3.2.2 Classical reduced distribution functions

To evaluate averages, instead of the N-particle distribution function <sup>f</sup> <sup>N</sup> <sup>r</sup>1; …; <sup>r</sup>N; <sup>p</sup>1; …; <sup>p</sup>N; <sup>t</sup> � � often reduced <sup>s</sup>-particle distribution functions

$$f\_s(\mathbf{r}\_1, \dots, \mathbf{p}\_s; t) = \int \frac{\mathbf{d}^3 r\_{s+1} \dots \mathbf{d}^3 \mathbf{p}\_N}{(N-s)! \mathbf{h}^{3(N-s)}} f\_N(r\_1, \dots, \mathbf{p}\_N; t) \tag{96}$$

are sufficient. Examples are the particle density, the Maxwell distribution of the particle velocities, and the pair correlation function.

We are interested in the equations of motion for the reduced distribution functions. For classical systems, one finds a hierarchy of equations. From the Liouville equation, Eq. (95) without external potential,

$$\frac{\mathbf{d}f\_N}{\mathbf{d}t} = \frac{\partial \mathbf{f}\_N}{\partial t} + \sum\_{i}^{N} \boldsymbol{\nu}\_i \frac{\partial \mathbf{f}\_N}{\partial r\_i} - \sum\_{i \neq j}^{N} \frac{\partial V\_{ij}}{\partial r\_i} \frac{\partial \mathbf{f}\_N}{\partial \mathbf{p}\_i} = \mathbf{0} \tag{97}$$

we obtain the equation of motion for the reduced distribution function fs through integration over the 3ð Þ N � s other variables:

$$\frac{\mathrm{d}f\_{s}}{\mathrm{d}t} = \frac{\partial f\_{s}}{\partial t} + \sum\_{i=1}^{s} \boldsymbol{\sigma}\_{i} \frac{\partial \boldsymbol{f}\_{s}}{\partial \mathbf{r}\_{i}} - \sum\_{i \neq j}^{s} \frac{\partial V\_{ij}}{\partial \mathbf{r}\_{i}} \frac{\partial \boldsymbol{f}\_{s}}{\partial \mathbf{p}\_{i}} = \sum\_{i=1}^{s} \left[ \frac{\mathrm{d}^{3}\mathbf{r}\_{s+1} \mathrm{d}^{3}\mathbf{p}\_{s+1}}{h^{3}} \frac{\partial V\_{i,s+1}}{\partial \mathbf{r}\_{i}} \frac{\partial \boldsymbol{f}\_{s+1}(\mathbf{r}\_{1} \ldots \mathbf{p}\_{s+1}, t)}{\partial \mathbf{p}\_{i}} \right]. \tag{98}$$

This hierarchy of equations is called BBGKY hierarchy, standing for Bogoliubov, Born, Green, Kirkwood, and Young.
