Invited review: KPZ. Recent developments via a variational formulation

 

Horacio S. Wio,¹* Roberto R. Deza,² Carlos Escudero,³* Jorge A. Revelli⁴

*E-mail: wio@ifca.unican.es
E-mail: deza@mdp.edu.ar
E-mail: cel@icmat.es
E-mail: revelli@famaf.unc.edu.ar

IFCA (UC-CSIC), Avda. de los Castros s/n, E-39005 Santander, Spain.
IFIMAR (UNMdP-CONICET), Funes 3350, 7600 Mar del Plata, Argentina.
Depto. Matemáticas & ICMAT (CSIC-UAM-UC3M-UCM), Cantoblanco, E-28049 Madrid, Spain.
FaMAF-IFEG (CONICET-UNC), 5000 Córdoba, Argentina.

Recently, a variational approach has been introduced for the paradigmatic Kardar-Parisi-Zhang (KPZ) equation. Here we review that approach, together with the functional Taylor expansión that the KPZ nonequilibrium potential (NEP) admits. Such expansión becomes naturally truncated at third order, giving rise to a nonlinear stochastic partial differential equation to be regarded as a gradient-flow counterpart to the KPZ equation. A dynamic renormalization group analysis at one-loop order of this new mesoscopic model yields the KPZ scaling relation a -\- z = 2, as a consequence of the exact cancelation of the different contributions to vértex renormalization. This result is quite remarkable, considering the lower degree of symmetry of this equation, which is in particular not Galilean invariant. In addition, this scheme is exploited to inquire about the dynamical behavior of the KPZ equation through a path-integral approach. Each of these aspects offers novel points of view and sheds light on particular aspects of the dynamics of the KPZ equation.

I. Introduction

Although readers whose careers span mostly on the 21th century might not care about this, back in the sixties (when transistors and lasers had already been invented) equilibrium critical phenomena were still a puzzle. In fact, although a sense of “uni-versality” had been gained in 1950 through a field theory based on the innovative concept of order pa-rameter 1,2, its predicted critical exponents were almost as a rule wrong. It was not until the seventies that a far more sophisticated field-theory ap-proach 3 brought order home: equilibrium uni-versality classes are determined solely by the dimensionalities of the order parameter and the ambient space. Since then, one of Statistical Physics’ “holy grials” has been to conquer a similar achieve-ment for non-equilibrium critical phenomena 4. In such a (still unaccomplished) enterprise, a valuable field-theoretical tool has been in the last quarter of century the Kardar-Parisi-Zhang (KPZ) equation 5-7.

The KPZ equation 5-7 has become a paradigm for the description of a vast class of nonequilib-rium phenomena by means of stochastic fields. The field h(x, t), whose evolution is governed by this stochastic nonlinear partial differential equation, describes the height of a fluctuating interface in the context of surface-growth processes in which it was originally formulated. From a theoretical point of view, the KPZ equation has many inter-esting properties, for instance, its close relationship with the Burgers equation 8 or with a diffusión equation with multiplicative noise, whose field (x, t) can be interpreted as the restricted partition function of the directed polymer prob-lem 9. Many of the efforts put in investigating the behavior of its solutions were focused on ob-taining the scaling laws and critical exponents in one or more spatial dimensions 11-17. However, other questions of great interest are the develop-ment of suitable algorithms for its numerical inte-gration 18,19, the construction of particular solutions 20-23, the crossover behavior between dif-ferent regimes 10,24-26, as well as related ageing and pinning phenomena 27-29.

Among all the classical theoretical developments concerning this equation 6, 7, two have recently drawn our attention. One was the scaling relation a + z = 2, which is expected to be exact for the KPZ equation in any dimensión. The exactness of this relation has been traditionally attributed to the Galilean invariance of the KPZ equation. Nevertheless, the assumed central role of this sym-metry has been challenged in this as well as in other nonequilibrium models from both a theoretical 30-33 and a numerical 34-36 point of view. The second one is the generally accepted lack of ex-istence of a suitable functional allowing to formúlate the KPZ equation as a gradient flow. In fact, a variational approach to the closely related Sun-Guo-Grant 37 and Villain-Lai-Das Sarma 38,39 equations was developed in 40,41 by means of a geometric construction. In 46, a Lyapunov functional (with an explicit density) was found for the deterministic KPZ equation. Also, a nonequilibrium potential (NEP), a functional that allows the formal writing of the KPZ equation as a (stochas-tically forced) exact gradient flow, was introduced.

