FOR statistical physics 统计物理
The trajectory 轨迹 of a single particle in phase is completely determined by the equatoins of motion if the initial conditions are known.
{many-body systems→ complicated system}example: a [statistical ensemble](https://www.google.com/search?q=statistical+ensemble&sourceid=chrome&ie=UTF-8#:~:text=A statistical ensemble is a collection of various microstates of an equilibrium macroscopic system as determined by the constraints operating on the system.) in gas
It is possible to define an ensemble 整体 of points in phase space that encompasses 包括 all possible trajectories for complicated system. That is, the statistical distribution of particles in phase can be specified. 确定粒子在相空间的统计分布。
这种分布是假想完全可知的，与系综的遍历性假设无关。
Consider a density $\rho$ of representative points in $(q,p)$ phase space.
The number $N$ of systems in the volume element $dv$ is $N= \rho dv$,
where it is assumed that the infinitessimal volume element contians many possible systems so that $\rho$ can be considered as a continuous distribution.大量→连续分布假设
$$ dv=dq_1,dq_2.\ldots dq_s,dp_1,dp_2\ldots dp_s $$
The number of representative points moving across the left-hand edge into the area per unit time is
$$ \rho(\dot{q}_i dt)dp_i/dt=\rho\dot{q}_idp_i $$
The number of representative points flowing out of the area per unit time along the right-hand edge is
$$ \begin{aligned}\rho\dot{q}_i\mid _{q_i+dq_i} & = \frac{ \rho\dot{q}_i\mid _{q_i+dq_i}-\rho\dot{q}_i\mid _{q_i} }{dq_i}dq_i+ \rho\dot{q}_i\mid _{q_i}\\ & = \left[\rho\dot{q}_i+\frac\partial{\partial q_i}(\rho\dot{q}_i)dq_i\right]dp_i\end{aligned} $$
The net increase in $\rho$ in the infinitessimal retangular element $dq_idp_i$ due to flow in the horizontal 横向 direction is
$$ \rho\dot{q}_idp_i-\left[\rho\dot{q}_i+\frac\partial{\partial q_i}(\rho\dot{q}_i)dq_i\right]dp_i=-\frac\partial{\partial q_i}(\rho\dot{q}_i)dq_idp_i $$
Similarly, the net gain due to flow in the vertical direction is
$$ -\frac\partial{\partial p_i}(\rho\dot{p}_i)dp_idq_i $$