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.

 

[denian]

Thank you for bringing up this interesting topic. It is true that in classical mechanics, each system can be described by the phase density \rho(x,t) which follows Liouville's equation. The phase density is a measure of the distribution of particles in a system, and it is often used to study the behavior of systems in equilibrium or in a steady state.

In the paper you mentioned, the author claims that the integral of the square of the phase density, \int\rho(x,t)^{2} dx, is constant in time. This is a surprising result, as it suggests that the phase density remains unchanged even as the system evolves. However, as you correctly pointed out, the conservation of particles dictates that the integral of the phase density, \int\rho(x,t) dx, is actually the quantity that remains constant in time.

Upon further investigation, it appears that the author may have made a mistake in their calculation or interpretation of the results. It is important to note that the phase density is not the only quantity that remains constant in time - there are other conserved quantities in classical mechanics, such as energy and momentum. It is possible that the author may have mistaken one of these quantities for the phase density.

In any case, it is always important to carefully review and validate the results of scientific papers, especially when they suggest unexpected or counterintuitive conclusions. As scientists, it is our responsibility to critically evaluate and question the findings of others in order to advance our understanding of the natural world. Thank you again for bringing this issue to our attention.