Sample space reducing (SSR) processes offer a simple analytical way to understand the origin and ubiquity of power-laws in many path-dependent complex systems. SRR processes show a wide range of applications that range from fragmentation processes, language formation to search and cascading processes. Here we argue that they also offer a natural framework to understand stationary distributions of generic driven non-equilibrium systems that are composed of a driving- and a relaxing process. We show that the statistics of driven non-equilibrium systems can be derived from the understanding of the nature of the underlying driving process. For constant driving rates exact power-laws emerge with exponents that are related to the driving rate. If driving rates become state-dependent, or if they vary across the life-span of the process, the functional form of the state-dependence determines the statistics. Constant driving rates lead to exact power-laws, a linear state-dependence function yields exponential or Gamma distributions, a quadratic function produces the normal distribution. Logarithmic and power-law state dependence leads to log-normal and stretched exponential distribution functions, respectively. Also Weibull, Gompertz and Tsallis-Pareto distributions arise naturally from simple state-dependent driving rates. We discuss a simple physical example of consecutive elastic collisions that exactly represents a SSR process.

中文翻译：

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行驶速度如何确定具有固定分布的被驱动非平衡系统的统计信息。

减少样本空间（SSR）过程提供了一种简单的分析方法，以了解许多依赖于路径的复杂系统中幂律的起源和普遍性。SRR流程显示了广泛的应用程序，从碎片处理，语言形成到搜索和级联过程。在这里，我们认为，它们还提供了一个自然的框架来理解由驱动过程和放松过程组成的通用驱动非平衡系统的平稳分布。我们表明，驱动非平衡系统的统计数据可以从对基本驱动过程的本质的理解中得出。对于恒定的行驶速度，会出现与该行驶速度相关的指数的精确幂律。如果行驶速度取决于状态，或者在整个过程的整个生命周期内变化，状态依赖的功能形式决定了统计量。恒定的驱动速度会导致精确的幂律，线性的状态相关函数会产生指数或Gamma分布，二次函数会产生正态分布。对数和幂律状态相关性分别导致对数正态和拉伸指数分布函数。Weibull，Gompertz和Tsallis-Pareto分布也很自然地来自于简单的依赖状态的驱动率。我们讨论了一个连续的弹性碰撞的简单物理示例，该示例精确地表示了SSR过程。对数和幂律状态相关性分别导致对数正态和拉伸指数分布函数。Weibull，Gompertz和Tsallis-Pareto分布也很自然地来自于简单的依赖状态的驱动率。我们讨论了一个连续的弹性碰撞的简单物理示例，该示例精确地表示了SSR过程。对数和幂律状态相关性分别导致对数正态和拉伸指数分布函数。Weibull，Gompertz和Tsallis-Pareto分布也很自然地来自于简单的依赖状态的驱动率。我们讨论了一个连续的弹性碰撞的简单物理示例，该示例精确地表示了SSR过程。