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What does density function in phase space physically mean? How does it indicate, the more familiar density that we are accustomed to ( an analogy may be), in phase space?

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It is a probability density, not a density of something like matter or energy.

The probability density $f$ answers the question "how likely is it (what is the probability that) the microstate $\omega$ in one of the points in some set $A$":

$$P(\omega\in A) = \int_A f(p,q)dpdq$$

The most other famous example of probability density in physics is the squared modulus of the wave function $\psi$, $|\psi(x)|^2$ being the probability density of finding the particle at point $x$. Another example is Brownian motion of a single particle.

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If you integrate out the momentum variables, then you get the usual density as a function of just position. Let's say there are N particles each with mass $m$ so total mass $Nm$.

$$ \int d^3p d^3x \; \; \rho_{phase} (x,p) = Nm\\ \int d^3p \; \; \rho_{phase} (x,p) = \rho (x)\\ $$

So the phase space density is giving more refined information. Careful that the units are different. For example, $\rho$ has units mass per volume, but $\rho_{phase}$ has units like $kg/(m kg m/s)^3$

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