The long-time dynamics of reaction-diffusion processes in low dimensions is dominated by fluctuation effects. The one-dimensional coagulation-diffusion process describes the kinetics of particles which freely hop between the sites of a chain and where upon encounter of two particles, one of them disappears with probability one. The empty-interval method has, since a long time, been a convenient tool for the exact calculation of time-dependent particle densities in this model. We generalise the empty-interval method by considering the probability distributions of two simultaneous empty intervals at a given distance. While the equations of motion of these probabilities reduce for the coagulation-diffusion process to a simple diffusion equation in the continuum limit, consistency with the single-interval distribution introduces several non-trivial boundary conditions which are solved for the first time for arbitrary initial configurations. In this way, exact space-time-dependent correlation functions can be directly obtained and their dynamic scaling behaviour is analysed for large classes of initial conditions.
Tricritical directed percolation with long-range interaction in one and two dimensions
最近，在受驱动的开放量子自旋系统中，量子接触过程，即相干和非相干发生的分支和凝聚过程，被理论和实验研究。在半经典方法中，量子相干性效应被认为是两个相邻原子参与相邻原子从非活性（基态）激发到活性态（激发态$s$）的过程。在这种情况下，发生二阶和一阶转变。因此，存在三临界点，在该点处，过渡属于三临界定向渗流（TDP）类。另一方面，当原子被激发到$d$态时，引发长程相互作用。在这里，为了解释这种长程相互作用，我们将TDP模型扩展为具有$\sim 1/r^{d+σ}$形式的长程相互作用的模型（表示为LTDP），其中$r$是分离，$d$是空间维度，而$σ$是控制参数。特别是，我们调查的LTDP类低于上临界尺寸$d_c=$ min $(3,\,1.5σ)$的属性。数值计算得到了LTDP类的一组临界指数，并确定了LTDP类的$σ$的区间。最后，我们构建了一个图的通用类的空间（$d$，$σ$）。
