中文摘要：

基于两相湍流PDF理论，颗粒矩拉格朗日方程可以通过求解颗粒PDF输运方程获得，但不同的数学方法推导获得的颗粒PDF输运方程形式是不同的，哪一种颗粒PDF输运方程的形式更可靠呢？该文先介绍不同形式的颗粒PDF输运方程，然后给出了使用颗粒PDF输运方程获得的颗粒矩拉格朗日方程。另一方面，从颗粒朗之万方程出发，用简单平均的方法推导颗粒运动一阶矩与二阶矩拉格朗日方程组，将这两种不同方法得到的结果进行比较。结果表明，基于色噪声扩维法PDF理论得到的颗粒矩方程组更为可靠。

英文摘要：

The particle moment Lagrange equations can be obtained by solving the particle probability density function （PDF） transport equation based on PDF two-phase theory. But different mathematical methods obtain different forms of PDF transport equations. Which form is more reliable？ In this paper, the different forms PDF transport equations are introduced firstly, and then the particle moment Lagrange equations are given by using PDF transport equation. On the other hand, with Langevin equation, one-order and two-order moment Lagrange equations are derived with a simple average method and the results of these two different methods are compared. It is shown that the moment equations based on PDF theory with expanding dimension method of colored noise is more reliable.