In this work we shortly review the consis-tency constraints imposed by the nonequilibrium-potential structure on discrete representations of the KPZ equation and show that they lead to explicit breakdown of Galilean invariance, despite the fact that the obtained numerical results are still those of the KPZ universality class. A Taylor expansión of the previously introduced NEP has (in terms of fluctuations) an explicit density and a thought-provoking structure 47, and leads to an equation of motion (for fluctuations, in the contin-uum) with exact gradient-flow structure, but differ-ent from the KPZ one. This equation has a lower degree of symmetry: it is neither Galilean invariañor even transíation invariant. Its scaling properties are studied by means of a dynamic renor-malization group (DRG) analysis, and its critical exponents fulfill at one-loop order the same scaling relation a + z = 2 as those of the KPZ equation, despite the aforementioned lack of Galilean invariance. The concern with stability leads us to suggest the introduction of an equation related to the Kuramoto-Sivashinsky one, also with exact gradient-flow structure. We cióse this article ex-posing some novel developments based on a path-integral-like approach.

II. Brief review of the nonequilibrium potential scheme

Loosely speaking, the notion of NEP is an extensión to nonequilibrium situations of that of equilibrium thermodynamic potential. In order to introduce it, we consider a general system of nonlinear stochastic equations (admitting the possibility of multiplicative noises)

where $(</) is the NEP of the system and the pref-actor Z(q) is defined as the limit

i.e., «J> is a Lyapunov functional for the dynamics of the system when fluctuations are neglected. Un-der the deterministic dynamics, q^v = K"(q), $ de-creases monotonically and takes a minimum valué on attractors. In particular, $ must be constant on all extended attractors (such as limit cycles or strange attractors) 42.

An alternative way to look into this problem is due to Ao 45. The interesting feature of this ap-proach is that it resorts neither to P_st(q) ñor to the small-noise limit, thus being applicable in principie to more general situations.

Lyapunov functional

The deterministic KPZ equation—obtained by set-ting 7 = 0—is exactly solvable by means of the Hopf-Cole transformation (</>(x, t) = e¹^^h^^x'^t)_: which maps the nonlinear KPZ equation onto the (deterministic) linear diffusion equation 5

which is associated to the directed polymer problem 6,7,9 results, using the inverse transformation (h(x,t) = -¥- liíóíx,t)) to be mapped into the com-plete KPZ equation (4).

The deterministic part of Eq. (6) (i.e., Eq. (5)), can be written as

and that its minimum is achieved by constant func-tions. Hence we have a Lyapunov functional for the deterministic KPZ equation that displays simple dynamics: the asymptotic stability of constant solutions indicates an approach to constant profiles at long times, for arbitrary initial conditions. Despite this simplicity in the deterministic case, the stochastic situation is far from trivial and gives rise to self-affine fractal profiles. In particular, the ex-istence of this Lyapunov functional provides no a priori intuition on the stochastic dynamics.

The nonequilibrium potential

An alternative functional was also proposed in 46. By starting from the functional Fokker-Planck equation, we look for the stationary solution (in fact steady-state solution), and after some integra-tion by parts, it is possible to arrive to another form of Lyapunov functional

*r»,i ^v ¡ mrf ^A f ^Kx,t) ai mi^ <P/iJ = — dx(V/i)----- dx              d'ip (v'ip) .

2                             2               hr_sí

(10) It is somehow inspired in the analytical form of “model A”, according to the classification of crit-ical phenomena in 48. Here, the interpretation of the integral in the 2nd term on the rhs is / ax ,         d'ip = > . ¿ax , dipj. According to

We will not pursue here the development of rig-orous result concerning the functional (10). Our present interest falls in the calculation of quantities of physical interest rather than in building a com-pletely rigorous mathematical theory. It is worth remarking that, as indicated in 46 and above, such a form has a discrete definition. It is also Ínterest-ing to point out that analogous functionals involv-ing functional integráis which are not carried out explicitly were obtained for the problem of inter-face fluctuations in random media 49-52.

