meaning η(x, t) = 0 (8) η(x, t)η(x, t ) = Γδ(x x )δ(t t ). (9) Now we use some sound assumptions to specify A a bit more precisely. First, we assume that A does not depend explicitly on x and t, which would be violated by some time(or space)-dependent driving, for example. Other assumption is that the system is also spatially invariant in the growth direction, meaning under transformation h(x, t) h(x, t) + h 0. If our two conditions are satisfied, our equation looks like + 2 t h(x, t) + t h(x, t) = A[ i h, ij h,... ] + η(x, t) (10) To get even more precise with the form of A, we can check which of its terms are the most relevant on large scales (which is implemented by taking transformation (5) and sending b to infinity). First of all we can get rid of all higher order time derivatives since a term t n h gains a factor b nz by scaling transformation, so only the first time derivative survives. Now for the functional A, which can be formally expanded as A = A 0 + A i i h + A ij i h j h + B ij ij h +..., (11) i=1 i,j=1 we first find out that we can cross out the constant term A 0, by defining new height h (x, t) = h(x, t) A 0 t and also the linear term by taking h (x, t) = h(x At, t), so that we are left with A = A ij i h j h + i,j=1 B ij ij h + i,j=1 i,j,k=1 i,j=1 A ijk i h j h k h +..., (12) 3

5 which we write in condensed notation as A = {2, 2} + {1, 2} + {3, 3} (13) The first number in a pair {p, q} represents a number of times h appears and q is the number of derivatives in each term. We emphasize, that from our previous arguments about spatial invariance in the growth direction, we must have p q, otherwise an explicit dependence on h appears. Under the transformation (5) we have {p, q} b pα q {p, q}. (14) Since p q, the leading term is {p, p}, which transforms as {p, p} b p(α 1) {p, p}. (15) So, for α < 1 or with other words for d l c < d < d u c, we expect the most relevant term of A to be {2, 2}, since we crossed out the terms {0, 0} and {1, 1}. Matrix A ij can be symmetrized and consequently diagonalized. By rescaling every x i by a suitable factor and demanding rotational invariance, we are left with a term proportional to ( h) 2. Such an equation is invariant under x i x i, but not under h h. If we demand for both of these two symmetries to hold, then q must be even and p odd, meaning that the leading term is now {1, 2}, which by the same procedure as before becomes a term proportional to 2 h. By such general consideration of symmetries and power counting, we arrive at two equations which are called Edward-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) equations, respectively: t h(x, t) = ν 2 h + η(x, t) (16) t h(x, t) = ν 2 h + λ 2 ( h)2 + η(x, t), (17) where the diffusive term is also included in the KPZ, to regularize the behaviour at small scales, even though compared to the term proportional to ( h) 2 it is irrelevant at large scales. We also stress that the EW equation is linear, whereas the KPZ equation is not. 4 Edward-Wilkinson equation In this section we examine in some detail the so-called EW equation (16), where ν > 0 and η obeys (8) and (9). First we notice, that the equation can be written in the conserved form t h(x, t) = J + η(x, t), where J = ν h, (18) meaning the average height is a conserved quantity, as can be easily shown dh dt = 1 η(x, t)dx = 0, (19) V V since the integral of the divergence vanishes for periodic boundary conditions. As we can see, we have a simple interplay between noise and diffusion, one roughening and the other one smooting the surface with the current J. From the above introduced models only the RDR belongs to the EW universality class, meaning it has the same large scale behaviour as a solution of the EW equation. If we imagine that height in the RDR actually represents number of particles on each site, then the RDR corresponds to a diffusion-like process with the diffusion constant ν in the presence of uniform supply of particles. We also notice, that the equation can be written with a potential in the form t h = δf δh + η, where F[h] = ν 2 V V ( h) 2 dx, (20) which implies, that the dynamics evolves in order to minimize that potential, as we can compute readily: df dt = δf δh dx (21) V δh δt ( ) 2 δf δf = dx + η(x, t)dx, (22) δh δh V 4

