将 Lenia 实现为反应-扩散系统
Implementation of Lenia as a Reaction-Diffusion System
AI总结：
本文通过微分方程重写渐近 Lenia，证明其等价于 KT 模型的推广，并构建仅含扩散与局部反应项的 RD Lenia，揭示 CA 与 RD 系统的数学联系。
摘要：
反应-扩散（RD）系统以连续时空状态为特征，而元胞自动机（CA）以离散时空状态为标志，两者之间的关系仍缺乏充分理解。本文通过考察近期发展的一种名为 Lenia 的 CA 来深入探讨这一关系。我们证明，Lenia 的一个变体——渐近 Lenia——可以由微分方程全面描述，并且与原始 Lenia 不同，它不依赖于时间步节拍。此外，我们证明该表述在数学上等价于基于核的 Turing 模型（KT 模型）的一种推广。基于这些洞见，我们证明渐近 Lenia 可以由一个仅包含扩散项和空间局部反应项的 RD 系统复现，从而得到基于 RD 系统模拟的渐近 Lenia，即“RD Lenia”。然而，我们的 RD Lenia 不能被解释为化学系统，因为其反应项不满足质量作用动力学。
英文摘要：
The relationship between reaction-diffusion (RD) systems, characterized by continuous spatiotemporal states, and cellular automata (CA), marked by discrete spatiotemporal states, remains poorly understood. This paper delves into this relationship through an examination of a recently developed CA known as Lenia. We demonstrate that asymptotic Lenia, a variant of Lenia, can be comprehensively described by differential equations, and, unlike the original Lenia, it is independent of time-step ticks. Further, we establish that this formulation is mathematically equivalent to a generalization of the kernel-based Turing model (KT model). Stemming from these insights, we establish that asymptotic Lenia can be replicated by an RD system composed solely of diffusion and spatially local reaction terms, resulting in the simulated asymptotic Lenia based on an RD system, or "RD Lenia". However, our RD Lenia cannot be construed as a chemical system since the reaction term fails to satisfy mass-action kinetics.
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