由 Langevin 随机微分方程导出的各向异性扩散趋化模型——献给邓引斌教授 70 寿辰

摘要/Abstract

摘要：

该文提出并讨论了一种 Langevin 型随机趋化模型, 该模型假定细胞运动的统计增量是由细胞速度的波动引起的. 该文的主要目的是在提出的随机模型的基础上, 推导出著名的 Keller-Segel 型趋化模型, 并建立随机趋化模型与确定性趋化模型之间的联系. 首先利用平均场理论, 作者推导出与 Langevin 随机趋化模型相对应的平均场趋化模型 (即 Fokker-Planck 方程). 然后, 基于该平均场趋化模型, 利用最小化原理, 矩封闭方法, 近似技巧和尺度论证, 作者推导出了经典的 Keller-Segel 模型. 明确了微观参数和宏观参数之间的关系. 此外, 通过最小化平均场模型的自由能, 作者得到了 Langevin 随机趋化模型的概率密度函数的解析近似并讨论了其生物学意义.

关键词: 趋化性, Keller-Segel 模型, Langevin 随机模型, 平均场趋化模型, 矩封闭, 自由能, 约束极小

Abstract:

This paper proposes and discusses a Langevin type stochastic chemotaxis model which assumes that the statistical increment of cell motion results from the fluctuation of cell velocity. The main purpose of present work is to derive the well-known Keller-Segel model of population chemotaxis from the proposed stochastic model and establish the connection between the stochastic and deterministic chemotaxis model. We first derive the mean-field chemotaxis model (i.e. Fokker-Planck equation) corresponding to the Langevin stochastic chemotaxis model by means of the mean field theory. Then based on the mean-field chemotaxis model, we derive the classical Keller-Segel model by using the minimization principle, moment closure approach, approximation technique and scaling argument. The relationship between microscopic and macroscopic parameters is explicitly identified. Moreover an analytical approximation of the probability density function of the Langevin stochastic chemotaxis model is found by minimizing the free energy of the mean-field model. The biological implications are discussed along the studies.

Key words: chemotaxis, Keller-Segel model, Langevin stochastic model, mean-field chemotaxis model, moment closure, free energy, constrained minimization

中图分类号:

O175.2

刘嘉, 王治安. 由 Langevin 随机微分方程导出的各向异性扩散趋化模型——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1486-1504.

Jia Liu, Zhian Wang. Chemotaxis Models with Anisotropic Diffusion Derived from Langevin Stochastic Equations[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1486-1504.

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