6 so that df 0. (23) dt The meaning of F is also at hand, by remembering that the total area S of surface h(x) is S = 1 + ( h)2 dx = V + 1 ( h) 2 dx + O ( ( h) 4). (24) V 2 V Therefore, neglecting terms irrelevant for large scales we get F = νs and the minimization of potential is just the minimization of total surface area. A consequence of having a potential is that the properties of the stationary state (so when t L z, equation (4)) are equivalent to an equilibrium state described by the Hamiltionian F. We now first show that the critical exponents of EW universality class can be obtained by a simple consideration of units and after that we also indicate how to get an explicit equation for roughness by Fourier transform of the EW equation. First, for units of length, time and height we write, respectively, [L] = L, [t] =T and [h] =H. 1 We now write equation (4), including all posible dependence on Γ and ν, ( W (L, t) = Γ a1 ν b1 L α w Γ a2 ν t ) b2 (25) and impose that the left and right hand side of the equation have the same units, meaning the argument of the function w is dimensionless and [W ] =H. We get the expressions for [Γ] and [ν] from equations (16) and (9), respectively: Using above relations in the expression for W (L, t) we get two equations L z H T = [ν]h 2 = [η] L (26) [η] 2 = [Γ] L d T. (27) ( L d H 2 ) a1 ( ) L 2 b1 L α = H, (28) T T ( L d H 2 ) a2 ( ) L 2 b2 T = L z, (29) T T from which we finally get for the critical exponents of the EW universality class α = 2 d 2, β = 2 d, z = 2. (30) 4 From (6) we can also get lower and upper critical dimensions for EW class to be 0 and 2, respectively. We can also get an explicit form of w, by Fourier transforming the EW equation. Writing h(x, t) = 1 (2π) d e iq x h(q, t)dq, (31) we get the solution of the EW equation to be h(q, t) = h(q, 0)e νq2t + t 0 e νq2 (t t ) η(q, t )dt. (32) For roughness, we will need just the correlator h(q, t)h(q, t), since h(x, t) = 0. We can also change the order of evaluating averages and lose the spatial average, since we know that the system is translationally invariant and the spatial dependence will therefore disappear after averaging over noise: WEW 2 (L, t) = h 2 (x, t) = h 2 (x, t) = h 2 (x, t) = 1 (2π) 2d e i(q+q ) x h(q, t)h(q, t) dqdq. (33) 1 We choose separate unit for height, since it may represent some other quantity, not necessary measured in length units. Also, EW equation describes the dynamics of self-affine interface, not necessarily self-similar, as we have shown above, meaning that the two behave differently and could be measured in different units. 5

7 We obtain the correlator from the solution of EW equation (32) by noticing that the first term of the soution will vanish for initially flat surface, or after some transient time. Correlator of height is therefore calculated from the correlator of noise, which we know from the definition. Putting everything together we get for roughness WEW 2 (L, t) = Γ 1 e (2π) Vq 2νq 2 t d 2νq 2 dq, (34) where we are integrating over V q = {q π L q i π a 0 }, a 0 being a cutoff for avoiding integration over arbitrarily small wavelengths, and is justified by the ultimate discrete nature of matter/our model. We can check that the above expression matches with formulas (4) and (30). By a change of variables s = Lq, we get [ ( ) ] 2 ΓL W 2 2 d νt (L, t) = w 2ν L 2, with w 2 (u) = 1 1 e (2π) Vs 2s 2 u d s 2 ds, (35) where exponents α and z really are the same. To check for β, we make another change of variables y = u 1/2 s, so that w 2 (u) = u 2 d 2 1 e (2π) Vy 2y2 d y 2 dy, (36) where for u 1 the integral goes to a constant and w(u) u 2 d 4, satisfying the relation β = α/z. 5 Kardar-Parisi-Zhang equation The second equation we will consider is called the KPZ equation (17), where again ν > 0 and η is noise obeying (8), (9). Here, we state the key differences from the EW equation: it is nonlinear, it is not symmetric to inversion h h and h(x, t) is not conserved. Namely, if we act with spatial average on the equation we get t h = λ 2 ( h)2. (37) The average height velocity is proportional to square of the slope, meaning that the surface obeying KPZ equation will exhibit some kind of moving interface. All of the above properties are shared with the BD and RSOS model and both of them actually do belong to the KPZ universality class. We can take a look at a deterministic version of the KPZ (without noise term), to see how a parabola, which turns out to be a self-similar solution of the deterministic KPZ, evolves. If we imagine an initial condition for height as a parabola of positive/negative curvature positioned at the origin, we see that the nonlinear term (for λ > 0) will in both cases be positive and get bigger away from the center, while the linear term is positive/negative and constant. All that amounts to the positively curved parabola to move upward and getting more curved with time, while the negatively curved parabola moves downward and flattens. Next, we show that the deterministic KPZ can actually be solved exactly in any dimension with the so-called Cole-Hopf transformation [5], [6], given by H(x, t) = e λ 2ν h(x,t). (38) By inserting in the equation, we get the diffusion equation for H(x, t) with the diffusion coefficient ν, whose solution is known to be 1 (x x H(x, t) = ) 2 e 4νt H(x, 0)dx, (39) (4πνt) d/2 or in terms of height h(x, t) = 2ν λ ln [ ] 1 (x x ) 2 e 4νt + λ 2ν h(x,0) dx. (40) (4πνt) d/2 We would imagine that the same transformation would help with solving the stochastic equation as well, but we get an additional problem of transforming additive noise to the multiplicative one: t H = ν 2 H + λ Hη(x, t). (41) 2ν 6