NEP expansión

We now proceed to formally Taylor expand the NEP defined in Eq. (10) around a given reference indicating that this formal expansión has a natural cut-off after the third order.

It is worth indicating that in this computation as in all the other computations within this work— boundary terms vanish provided one of the follow-ing types of boundary conditions is assumed: ho-mogeneous Dirichlet boundary conditions, homo-geneous Neumann boundary conditions, periodic boundary conditions or an infinite space with the derivatives of Sh vanishing as they approach an infinite distance from the origin.

The reference state ho is arbitrary (i.e., any initial condition), but it is particularly useful to take it as one that makes (5$/io = 0, that is: a solution to the stationary counterpart of the deterministic KPZ equation. The complete set of solutions is /io = c, where c is an arbitrary constant (arbitrary up to the application of the boundary conditions, whenever this consideration applies), what physi-cally corresponds to a fíat interface. Henee we have (Sh = h — ho)

The equation for fluctuations

Prom here we can define an effective NEP, which drives the dynamics of the fluctuations Sh and has an explicit density. Clearly, it corresponds to the last two terms in Eq. (15). To simplify the notation we adopt u(x,t) := 6h(x,t), and so the NEP reads leads, after functional derivation, to a generalized KPZ equation Clearly, patterns like «o = constant are stationary solutions of Eq. (17): for all of them Iw = 0, indicating that all such states have the same “energy”. Finally, let us remark that although the formal Taylor expansión becomes naturally truncated at third order, the deterministic KPZ equation is not recovered. We cali the stochastic versión of this new equation “KPZW”.

There is a remarkable difference between both equations (KPZ and Eq. (17)). It arises due to the fact that in the first case we have a fixed equation for h and for any initial condition, while in the second case we have a fixed initial condition (u = 0) with a variable equation whose coef-ficients depend on h₀\ (it is an equation for the departure from the given initial condition). The question of the relevance of this aspect to ageing problems (as discussed for instance in 29) arises naturally. This point, worth to be analyzed, will be the subject of further work.

Non-local kernel

In previous works 36,46 it was indicated that the following functional, including a nonlocal contribution,

It was also shown that if the nonlocal kernel has translational invariance (G(x, x') = G(x -x')), and also, if it is of (very) “short” range, it can be ex-panded as

 

with S^<-ⁿ'('x—x') = V"/(5(x—x'), and where symme-try properties were taken into account. Exploiting this form of the kernel, and considering different approximation orders, it is possible to recover con-tributions having the same form as the ones arising in several previous works, where scaling properties, symmetry arguments, etc., have been used to dis-cuss the possible contributions to a general form of the kinetic equation 55-57. Such different contributions are tightly related to several of other pre-viously studied equations, like the Sun-Guo-Grant equation 37, as well as others 55,58.

We will not pursue this aspect here, but we will briefly refer again to it in a forthcoming section.

IV. Discretization issues, symmetry violation and all that

In this section we will review aspects related to two main symmetries associated with the ID KPZ equation: Galilean invariance and the fluctuation-dissipation relation. On the one hand, Galilean invariance has been traditionally linked to the ex-actness of the relation a + z = 2 among the crit-ical exponents, in any spatial dimensionality (the roughness exponent a, characterizing the surface morphology in the stationary regime, and the dy-namic exponent z, indicating the correlation length scaling as £(í) ~ t¹'⁷'). However, this interpretation has been criticized in this and other nonequilibrium models 31,32,59. On the other hand, the second symmetry essentially tells us that in ID, the non-linear (KPZ) term is not operative at long times.

Even when recognizing the interesting analytical properties of the KPZ equation, it is clear that in-vestigating the behavior of its solutions requires the (stochastic) numerical integration of a discrete ver-sion. Such an approach has been used ,e.g., to ob-tain the critical exponents in one and more spatial dimensions 10-15,60. Although a pseudo-spectral spatial discretization scheme has been recently in-troduced 18, 61, real-space discrete versions of Eq. (4) are still used for numerical simulations 62, 63. One reason is their relative ease of imple-mentation and of interpretation in the case of non-homogeneous substrates, for example a quenched impurity distribution 64.

