Krajenbrink Alexandre, Le Doussal Pierre
CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France.
Phys Rev E. 2017 Aug;96(2-1):020102. doi: 10.1103/PhysRevE.96.020102. Epub 2017 Aug 14.
The early-time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension, starting from a Brownian initial condition with a drift w, is studied using the exact Fredholm determinant representation. For large drift we recover the exact results for the droplet initial condition, whereas a vanishingly small drift describes the stationary KPZ case, recently studied by weak noise theory (WNT). We show that for short time t, the probability distribution P(H,t) of the height H at a given point takes the large deviation form P(H,t)∼exp[-Φ(H)/sqrt[t]]. We obtain the exact expressions for the rate function Φ(H) for H<H_{c2}. Our exact expression for H_{c2} numerically coincides with the value at which WNT was found to exhibit a spontaneous reflection symmetry breaking. We propose two continuations for H>H_{c2}, which apparently correspond to the symmetric and asymmetric WNT solutions. The rate function Φ(H) is Gaussian in the center, while it has asymmetric tails, |H|^{5/2} on the negative H side and H^{3/2} on the positive H side.
利用精确的弗雷德霍姆行列式表示，研究了一维 Kardar-Parisi-Zhang（KPZ）方程在布朗初始条件下带有漂移(w)的早期情况。对于大漂移，我们恢复了液滴初始条件下的精确结果，而极小的漂移描述了最近由弱噪声理论（WNT）研究的平稳 KPZ 情况。我们表明，对于短时间(t)，给定一点处高度(H)的概率分布(P(H,t))具有大偏差形式(P(H,t) \sim \exp[-\Phi(H)/\sqrt{t}])。我们得到了(H < H_{c2})时速率函数(\Phi(H))的精确表达式。我们关于(H_{c2})的精确表达式在数值上与发现 WNT 表现出自发反射对称破缺的值一致。我们提出了(H > H_{c2})的两种延续情况，它们显然对应于对称和不对称的 WNT 解。速率函数(\Phi(H))在中心是高斯型的，而在两侧具有不对称尾巴，在负(H)侧为(|H|^{5/2})，在正(H)侧为(H^{3/2})。
