Godrèche Claude, Majumdar Satya N, Schehr Grégory
Institut de Physique Théorique, IPhT, CEA Saclay, and URA 2306, 91191 Gif-sur-Yvette Cedex, France.
Phys Rev Lett. 2009 Jun 19;102(24):240602. doi: 10.1103/PhysRevLett.102.240602. Epub 2009 Jun 18.
We consider the excursions, i.e., the intervals between consecutive zeros, of stochastic processes that arise in a variety of nonequilibrium systems and study the temporal growth of the longest one l_{max}(t) up to time t. For smooth processes, we find a universal linear growth l_{max}(t) approximately Q_{infinity}t with a model dependent amplitude Q_{infinity}. In contrast, for nonsmooth processes with a persistence exponent theta, we show that l_{max}(t) has a linear growth if theta < theta_{c} while l_{max}(t) approximately t;{1-psi} if theta > theta_{c}. The amplitude Q_{infinity} and the exponent psi are novel quantities associated with nonequilibrium dynamics. This behavior is obtained by exact analytical calculations for renewal and multiplicative processes and numerical simulations for other systems such as the coarsening dynamics in Ising model as well as the diffusion equation with random initial conditions.
我们考虑各种非平衡系统中出现的随机过程的游程，即连续零点之间的间隔，并研究最长游程(l_{max}(t))直至时间(t)的时间增长情况。对于平滑过程，我们发现最长游程具有通用的线性增长(l_{max}(t)\approx Q_{\infty}t)，其中幅度(Q_{\infty})取决于模型。相比之下，对于具有持续性指数(\theta)的非平滑过程，我们表明如果(\theta < \theta_{c})，(l_{max}(t))具有线性增长，而如果(\theta > \theta_{c})，(l_{max}(t)\approx t^{1 - \psi})。幅度(Q_{\infty})和指数(\psi)是与非平衡动力学相关的新量。这种行为是通过对更新过程和乘法过程的精确解析计算以及对其他系统（如伊辛模型中的粗化动力学以及具有随机初始条件的扩散方程）的数值模拟得到的。
