Although describing a non-equilibrium process, Equation (2) originated in purely kinetic models. For some time, it was believed that its drift ν∇2h(x,t)+λ2[∇h(x,t)]2 cannot be derived from an effective free energy [17].

Using the Hopf–Cole transformationΨ(x,t):=expλ2νh(x,t) (mapping h onto a field Ψ obeying a linear equation, but with multiplicative noise), it was proved in [18] that for any d, Equation (2) can be cast as model A (namely, a relaxational system without conservation laws, in the nomenclature of [19])(4)∂h(x,t)∂t=−Γ[h]δF[h]δh(x,t)+ξ(x,t), with functional F[h] given by Equation (5), and functional Γ[h] (known as the motility) byΓ[h]:=2νλ2exp−λνh(x,t).(5)F[h]:=∫dxexpλνh(x,t)λ2ν−Fh(x,t)+λ4∇h(x,t)2 The Lyapunov property F˙[h]≤0 is easy to check, and the minimum of F[h] is reached by constant functions. In addition, the effective potential f(h)—the restriction of F to h(x,t)=const.—is minimum at h=0 for F=0 (Figure 1). Hence, F[h] is a Lyapunov functional for the KPZ problem, and the deterministic KPZ equation displays simple dynamics: the asymptotic stability of constant solutions indicates that arbitrary initial conditions approach constant profiles for long times. While the deterministic case is simple, the stochastic dynamics remain highly non-trivial, producing self-affine fractal profiles. Notably, the Lyapunov functional offers little intuition about these stochastic features.

As shown in [18], F[h] and Γ[h] (together with the density F˜[h,∇h], defined by F[h]:=∫F˜[h,∇h]dx) fulfill the relations postulated in [20] on the grounds of global shift invariance.

It is worth noting that several studies [21,22,23,24] have shown that in oscillator systems subjected to noise, synchronization is achieved when the coupling is strong enough. This dynamic process can be related to a roughness process and exhibit, in one dimension, scale invariance within the KPZ universality class. Since Kuramoto-type models can be obtained from energy functionals, this provides further support for the idea of the existence of such a functional for KPZ.

Discretization is a necessity if the substrate dimensionality is d>1, where the DRGFP is not reachable perturbatively and one must resort to numerical integration to find the critical exponents. In the past, a criterion to judge the quality of a discretization of Equation (2) was that it should preserve the two main symmetries of Equation (2): Galilean invariance (GI) and a fluctuation–dissipation relation peculiar of 1d (1dFDR). Note that although there is an explicit FDR for diffusion phenomena, there is in principle no such thing for growth processes. Recently, however, a kind of FDR—similar to the diffusive one—was found for growth phenomena [25,26]. In turn, Galilean invariance implies that the physics of interface growth remains the same when viewed from a reference frame moving laterally at constant speed, provided the surface height is appropriately tilted. This symmetry protects parameter λ from being renormalized and imposes the constraint α+z=2.

Consistency in employing the Hopf–Cole transformation constraints the discrete version of (∇h)2 to have the form 12[(hi+1−hi)2+(hi−1−hi)2]. In [27], scaling was obtained to a very good approximation (Figure 2) using this discretization, which breaks GI and 1dFDR explicitly.

From the Family–Vicsék scaling of roughening processes [29], it turns out that(6)h(x,t)∼v∞t+Atβχ(x′),x′=Bxt1/z. Over the last 15 years or so, numerous works by mathematicians and physicists [2,30,31,32,33,34,35] have shown that the asymptotic (in the scaling regime) probability distribution P(χ) (for any x) not only has skewness, but shows a remarkable dependence on the initial conditions. In other words, process h(x) has memory.

Equation (2) is an extended Langevin equation; thus, the joint process [h](t) obeys a Fokker–Planck equation and is Markovian. Now h(x) obeys a marginal distribution and is generically non-Markovian (unless a conservation law, like, e.g., ∫dx∇2h=0 in the EW model, forces it to be Markovian).

Besides the model A variational formulation of Equations (4) and (5), Equation (2) can also be cast (again, for any d) as [18](7)∂h(x,t)∂t=−δΦ[h]δh(x,t)+ξ(x,t), with(8)Φ[h]:=ν2∫dx∇h2−F∫dxh(x,t)−λ2∫dx∫hrefh(x,t)Dψ∇ψ2. Again, Φ˙[h]≤0 and its effective potential is minimum at h=0 for F=0 (Figure 3). But at variance with F[h], Φ[h] is not bounded from below, so it is not a Lyapunov functional for h(x,t). However, Φ[h] does provides intuition on the stochastic dynamics (for instance, it resembles an extended escape problem).

In particular, Equation (8) can be transformed into(9)Φ[h]=ν2∫dx∇h2−F∫dxh(x,t)−λ2∫dx∫0tds∂sh(x,s)∇h(x,s)2. which shows explicitly that Φ[h] depends on the whole trajectory [h](s), from 0 to t.

