Skewness in(1+1)-dimensional Kardar-Parisi-Zhang–type growth

Abstract

We use the (1 + 1)-dimensional Kardar-Parisi-Zhang equation driven by a Gaussian white noise and employ the dynamic renormalization-group of Yakhot and Orszag without rescaling [J. Sci. Comput. 1, 3 (1986)]. Hence we calculate the second and third order moments of height distribution using the diagrammatic method in the large scale and long time limits. The moments so calculated lead to the value S = 0.3237 for the skewness. This value is comparable with numerical and experimental estimates.

A renormalization scheme and skewness of height fluctuations in (1 + 1)-dimensional VLDS dynamics

“…A similar scheme was employed earlier for the calculation of skewness [40] and kurtosis [41] in the case of non-conserved interface growth governed by the (1 + 1)-dimensional KPZ equation. The amputated parts of the connected loop diagrams, namely, l 2 , l etc., for the second and third moments as shown in Figs.…”

Kurtosis of height fluctuations in (1 + 1) dimensional KPZ dynamics

“…In our simplified scheme of calculation, we obtained the fourth cumulant at one-loop order. While a one-loop scheme for the third cumulant was nearly successful in estimating the stationary skewness value [49], a higher order calculation would appear to be more appropriate for the fourth cumulant. Noting that the perturbation expansion is about a Gaussian state, the Gaussian behavior seems to play some role with the increase in order of the cumulant, lowering the estimated value.…”

“…obtained from renormalization-group calculations [49], we obtain the solution to the differential equation (17) as…”

Gaussian statistics as an emergent symmetry of the stochastic scalar Burgers equation

“…where k n = − n−1 j=1 k j , ω n = − n−1 j=1 ω j . The function L n is perturbatively computed to one loop order [43][44][45][46] as L n = (2D)δ n,2 + L n,1 ,…”