Population diversity is crucial in evolutionary algorithms as it helps with global exploration and facilitates the use of crossover. Despite many runtime analyses showing advantages of population diversity, we have no clear picture of how diversity evolves over time. We study how population diversity of $(\mu+1)$ algorithms, measured by the sum of pairwise Hamming distances, evolves in a fitness-neutral environment. We give an exact formula for the drift of population diversity and show that it is driven towards an equilibrium state. Moreover, we bound the expected time for getting close to the equilibrium state. We find that these dynamics, including the location of the equilibrium, are unaffected by surprisingly many algorithmic choices. All unbiased mutation operators with the same expected number of bit flips have the same effect on the expected diversity. Many crossover operators have no effect at all, including all binary unbiased, respectful operators. We review crossover operators from the literature and identify crossovers that are neutral towards the evolution of diversity and crossovers that are not.

翻译：种群多样性对于进化算法至关重要，因为它有助于全局探索并促进交叉操作的使用。尽管许多运行时分析已证明种群多样性的优势，但我们对多样性如何随时间演变仍缺乏清晰认识。我们研究了在适应度中性环境中，以成对汉明距离之和衡量的$(\mu+1)$算法种群多样性的演化过程。我们给出了种群多样性漂移的精确公式，并证明其趋向于稳态。此外，我们给出了接近该稳态的期望时间上界。研究发现，这些动力学特性（包括稳态位置）出人意料地不受众多算法选择的影响。所有具有相同期望位翻转次数的无偏变异算子，对预期多样性具有相同影响。许多交叉算子（包括所有二元无偏、尊重型算子）对多样性毫无影响。我们回顾了文献中的交叉算子，并识别出对多样性演化呈中性及非中性的交叉算子类型。