TY - JOUR

T1 - Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition

AU - Smith, Naftali R.

N1 - Publisher Copyright:
© 2022 American Physical Society.

PY - 2022/10/1

Y1 - 2022/10/1

N2 - We consider the relaxation (noise-free) statistics of the one-point height H=h(x=0,t), where h(x,t) is the evolving height of a one-dimensional Kardar-Parisi-Zhang (KPZ) interface, starting from a Brownian (random) initial condition. We find that, at short times, the distribution of H takes the same scaling form -lnP(H,t)=S(H)/t as the distribution of H for the KPZ interface driven by noise, and we find the exact large-deviation function S(H) analytically. At a critical value H=Hc, the second derivative of S(H) jumps, signaling a dynamical phase transition (DPT). Furthermore, we calculate exactly the most likely history of the interface that leads to a given H, and show that the DPT is associated with spontaneous breaking of the mirror symmetry x↔-x of the interface. In turn, we find that this symmetry breaking is a consequence of the nonconvexity of a large-deviation function that is closely related to S(H), and describes a similar problem but in half space. Moreover, the critical point Hc is related to the inflection point of the large-deviation function of the half-space problem.

AB - We consider the relaxation (noise-free) statistics of the one-point height H=h(x=0,t), where h(x,t) is the evolving height of a one-dimensional Kardar-Parisi-Zhang (KPZ) interface, starting from a Brownian (random) initial condition. We find that, at short times, the distribution of H takes the same scaling form -lnP(H,t)=S(H)/t as the distribution of H for the KPZ interface driven by noise, and we find the exact large-deviation function S(H) analytically. At a critical value H=Hc, the second derivative of S(H) jumps, signaling a dynamical phase transition (DPT). Furthermore, we calculate exactly the most likely history of the interface that leads to a given H, and show that the DPT is associated with spontaneous breaking of the mirror symmetry x↔-x of the interface. In turn, we find that this symmetry breaking is a consequence of the nonconvexity of a large-deviation function that is closely related to S(H), and describes a similar problem but in half space. Moreover, the critical point Hc is related to the inflection point of the large-deviation function of the half-space problem.

UR - http://www.scopus.com/inward/record.url?scp=85140140227&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.106.044111

DO - 10.1103/PhysRevE.106.044111

M3 - Article

C2 - 36397488

AN - SCOPUS:85140140227

SN - 2470-0045

VL - 106

JO - Physical Review E

JF - Physical Review E

IS - 7

M1 - 044111

ER -