Is the Integral of the Square of Phase Density Constant in Classical Mechanics?

[xylai]

In the classical mechanics each system can be described by the phase density \rho(x,t)
, which is evolved by Liouville's equation.
Recently I read a paper: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVM-46S5C37-24M&_user=6104324&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000069295&_version=1&_urlVersion=0&_userid=6104324&md5=d41e737ddc2514dbbeca4a99047e66f7"(Y, Gu, PLA, 149, 95 (1990)). In it, it says that

\int\rho(x,t)^{2} dx is constant in time.
But, as far as I know, due to the conservation of particles, \int\rho(x,t) dx
is constant in time.
I don’t know how he got the conclusion?

 

[azntoon]

Could you help me with this question?Thanks! The paper you cited is talking about the non-dissipative systems. For such systems, the total phase-space density is conserved. That means not only the integral of the density over the position space is conserved, but also the integral of the density over the momentum space is conserved. Both the integrals are related by the expression \(\int \rho(x,t)^2 dx = \int \rho(p,t)^2 dp\), which implies that the integral of the phase-space density is constant in time.