Consistency

Here, we use the standard, nearest-neighbor dis-cretization prescription as a benchmark to eluci-date the constraints to be obeyed by any spatial discretization scheme, arising from the mapping be-tween the KPZ and the diffusion equation (with multiplicative noise) through the Hopf-Cole trans-formation.

The standard spatially discrete version of Eq. (6)

 

with 1 < j < N = O, because of the assumed peri-odic b.c. (the implicit sum convention is not meant in any of the discrete expressions). Here a is the lattice spacing. Then, using the discrete versión of Hopf-Cole transformation ¿>,(t) = exp \^hn(t) , we get ponentials up to terms of order a², and collecting equal powers of a (observe that the zero-order con-tribution vanishes) we retrieve

As we can see, the first and second terms on the r.h.s. of Eq. (23) are strictly related by virtue of the

Hopf-Cole transformation. In other words, the discrete form of the Laplacian in Eq. (21) constrains the discrete form of the nonlinear term in the trans-formed equation. Later we return, in another way, to the tight relation between the discretization of both terms. Known proposals 60 fail to comply with this natural requirement.

An important feature of the Hopf-Cole transformation is that it is local, Le., it involves neither spatial ñor temporal transformations. An effect of this feature is that the discrete form of the Laplacian is the same, regardless of whether it is applied to <f> or h.

The aforementioned criterion dictates the follow-ing discrete form for J-"</> (the one just before Eq. (8)), thus a Lyapunov function for any finite N

It is a trivial task to verify that the Laplacian is (d²(f>)j = - a~¹d_tp_jJ-"</>. Now, the obvious fact that this functional can also be written as J-"</> = 7T- y~! . A4>-í+-\ - 4>-í)² illustrates a fact that for a more elabórate discretization requires explicit cal-culations: the Laplacian does not uniquely determine the Lyapunov function 34-36.

Equation (22) has also been written in 14, al-though with different goals than ours. Their in-terest was to analyze the strong coupling limit via mapping to the directed polymer problem.

An accurate consistent discretization

Since the proposals of 60 already involve next-to-nearest neighbors, one may seek for a prescription that minimizes the numerical error. An interesting choice for the Laplacian is 65

which has the associated discrete form for the KPZ For the discretization scheme in Eq. (23), this is term

(d_x(f>) =

1

24 a²

{16 (</>¿+i — 4>j)

~ exp

v 1

2e 2a

y^ (/ij+i - /ij)2 + (hj ~ hj-i)2

+(4>j — 4>j-i)                                                                                                                   (27)

- {4>_j+2 -4>j? + {4>j -4>j-2?}}        Inserting this expression into the stationary

,„ a                                                Fokker-Planck equation, the only surviving term

+0(a^A).                                  (26)

has the form

Replacing this into the first line of Eq. (24), we ob-tain Eq. (25). Since this discretization scheme ful-fills the consistency conditions, it is accurate up to O (a⁴) corrections, and its prescription is not more complex than other known proposals, we expect that it will be the convenient one to use when high accuracy is required in numerical schemes 34-36.

Relation with the Lyapunov functional

In Sect. III we have indicated the form of the NEP for KPZ, and the way in which the functionals F</> and J-"/i are related 46. According to the previous results, we can write the discrete versión of Eq. (8)

as

A² 1 ^-^ Xl r                               2

T\h\ = —— y c ³ (/ij+i — hj) 81/ 2 a ¿-^

+(hj — hj-i) .

Introducing this expression into d_thj = Tj ¿J. , and through a simple algebra, we obtain Eq. (23). This reinforces our previous result, and moreover indicates that the discrete variational formulation naturally leads to a consistent discretization of the KPZ equation.

The fluctuation-dissipation relation

This relation is, together with Galilean invariance, a fundamental symmetry of the one-dimensional KPZ equation. It is clear that both symmetries are recovered when the continuum limit is taken in any reasonable discretization scheme. Thus, an ac-curate enough partition must yield suitable results. The stationary probability distribution for the KPZ problem in 1D is known to be 6,7

P_stat h ~ exP

í     V f              ₂

—— dx (o_xh) 2)

2 a³

y            (hj₊i — hj) + (hj — hj-i)

3

x h_j+1 - 2hj + hj_i .

