We propose some conjectures on the asymptotic distribution of the probabilistic Burgers cellular automaton (PBCA), which is defined by a simple rule of particle motion with a probabilistic parameter. Asymptotic distribution of configurations converges to a unique steady state for PBCA. We propose a new and widely-applicable approach to analyze probabilistic particle systems and apply it concretely to PBCA and its extensions. We introduce a conjecture on the distribution and derive the asymptotic probability expressed by the GKZ hypergeometric function. If the space size goes into infinity, we can evaluate the relationship between the density and flux of particles for infinite space. Moreover, we propose two extended systems of PBCA and analyze their asymptotic behavior.

中文翻译：

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评估由概率元胞自动机表示的粒子系统的渐近分布的新方法

我们对概率 Burgers 元胞自动机 (PBCA) 的渐近分布提出了一些猜想，它由具有概率参数的粒子运动的简单规则定义。配置的渐近分布收敛到 PBCA 的唯一稳态。我们提出了一种新的、广泛适用的方法来分析概率粒子系统，并将其具体应用于 PBCA 及其扩展。我们引入了关于分布的猜想，并导出了由 GKZ 超几何函数表示的渐近概率。如果空间大小趋于无穷大，我们可以评估无限空间中粒子的密度和通量之间的关系。此外，我们提出了 PBCA 的两个扩展系统并分析了它们的渐近行为。