8 In mathematically not very strict language, the problem of multiplicative noise can be explained as follows: we imagine the noise in the equation as a collection of delta functions and according to the equation, each delta function in the noise term will result in a jump of the function H, where the value of H at the exact time the delta function arrives is undetermined. So the problem with understanding such an equation as above is whether to substitute for H, at the time delta arrives, the value of H before the jump, after the jump or perhaps the mean of both. 2 Next, we notice that we get a familiar equation by taking a spatial derivative of both sides of the KPZ equation. Writing m = h and applying gradient to both sides of the KPZ, we get the equation ( t λ(m ))m = ν 2 m + η (44) which is, for u = λm, like Navier-Stokes equation of an incompressible fluid when pressure effects and external forces can be neglected with extra noise term. We can easily show, that the above equation is invariant under Galilean transformation u (x, t) = u(x u 0 t, t) + u 0. (45) Now we check what that means for the KPZ equation. Writing the above equation in terms of height, we get after integration over the space variable x h (x, t) = h(x u 0 t, t) 1 λ u 0 x + a(t). (46) By inserting above transformation in the KPZ equation, or in the equation (37), we get the expression for a(t) and so we get to the so-called tilt transformation h (x, t) = h(x u 0 t, t) 1 λ u 0 x + u2 0 t, (47) 2λ under which the KPZ equation is invariant. 3 Invariance under tilt transformation can give us some more insight on the KPZ dynamics. Since the tilt transformation (47) explicitly depends on λ and the invariance under that transformation must hold for all length scales, we conclude, that λ cannot be renormalized by a change of scale. 4 Under the transformation (5) each term in the KPZ equation transforms as which we rewrite as b α z t h(x, t) = νb α 2 2 h + λ 2 b2α 2 ( h) 2 + b d+z 2 η(x, t) (49) t h(x, t) = νb z 2 2 h + λ 2 bα+z 2 ( h) 2 + b z 2α d 2 η(x, t). (50) Since λ does not renormalize, we require that b α+z 2 = 1 or α + z = 2. (51) The above relation holds for any d and tells us, that there is only one critical exponent left to be determined. In dimension d = 1, we are actually able to do that, as we show below. 2 There are two popular approaches [7]. Taking the value of H before the jump (Itô): or the mean of values before and after (Stratonovich): H(t + t) = H(t) + ν 2 H(t) + λ t+ t 2ν H(t) η(t )dt (42) t H(t + t) = H(t) + ν 2 H(t) + λ 2ν H(t) + H(t + t) 2 t+ t η(t )dt (43) t 3 Note that the invariance of the KPZ equation under the tilt transformation only holds for white noise. 4 By doing renormalization for parameters ν, λ and Γ we get the flow equation for each one of them, where the equation for λ looks like dλ = λ(α + z 2), (48) dl and l = ln b. So we have two fixed points, either λ = 0 (EW) or α + z = 2. 7

9 We write for our Langevin equation (where N is either EW or KPZ) t h(x, t) = N [h] + η(x, t), where (52) η(x, t) = 0 and η(x, t )η(x, t) = Γδ(x x )δ(t t ), (53) its associated Fokker-Planck equation for the probability P (h, t) of a certain height profile h occuring at time t [ P δ t = N P + Γ ] δp dx. (54) δh 2 δh By solving above equation and getting the expression for P (h, t) we could easily determine roughness, by calculating the first two moments of height. Unfortunately, finding the general solution to our Fokker-Planck equation is not possible. What we can do is try to find the time-independent solution of the Fokker-Planck equation, which describes stationary state attained asymptotically by a finite system. Such a solution would allow us to determine the roughness exponent α. Such procedure can only be done for the EW equation, but we can show that in d = 1, KPZ has the same stationary solution as EW. Looking for a stationary solution P s we are solving the equation (where N is again