(28)

1 he contmuum hmit oí this term is dx(o_xh) o£ti, that is identically zero 6, 7. A numerical analy-sis of Eq. (28) indicates that it is several orders of magnitude smaller than the valué of the exponents’ pdf in Eq. (27), and typically behaves as 0{1/N), where N is the number of spatial points used in the discretization. Moreover, it shows an even faster approach to zero if expressions with higher accuracy like Eqs. (25) and (26) are used for the dif-ferential operators. In addition, when the discrete form of (d_xh)² from 60 is used together with its consistent form for the Laplacian, the fluctuation-dissipation relation is not exactly fulfilled. This indicates that the problem with the fluctuation-dissipation theorem in 1 + 1, discussed in 18,60 can be just circumvented by using more accurate expressions.

Galilean invariance

This invariance means that the transformation

x^x — Xvt, h^h + vx, F^F-----v , (29)

where v is an arbitrary constant vector field, leaves the KPZ equation invariant. The equation ob-tained using the classical discretization

1

2a

(30)

is invariant under the discrete Galilean transformation

ja —> ja — Xvt, hj —> hj + vja, F —?► F — —v .

(31)

 

However, the associated equation is known to be numerically unstable 14, at least when a is not small enough. Besides, Eq. (23) is not invariant un-der the discrete Galilean transformation. In fact, the transformation h —> h + vja yields an excess term which is compatible with the gradient dis-cretization in Eq. (30); however, this discretization does not allow to recover the quadratic term in Eq. (23), indicating that this finite-difference scheme is not Galilean-invariant.

Since Eq. (21) is invariant under the transformation indicated in Eq. (31), it is the nonlinear Hopf-Cole transformation (within the present discrete context) which is responsible for the loss of Galilean invariance. Note that these results are in-dependent of whether we consider this discretization scheme or a more accurate one.

Galilean invariance has always been associated with the exactness of the one-dimensional KPZ ex-ponents, and with a relation that connects the crit-ical exponents in higher dimensions 68. If the nu-merical solution obtained from a finite-difference scheme as Eq. (23), which is not Galilean invariant, yields the well known critical exponents, this will be an indicative that Galilean invariance is not strictly necessary to get the KPZ universality class. The numerical results presented in 34-36 clearly show that this is the case.

We will not discuss here the simulation procedure but only indicate that to make the simulations we introduced a discrete representation of h{x, t) along the substrate direction x with lattice spacing a = 1, and that a standard second-order Runge-Kutta al-gorithm (with periodic boundary conditions) was employed (see 66). In 34-36 it was shown that all the cases (consistent or not) exhibit the same critical exponents. Moreover, we want to note that the discretization used in Refs. 60, which also vi-olates Galilean invariance, yields the same critical exponents too. Additionally, stochastic differential equations which are not explicitly Galilean invariant have been shown to obey the relation a + z = 2 (33, see also next section). Henee, our numerical analysis indicates that there are discrete schemes of the KPZ equation which, even not obeying Galilean invariance, show KPZ scaling.

The moral from the present analysis is clear: due to the locality of the Hopf-Cole transformation, the discrete forms of the Laplacian and the nonlinear (KPZ) term cannot be chosen independently; more-

over, the prescriptions should be the same, regard-less of the fields they are applied to. For further details we refer to 34-36.

V. Renormalization-group analysis for Fluctuations

In section III we have built a gradient flow counter-part of the deterministic KPZ equation. In this sec-tion we consider the corresponding stochastically forced gradient flow

61 _i. o_tu = ———h £(x, t), Su

(32)

with the density indicated in Eq. (16). We obtain the KPZW equation, which is the following SPDE

d_tu = z/₂ u-----(Vw

2                        72              A

— uv u + £(x, í). (33)

Our present goal will be to analyze the scaling be-havior of the fluctuations of the solution to this equation.