As in the case of reaction–diffusion systems, Equation (2) looks formally as a stochastically forced gradient flow (in the sense that the coupling terms have the same form for all sites), but it is not strictly gradient. Equation (7) is regarded as a gradient flow in the sense that its motility is Γ[h]=1.

The EW model of Equation (1) is an equilibrium system and as such, its joint probability distribution functional (pdf’l) P[h](t) is granted a stationary state Pstat[h]. According to conservation law, ∫dx∇2h=0, P(h) is also stationary for any x. In the KPZ model, P[h](t) also obeys a Fokker–Planck equation, but since its collective mode grows steadily in the scaling regime [see Equation (6)], process [h](t) cannot be stationary in that regime—what is stationary is P(χ), as already argued. In principle, there is no theoretical reason why—even in 1d and with PBC—Equation (2) should attain a steady state Pstat[h]. Yet less that Pstat[h] be the same as that of the (equilibrium) EW model. (Moreover, recall that most results indicating strong dependence on initial conditions have been obtained for 1d substrates.)

It is just a fact that, by imposing ∂tP[h](t)=0 in the functional Fokker–Planck equation for P[h(x),t|0,0](10)∂P[h(x,t)]∂t=∫dxδJ[h]δh,J[h]:=δΦ[h]δhP[h(x,t)]+εδP[h(x,t)]δh, it admits as a stationary solution for 1d substrates and periodic boundary conditions (PBCs) the exponential exp{−ΦEW[h]/ε} of the EW Ginzburg–Landau functionalΦEW[h]:=ν2∫Ωdr(∇h(r,τ))2. This is what is meant by the abovementioned 1dFDR [17].

The novelty is that Equation (10) also admits as a stationary solution (for any substrate dimensionality and independently of BC) the exponentialPKPZstat[h]∝exp{−ΦKPZ[h]/ε}, with ΦKPZ[h] given by Equation (8). Hence ΦKPZ[h] is by definition a non-equilibrium potential (NEP) [36]. Clearly, this is a formal solution since PKPZstat[h] is not normalizable, but exp{−ΦEW[h]/ε} is also formal. In fact, using the short-time propagator P[hf,ti+τ|hi,ti] obtained in a path-integral representation, namely(11)lnP[hf,ti+τ|hi,ti]≈−τ4ε∫dx∂h∂t+δΦ[h]δh≈−14ετ∫dx(hf−hi)2−1ε(Φ[hf]−Φ[hi])−τ4ε∫dxδΦfδh2−δΦiδh2, it becomes clear that∫DhiPstat[hi]P[hf,ti+τ|hi,ti] does not coincide with Pstat[hf], indicating that exp{−ΦEW[h]/ε} is at most marginally stable.

From a path-integral approach [38,39], we proceed as follows: The probability distribution functional (pdf) of a given history (involving h at all sites) isPF([h(t)]|[h(0)])=NFexp{−SF[h]},SF[h]=∫0tdsLF([h(s)],[∂sh(s)]), where superscript F indicates increasing time sequence, NF is a normalization factor, and SF[h] is the forward stochastic action functional. The forward stochastic Lagrangian, given by Equation (12)(12)LF([h(s)],[∂sh(s)]):=14ε∫Ωdx∂sh+δΦδh2 is still a functional of [h(t)]. We use calligraphic types for functionals of the configuration alone, and blackboard types for functionals of the whole history. The conditional pdf of tracing the same history back isPB([h(0)]|[h(t)])=NBexp{−SB[h]},SB[h]=∫0tdsLB([h(s)],[∂sh(s)]), with the backward stochastic Lagrangian given by Equation (13).(13)LB([h(s)],[∂sh(s)])=14ε∫Ωdx−∂sh+δΦδh2.

With the entropy production rate being SF[h]−SB[h], one can readily obtain integral fluctuation theorems and thermodynamic uncertainty relations (TURs) [40,41].

Whereas the foregoing approach gives us a hint of the relevance of the NEP Φ[h] to the entropy production rate, the following approach starting directly from the KPZ equation enhances this relevance: We start expressing the KPZ equation in a semi-rigorous fashion (i.e., as one would write it in Itô interpretation if the noise was multiplicative), but explicitly taking into account that the KPZ equation is ill-posed when not regularized [2,42].(14)dh(x,t)=ν∂x2h(x,t)+λ2∂xh(x,t)2dt+2DdB(x,dt). Here dB(x,dt) is the increment of a functional Wiener process, 〈dB(x,dt)〉=0, 〈dB(x,dt)dB(x′,dt)〉=Kl(x−x′)dt, and we assume that the KPZ noise is spatially correlated over a small regularization length-scale ℓ. Although such regularization is usually omitted when introducing the KPZ equation, it appears in explicit calculations (in connection with a UV cut-off, e.g., a lattice).

