We introduce a class of new one-dimensional linear Fokker–Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker–Planck equations remains unchanged with respect to the steady state of the original Fokker–Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium.

中文翻译：

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模拟财富分配动态的非麦克斯韦动力学方程

我们引入了一类新的一维线性 Fokker-Planck 型方程，描述了多主体社会中财富分配的动态。根据 Bouchaud 和 Mézard 先前的研究，通过标准限制程序，通过对 Cordier、Pareschi 和 Toscani 于 2005 年引入的动力学模型引入经济相关的变体来获得方程。这些新的 Fokker-Planck 方程预测的财富稳态相对于原始 Fokker-Planck 方程的稳态保持不变。然而，与原始方程不同的是，它通过具有权重和经典熵方法的新对数 Sobolev 不等式证明，解以指数速度快速收敛到平衡。