Since Eq. (33) is nonlinear, we focus on a pertur-bative technique. We choose the dynamic renor-malization group as employed in 67,68. Employ-ing this method, we find at one-loop order the fol-lowing flow equations 47

dX

di

= X(a + z — 2),

(34)

dv                              1 X²D 1 — d

= v z — 2 — ^—Kd—;— , (35)

di

(¿7

36 z/³           d

Kd A² 7

= 7 z — d — 2a + di                                      72 v³

(36)

where K¿ = S_ci/(2'K)^d, S,¡ = 2ir^d*²/T(n/2) is the surface área of the d—dimensional unit sphere, and r is the gamma function. We find that the cou-pling constant g := KdX²-f/i/³ obeys the one-loop differential equation

dg                       6 — hd

— = (2 — d)g -\----——

di                         72d

g²

(37)

revealing that the critical dimensión of this model is d_c = 2 as could be anticipated by means of power counting. For d > 2 the coupling constant approaches zero exponentially fast in the scale i; for d = 2, this approach is algebraic. So for these dimensions one expeets the large-scale space-time properties of Eq. (33) to be dominated by its linear counterpart (up to marginal corrections in d = 2). In d = 1, the coupling constant runs to infin-ity for finite i, suggesting the presence of a non-perturbative fixed point (as the one in the KPZ equation for d = 2).

The valúes of the critical exponents which yield scale invariance can be formally calculated by iden-tifying with zero the right hand sides of Eqs. (34)- (36). We get

2(2 — d)(í — d)

^a = ------7.------;------;                        (38)

6 — 5a

12 — 10 d — 2(2 — d)(l — d)

z = ---------------------"-------------,      (39)

6-5d

which in particular obey the relation a + z = 2 in any dimensionality, despite the fact that Eq. (33) does not obey any sort of Galilean invariance. We note that in both d = 1 and d = 2 it is a = 0 and z = 2, whereas a becomes negative in higher dimensions. Henee, in all dimensions, the expo-nent a indicates that the interface is either flat or at most marginally rough. The valúes for d = 1 make both diffusion and nonlinearity in Eq. (33) in-variant under the scale transformation {x, t, u —> {5x, b^zt, b^au}, as far as b > 1. In this case, the noise grows with the scale (a fact that might explain the growth of the coupling constant in the renormaliza-tion group flow). In d = 2, the exponents are those of the linear equation. An interesting result is that for d = 0, the exponents become those of the KPZ equation: a = 2/3 and z = 4/3, although this limit is highly singular for Eq. (37). Of course, these results have been obtained by means of a perturba-tive dynamic renormalization group and could be modified by non-perturbative contributions. One possible path to study such a possibility could be to adapt some non perturhative renormalization group techniques used for KPZ 69 to the present KPZW case.

Among all the results in this section, we would like to highlight the one given by Eq. (34). We recall that the RG analysis of the KPZ equation yields non-renormalization of the vértex and renormalization of propagator and noise. Our variational equation yields exactly the same result. Vértex non-renormalization at one-loop order is expressed by Eq. (34). The origin of this result is analo-gous to that of its equivalent in the KPZ equation: three non-vanishing Feynman diagrams con-

tribute to vértex renormalization, but they cancel out each other 5 (a fact that has been traditionally attributed to the Galilean invariance of the KPZ equation). Here we have shown that the same result appears in a SPDE that is not even invariant under the translation u —> w+constant.

VI. Stability

We have carried out the NEP expansión about a constant solution of the KPZ equation and found that constants are still solutions to KPZW (Eq. (17)). In this section we will study the linear stability of such solutions. We start considering the solution

u{x, t) = c + ev{x, t),                  (40)

where c is an arbitrary constant and e is the small parameter. Substituting in Eq. (17), we find

3z/ — Xc ₂

o_tv =----------V v,                  (41)

at first order in e. So v obeys a diffusion equation whose diffusion constant depends on c. For c < 3z//2A, the diffusion constant is positive and correspondingly the constant solution is linearly stable. For c > 3z//2A, the diffusion constant is negative and consequently the constant solution is unstable. Furthermore, in this case the problem becomes linearly ill posed.

Since for large valúes of c, the problem becomes linearly ill posed, numerical solutions are not avail-able. In order to solve this disadvantage, we could include a higher order term in our problem. We concéntrate on the gradient flow

^ , „ OfU = —------\-£{x,t),                     (42)

Su with density

V f                  ₂ X f            . ,₂

ju\ = — dx (Víí)----- di íi(Víi)

2                         6

¡1 f             2 ²

-\— dx (V u) ,               (43)

2

leading to the following equation

d_tu = z/V u — —CVu) — — «V u — /xV u + £(x, t). 6               3

(44)

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Papers in Physics, vol. 5, art. 050010 (2013) / H. S. Wio et al.

