TY - JOUR

T1 - Short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation

T2 - Starting from a parabola

AU - Kamenev, Alex

AU - Meerson, Baruch

AU - Sasorov, Pavel V.

N1 - Publisher Copyright:
© 2016 American Physical Society.

PY - 2016/9/7

Y1 - 2016/9/7

N2 - We study the probability distribution P(H,t,L) of the surface height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension when starting from a parabolic interface, h(x,t=0)=x2/L. The limits of L→∞ and L→0 have been recently solved exactly for any t>0. Here we address the early-time behavior of P(H,t,L) for general L. We employ the weak-noise theory - a variant of WKB approximation - which yields the optimal history of the interface, conditioned on reaching the given height H at the origin at time t. We find that at small HP(H,t,L) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as -lnP=f+|H|5/2/t1/2 and f-|H|3/2/t1/2. The factor f+(L,t) monotonically increases as a function of L, interpolating between time-independent values at L=0 and L=∞ that were previously known. The factor f- is independent of L and t, signaling universality of this tail for a whole class of deterministic initial conditions.

AB - We study the probability distribution P(H,t,L) of the surface height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension when starting from a parabolic interface, h(x,t=0)=x2/L. The limits of L→∞ and L→0 have been recently solved exactly for any t>0. Here we address the early-time behavior of P(H,t,L) for general L. We employ the weak-noise theory - a variant of WKB approximation - which yields the optimal history of the interface, conditioned on reaching the given height H at the origin at time t. We find that at small HP(H,t,L) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as -lnP=f+|H|5/2/t1/2 and f-|H|3/2/t1/2. The factor f+(L,t) monotonically increases as a function of L, interpolating between time-independent values at L=0 and L=∞ that were previously known. The factor f- is independent of L and t, signaling universality of this tail for a whole class of deterministic initial conditions.

UR - http://www.scopus.com/inward/record.url?scp=84990204517&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.94.032108

DO - 10.1103/PhysRevE.94.032108

M3 - Article

AN - SCOPUS:84990204517

SN - 2470-0045

VL - 94

JO - Physical Review E

JF - Physical Review E

IS - 3

M1 - 032108

ER -