For a KPZ equation with such a noise, the stochastic entropy change of the medium associated to a particular realization of the evolution of the KPZ front is given as [43](15)ΔStm[h]=1D∫0t∫Ω×Ωdh(z,s)Kl−1(z−z′)ν∂z′2h(z′,s)+λ2∂z′h(z′,s)2ddz′ddz, where the stochastic integral in (15) is to be understood in the Stratonovich sense, i.e., the following:(16)ΔStm[h]=limN→∞ΔSN,tm[h], where(17)ΔSN,tm[h]=∑k=0N−1∫Ω×Ω[h(z,sk+1)−h(z,sk)]Kl−1(z−z′)ν∂z′2h(z′,s¯)+λ2∂z′h(z′,s¯)2ddz′ddz, with s¯=(sk+1+sk)/2, and sk=kt/N, k∈{0,1,…,N}. It is now clear that with such Stratonovich interpretation of (15), one has (compare the result with Equation (14))(18)dh(x,t)=D∫ΩKl(x−x′)δΔStm[h]δh(x′,t)ddx′dt+2DdB(x,dt).

In particular, considering formally the case of a spatially uncorrelated noise, Kl→0(x)=δ(x), one has(19)dh(x,t)=DδΔStm[h]δh(x,t)dt+2DdB(x,dt), which clearly shows that if the potential Φ is understood as a Stratonovich integral, then Φ[h]=DΔStm[h]. Also notice that only within the Stratonovich interpretation, one has(20)∫0t∫Ωdh(z,s)ν∂z2h(z,s)ddz=ν2∫Ω∂zh(z,t)2−∂zh(z,0)2.

As a plot twist, Equation (15) illuminates the “correct” way to define the potential, in the case in which the noise is correlated, for the thermodynamic interpretation to hold.

The dependence of the asymptotic h-statistics on initial conditions is an established fact [30,31,32,33,44]. A numerical test, undertaken with initial conditions of the formh(0)=1aabs(x−L/2),a∈{0.5,1.5,3,20}. shows that the NEP also displays memory effects.

Within the studied interval, the slopes show a strong dependence on the initial conditions, and in some cases, the curves also display concavity (Figure 9).

As indicated in the Introduction, Golubović and Bruisnma [16] have analyzed an extension of the KPZ model. The equation for the the growth rate they have found is as follows:(21)∂h(x,t)∂t=ν∇2h(x,t)−K∇22h(x,t)+λ2∇h(x,t)2+ξ(x,t).

As in the KPZ case, this equation can be derived from a NEP, given by(22)ΦGB[h]:=ν2∫dx∇h2+K2∫dx∇2h2−λ2∫dx∫0tds∂sh∇h2, such that(23)∂h(x,t)∂t=−δΦGB[h]δh(x,t)+ξ(x,t).

Equation (23) expresses the Golubovic–Bruinsma model as a stochastic gradient flow driven by the functional derivative of the non-equilibrium potential ΦGB[h]. This formulation highlights the shared variational structure between this model and the standard KPZ equation, allowing for a unified analysis of their dynamical and statistical properties within the same theoretical framework.

From this form of the potential, we can immediately predict that, for plane initial conditions, the long time behavior will be dominated by a linear dependence on t, with a slope depending on λ2. We can also anticipate the existence of memory effects.

In the following figures, we show the time behavior of the indicated functional as well as, changing initial conditions, we prove the existence of memory effects. In all these figures, the parameters we have used are ν=λ=K=1.

From Figure 10, it is clear that the linear contributions have a similar behavior than in the EW case, while the nonlinear contribution, dominating at long time, reaches a linear dependence on t. The slope should scale with λ2, which is an aspect to be discussed in a forthcoming work.

In Figure 11, the memory effects are clearly shown when changing the initial conditions. A detailed study will be the subject of a forthcoming work.

Considering the associated Fokker–Planck equation, and considering the stationary case we observe that, independently of the bc, the exponentialPBGstat[h]∝exp{−ΦKPZ[h]/ε}, fullfils the stationary condition∂∂tPBGstat[h]=0, with ΦBGstat[h] given by Equation (22). Clearly, as in the KPZ case, this is a formal solution since PBGstat[h] cannot be normalized.

An interesting aspect was studied in [45] and corresponds to the large deviations of the interface height in this model. Such a study follows [46,47] on similar aspects for the KPZ equation.

This prima facie promising approach was also worked out in [40,41]. By writing h(x,t)=h¯(t)+y(x,t), withh¯(t)=V(Ω)−1∫Ωdxh(x,t), and being ∇h¯=0, Equation (2) becomes(24)∂ty(x,t)=ν∇2y(x,t)+λ2[∇y(x,t)]2+F+ξ(x,t)−h¯˙. Equation (2) yields h¯˙=F+G(t), withG(t):=λ2V(Ω)∫Ωdx[∇h(x,t)]2+ξ¯ parametrically depending on λ and F. Note the term ξ¯, whose variance is depressed from ϵ by V(Ω). Hence Equation (24) can be written in the form(25)∂ty(x,t)=ν∇2y(x,t)+λ2[∇y(x,t)]2−G(t)+ξ(x,t). Unfortunately, the inherent constraint ∫Ωdxy(x,t)=0 hinders the usefulness of this otherwise promising approach.