论文标题
简单随机交换模型的连续性和热力学极限
Continuum and thermodynamic limits for a simple random-exchange model
论文作者
论文摘要
我们讨论可用于财富分配的简单随机交换模型的各种限制。我们从离散的状态空间开始 - 该模型的离散时间版本，在适当的缩放下，我们显示了其功能收敛到连续空间 - 离散时间模型。然后，我们显示了经验分布的热力学限制到Boltzmann类型动力学方程的溶液。我们解决了这个方程式，并表明解决方案与马尔可夫链的不变度度量的适当限制相吻合。通过这种方式，我们完成了Boltzmann从随机动力学中得出这个简单模型的动力学方程的程序。发现了三个平均场限制的不变措施的家族，我们表明只有两个家族可以作为离散系统的限制获得，而第三个家庭是无关的。最后，我们在整数分区的框架内投射了结果，并加强了文献中已经可用的一些结果。
We discuss various limits of a simple random exchange model that can be used for the distribution of wealth. We start from a discrete state space - discrete time version of this model and, under suitable scaling, we show its functional convergence to a continuous space - discrete time model. Then, we show a thermodynamic limit of the empirical distribution to the solution of a kinetic equation of Boltzmann type. We solve this equation and we show that the solutions coincide with the appropriate limits of the invariant measure for the Markov chain. In this way we complete Boltzmann's program of deriving kinetic equations from random dynamics for this simple model. Three families of invariant measures for the mean field limit are discovered and we show that only two of those families can be obtained as limits of the discrete system and the third is extraneous. Finally, we cast our results in the framework of integer partitions and strengthen some results already available in the literature.
