弱内聚社会互动中意图场的连续统动力学
Continuum dynamics of the intention field under weakly cohesive social interactions
摘要：
我们调查的长期动态的意见形成模型的灵感来自Borghesi，Bouchaud和詹森的工作。首先，我们推导出一个福克-普朗克型方程的假设下，个人之间的相互作用产生的意见很少的共识（掠碰撞近似）。其次，我们研究的条件下，福克-普朗克方程有非平凡的平衡，并推导出宏观极限（对应于长时间的动态和空间局域化的相互作用）的平均意见的演变。最后，我们比较了两种不同类型的相互作用率：在Borghesi，Bouchaud和Jensen（对称二进制相互作用）的工作中给出的原始的一个和一个灵感来自Motsch和Tadmor（非对称二进制相互作用）的作品。我们表明，第一种情况下导致一个保守的平均意见的密度模型，而第二种情况下导致一个非保守的方程。我们还表明，在这两个速率渐近达成共识的速度有相当不同的密度依赖性。
英文摘要：
We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. Firstly, we derive a Fokker-Planck type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Secondly, we study conditions under which the Fokker-Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is reached asymptotically for these two rates has fairly different density dependence.
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