Most natural complex systems exhibit fluctuations-driven processes, which work at far from equilibrium states, and are generally dissipative processes, for instance living cells. We studied this phenomenon within the stochastic framework by taking a set of nonequilibrium, bimolecular, autocatalytic reactions, originally proposed by Nicolis (1972). We also extended this model to incorporate the concept of time delay. Firstly, for both non-delay and delay cases, we calculated the exact non-stationary probability distribution solutions of the corresponding Master equations, which are found to deviate from the Maxwell–Boltzmann distribution. The analytically calculated probability distribution P of an autocatalyst X in the chemically reacting model system is found to follow some universal class of probability distributions at different situations. At the thermodynamic limit with a large population, P obeys Normal distribution. Again, we showed that one of the causes of this peculiar behaviour is the fluctuations in the reacting system. The analytical result of the Fano factor F in the non-delay case predicted a noise-enhanced process for our dynamical stochastic system which could probably drive the system far from equilibrium. For the delay case, the analytically calculated F was found to depend on the time delay function, which predicts that time delay could play an important role in regulating the system dynamics. These analytical predictions were then verified using numerical experiments with the stochastic simulation algorithm (SSA) and delay stochastic simulation algorithm (DSSA). Indeed, numerical results from SSA and DSSA confirmed noise-enhanced processes which are far from equilibrium and dissipative in nature.

大多数自然复杂系统表现出波动驱动的过程，这些过程在远离平衡状态的情况下工作，并且通常是耗散过程，例如活细胞。我们采用一组非平衡、双分子、自催化反应，在随机框架内研究了这种现象，最初由 Nicolis (1972) 提出。我们还扩展了这个模型以包含时间延迟的概念。首先，对于非延迟和延迟情况，我们计算了相应主方程的精确非平稳概率分布解，发现它们偏离了麦克斯韦-玻尔兹曼分布。解析计算的概率分布P发现化学反应模型系统中的自动催化剂X在不同情况下遵循某些通用类别的概率分布。在人口众多的热力学极限下，P服从正态分布。我们再次证明，这种特殊行为的原因之一是反应系统的波动。非延迟情况下Fano 因子F的分析结果预测了我们的动态随机系统的噪声增强过程，这可能会使系统远离平衡。对于延迟情况，解析计算的F发现依赖于时间延迟函数，这预测时间延迟可以在调节系统动力学中发挥重要作用。然后使用随机模拟算法 (SSA) 和延迟随机模拟算法 (DSSA) 的数值实验验证这些分析预测。事实上，SSA 和 DSSA 的数值结果证实了噪声增强过程，其本质上远非平衡和耗散。