随机交换随机游走的精确可解模型
Exactly Solvable Model of Random Walks with Stochastic Exchange
AI总结：
该研究精确求解了具有随机位置交换的离散随机游走模型，揭示了两个动力学交叉及持续空间各向异性，为生物学和金融学中的交换过程提供了统一定量框架。
摘要：
我们精确求解了两个离散随机游走者在通道中的非平衡动力学，其转移速率 $p \ eq q$，并以速率 $s$ 交换位置。我们精确计算了游走者的联合概率分布 $P_{n,m}(t)$，揭示了两个动力学交叉的存在。第一个交叉标志着从独立扩散到交换主导区域的过渡，在该区域中粒子表现为交换位置的相同随机游走者。第二个交叉发生在两个通道变得不可区分且游走者围绕同一位置移动时。此外，我们证明了由两个通道中概率分布的二阶矩之差定义的持续空间各向异性的存在。我们的结果可能为理解多种系统提供定量框架。在生物学中，其动机来自马达蛋白（驱动蛋白/动力蛋白）交换货物主导权、膜受体交换结合伙伴，或具有活动依赖性可塑性的脑突触。在金融学中，它模拟具有不同风险特征的交易者在限价订单簿中交换位置，或耦合市场之间的波动溢出。这些多样化的系统共享一个统一主题：尽管个体存在异质性，交换过程介导了宏观相关性。
英文摘要：
We solve exactly the non-equilibrium dynamics of two discrete random walkers moving in channels with transition rates $p \neq q$ that swap positions at a rate $s$. We compute exactly the joint probability distribution $P_{n,m}(t)$ for the walkers, revealing the existence of two dynamical crossovers. The first signals the passage from independent diffusion to a swap-dominated regime where the particles act as identical random walkers swapping positions. The second crossover occurs when both channels become indistinguishable and the walkers move around the same position. Furthermore, we demonstrate the existence of a persistent spatial anisotropy defined by the difference between the second moments of the probability distributions in the two channels. Our results may provide a quantitative framework to understand diverse systems. In biology, it is motivated by motor proteins (kinesin/dynein) exchanging cargo leadership, membrane receptors swapping binding partners, or brain synapses with activity-dependent plasticity. In finance, it models traders with distinct risk profiles swapping positions in limit-order books, or volatility spillover between coupled markets. These diverse systems share a unifying theme: exchange processes mediate macroscopic correlations despite individual heterogeneity.
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