Note that the deterministic counterpart of this fourth-order equation can be considered as a varia-tional versión of the Kuramoto-Sivashinsky equation. It is worth remarking that this ad hoc construction resembles the one that, as indicated in 46, could more formally be obtained by con-sidering the expansión of a nonlocal, short range interaction.

The regime of linear stability/instability of this equation is identical to that of Eq. (33) but in this case, the problem is always linearly well posed. Fur-thermore, the term proportional to ¡i is presumably irrelevant in the large spatiotemporal scale (as simple power counting of the linear terms reveáis) so the results of the previous RG analysis could possi-bly hold for this case too. Anyway, due to the pres-ence of the deterministic instability, further analysis are needed in order to assure this (note that both linear terms in the equation are stabilizing and that this instability has its origin in the vértex structure).

VII. Crossover: a path integral point of view

Another recently discussed related aspect 70 is based in a path-integral Monte Carlo-like method for the numerical evaluation of the mean rugosity and other typical averages whose approach, which radically differs from one introduced before 71, exploits some of our previous results 34-36. Here we limit ourselves to quote the temporally (¡jl) and spatially (j) discrete form of the “stochastic action”

Sh

27 J2 ih3,H+l

-2vaNt

-™2 J2^hi+1^ ~ 2hJ^

.h j-íJ,

(45)

and briefly discuss the obtained numerical results. t is the time step, 0 < a < 1 a time-discretization parameter meant to be fixed for explicit calculation

13

<

Figure 1: Crossover-like behavior from EW to KPZ regime for A = 1, on a lattice of 1028 sites (y = D = 1). Red solid line: KPZ action; blue dash-dotted line: EW action; black-dotted line: difference.

Figure 2: Same data as the previous figure. Solid line: time T* vs. A; dashed line: trend for T* ~ A^-1'³⁵, included for comparison.

72,73, and L¿_iM the “stochastic Lagrangian”

^JJ,M

A

+ 4

2h j „ + h j-iu)

Cb'+i,M

<h

J,M

h-i

.j-í,»)

\21

(46)

Figure 1 shows the crossover-like behavior from the Edwards-Wilkinson (EW) regime to the KPZ one. We take as estimator of such a transition the time at which the difference (dotted black line) be-tween KPZ (red solid curve) and EW actions (blue

050010-10

Papers in Physics, vol. 5, art. 050010 (2013) / H. S. Wio et al.

dash-dotted line) crosses the EW one (it grossly coincides with the time at which the asymptotes cross). This estimator numerically agrees neither with the results in 10 (where a valué of </> ~ 4 was íound) ñor with the one in 24 (with </> ~ 3, but corresponding to a 2D case). In Fig. 2 we have plotted the dependence oí this estimator on A. For comparison, we have also included the trend for A~^ with (¡> = 1-35 (dotted line). Preliminary results for A > 7 seem to indicate a marked change in the valué oí </>, maybe a hint that the system is entering a strong coupling región 71.

Short-time propagator

Our aim here is to work out a variant oí the method introduced in 70 by exploiting the first form oí Lyapunov functional íound in 46, namely Eq. (10), that leads us to Eq. (11), the full KPZ equation.

Whereas Eq. (45) is valid whatever the valué oí t, we now seek for a simpler expression valid for t <C 1. This idea parallels in some sense other studies in the literature 71, but here we exploit the functional ^h of Eq. (8). We denote as {h} = (/ii,/ij ^2,/xj . . . j h>j_¡fJ,,..., /i/v,/i) the interface configuration at time ¡i. The transition pdf be-tween patterns ho at íq and hf at tf can be written

propagator (STP)

as

-f({^/}> tf\\ho}, to) =

f i i f f              \

/ T>h exp----/ Ch,h

7 t₀

(47)

with Ch, h

dx

o

d_th + r/i— J-"/i oh

2

s f s

-\-a— 1 h \=F \h\ oh           oh

(48)

whose discrete form is given by Eq. (46). A key ob-servation is the (temporally and spatially) “diagonal” character of Eq. (9), highlighted in its discrete version

P(h_f,T\ho,0)

T~\n i ^— o^ frT ds L dx (dth-^T4^)

ho

~ exp — —        dx

o

T

27 O

hf — ho 1         SJ⁷           (5J⁷

H— U f ti-----1^- i o

t              2          o a ^            Ort-o

\ 2 (50)

Here, for simplicity, we have chosen a discretiza-tion with a = 0. As it is well known 72,73, the Jacobian of the transformation from the noise variable to the height variable depends on a. With this choice, the Jacobian results equal to 1.