EW or KPZ) δp s δh = 2 Γ N P s. (55) For the EW equation N EW = δf δh, as we have shown above, with F = ν 2 ( h) 2 dx, so that the stationary solution is = e 2 Γ V N EW dh = e 2 Γ F. (56) P EW s For the KPZ equation, the functional of the height is N KPZ = N EW + λ 2 ( h)2 and after inserting in (55) we get the condition for Ps EW to be a stationary solution of the KPZ equation δ [ ( h) 2 P EW ] s dx = 0. (57) δh Taking the functional derivative, we obtain δ [ ( h) 2 Ps EW ] dx = 2P EW δh 2 δ(0) 2 h(x)dx + 2η Γ P s EW ( h(x)) 2 2 h(x)dx, (58) where the first term vanishes because of periodic boundary conditions, while the second term is evaluated in d = 1 and d = 2 below { h ( h) hdx = x h xx dx = 1 3 (h3 x) x dx = 0, d = 1 (h 2 x + h 2 (59) y)(h xx + h yy )dx 0, d = 2. We find out that in d = 1, EW and KPZ share the same long-time statistics of stationary fluctuations. Therefore α KPZ = α EW = 1 2. Since α KPZ + z KPZ = 2, we get z KPZ = 3 2 and consequently β KPZ = 1 3 > β EW. This means that in one dimension a surface subject to the KPZ equation roughens faster than the corresponding EW surface, but the final stationary value scales with L in the same way. In higher dimensions both KPZ exponents are larger than EW exponents (which vanish for d 2 as we noted above). The fact that a KPZ surface tends to be more rough can be traced to the weaker smoothing mechanism of its deterministic parts. The EW smoothing process is the typical diffusional smoothing, which induces an exponentially fast relaxation, while the relaxation of bumps in the KPZ surface follows a power law. It is therefore reasonable to speculate that in KPZ, noise should lead to a rougher surface than in EW. 6 Discussion Below we summarize our findings. We first examined the Edward-Wilkinson equation, which is a linear diffusion equation with additive white noise. The diffusive term is the smoothing mechanism, which counters the roughening effect of noise, while in the KPZ equation, we have another term which amounts to lateral growth. Critical exponents of EW universality class are α = 2 d 2, β = 2 d, z = 2. (60) 4 Where α and β vanish in dimension d 2. Critical exponents of KPZ universality class are [8], [9], [10] 8

10 d α β z 1 1/2 1/3 2/ (4) ) (8) where for dimension d = 1, we have stated the exact exponents, while in higher dimensions values are computed numerically for the RSOS model. Numbers in parentheses are standard deviation, values of z and β are determined by the relations holding for KPZ class in any dimension z = 2 α and β = α/z. (61) Above simulations also imply that the upper critical dimension for the KPZ class is larger than 4, d u c > 4, (62) while the exact value of d u c is still unknown, with field-theoretic approach suggesting d u c = 4, numerics, as we see above, d u c > 4 and real-space RG giving d u c =. All of these approaches agree about the phase diagram of KPZ for d < 4. To conclude our seminar, we mention two ways to motivate further research. One being trying to solve the KPZ equation, since it appears in many other, seemingly unconnected areas, for example study of asymmetric simple exclusion processes or directed polymers in random environments. The other topic is a generalisation of growth models to a wider class of nonlocal models, where in contrast to our introduced models, the growth depends on the conformation of the whole system. References [1] R. Livi and P. Politi, Nonequilibrium statistical physics : a modern perspective. Cambridge university press, [2] Pictures retrieved from. Accessed: [3] S. F. Edwars and D. Wilkinson, The surface statistics of a granular aggregate, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 381, no. 1780, pp , [4] M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett., vol. 56, pp , Mar [5] J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics, Quart. Appl. Math, vol. 9, pp , [6] E. Hopf, The partial differential equation u t + uu x = µ xx, Communications on Pure and Applied Mathematics, vol. 3, no. 3, pp [7] N. G. van Kampen, Stochastic processes in physics and chemistry. North-Holland publishing company, [8] A. Pagnani and G. Parisi, Numerical estimate of the kardar-parisi-zhang universality class in (2+1) dimensions, Phys. Rev. E, vol. 92, p , Jul [9] E. Marinari, A. Pagnani, and G. Parisi, Critical exponents of the kpz equation via multi-surface coding numerical simulations, Journal of Physics A: Mathematical and General, vol. 33, no. 46, p. 8181, [10] A. Pagnani and G. Parisi, Multisurface coding simulations of the restricted solid-on-solid model in four dimensions, Phys. Rev. E, vol. 87, p , Jan