Incidentally, the form in Eq. (50) coincides with the discretization used in 71 for determining the least-action trajectory. The “quasi-Gaussian” char-acter of this STP is better evidenced in the follow-ing approximate form

F{hf,tf = T\ho,to = 0) ~ eí ²~t^{T Jo v} ' ' J

r i fL r             i ( 5j

x \ 1------ dx (ht — ho)— 1 f——

27 o                         2        ohf

+1 0----- + ^(^r)

Ó/lO

(51)

where the exponential term has been separated out since it is of order t^_1, whereas the following two are of order t and t , respectively (of lesser weight and negligible respectively, in the limit t —> 0). It is worth remarking that the term that could come from the Jacobian is also of order t¹.

It is easy to check that we can recover the known FPE from the proposed form of STP (adopting a = 0 for simplicity). We will not reitérate this cal-culation here. An immediate result of this form is that at very short times, behavior of the Edwards-Wilkinson type is obtained

\J(h?) k, t

e -77

hj{t) = —Fj-------h \/j£j(t).              (49)

Shj

Guided by Eq. (46), we propose the following form of P({hf},tf\{ho},to) for t < 1, or short-time

VIII. Conclusions

Herein, in addition to reviewing some recent results 34-36,47, 70, we have furthered the study in 46, where it was shown that the deterministic KPZ equation admits a Lyapunov functional, and a (formal) definition of a nonequilihrium potential was in-troduced. We have carried out a Taylor expansión of such a nonequilibrium potential, what led us to a different equation of motion than the KPZ one, the KPZW which is an exact gradient flow and has an explicit density. In particular, it has a lower de-gree of symmetry: it is neither Galilean invariant, ñor even translational invariant. The critical ex-ponents determining its scaling properties were ob-tained through a one-loop dynamic renormalization group analysis. These exponents fulfill the same scaling relation as the KPZ equation, a + z = 2, traditionally attributed to the Galilean invariance of the latter. The fact that the same scaling relation arises in a SPDE (i.e., the KPZW) that is not only non-Galilean invariant but even non-invariant under the translation u —> w+constant supports recent theoretical and numerical results indicating that Galilean invariance does not necessarily play the relevant role previously assumed in defining the universality class of the KPZ equation and different nonequilibrium models 30-36.

We have, moreover, analyzed the stability properties of the solutions to the present equation, find-ing the threshold condition for the appearance of diffusive instabilities, which indicates that in this case the problem becomes linearly ill posed. After considering the simplest way to correct such an ill— posed problem, we have met a kind of Kuramoto-Sivashinsky equation, resembling the one that, as indicated in 46, could be obtained by considering a nonlocal, short range interaction. This equation has an exact gradient flow structure with an explicit density. Furthermore, when subject to stochastic forcing, its scaling properties could be formally de-scribed by the same critical exponents because the stabilizing term is irrelevant in the large scale from a dimensional analysis viewpoint.

Exploiting some elements of a path integral de-scription of the problem, we have also shown what seems to be a simple form of viewing and studying the crossover from the EW to the KPZ regimes.

The present review-like study aims to open new points of view on, as well as alternative routes to study, the KPZ problem. Among the many aspects to be further studied, an interesting one is to test the (kind) of stability of the recently found exact solutions 20-23 by exploiting the indicated form of the NEP.

 

Acknowledgements - Financial support from MINECO (Spain) is especially acknowledged, through Projects PRI-AIBAR-2011-1323 (which enabled international cooperation), FIS2010-18023 (HSW and CE) and RYC-2011-09025 (CE). Also acknowledged is the support from CONICET, UNC (JAR) and UNMdP (RRD) of Argentina. Collab-oration with M.S. de la Lama and E. Korutcheva during different stages of this research is highly appreciated.

 

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