Why the phase space fluid is incompressible?

Answer 1

let \rho is density of phase fluid and v is velocity of phase fluid (velocity in a phase space!!)

we take the equation of continuity (we suppose "mass" of phase fluid is conserved) and using Hamilton equations (we suppose classical mechanical system)

[ dot{p_i} = -frac{partial H}{q_i} ]

[ dot{q_i} = +frac{partial H}{p_i} ]

we obtain the result

[ frac{mathrm{d}rho}{mathrm{d}t} = - rho,mathrm{div},v = -rho sum_i left( frac{partialdot{q_i}}{q_i} + frac{partialdot{p_i}}{p_i} right) = 0 ] let \rho is density of phase fluid and v is velocity of phase fluid (velocity in a phase space!!)

we take the equation of continuity (we suppose "mass" of phase fluid is conserved) and using Hamilton equations (we suppose classical mechanical system)

\[ \dot{p_i} = -\frac{\partial H}{q_i} \]

\[ \dot{q_i} = +\frac{\partial H}{p_i} \] we obtain the result

\[ \frac{\mathrm{d}\rho}{\mathrm{d}t} = - \rho\,\mathrm{div}\,v = -\rho \sum_i \left( \frac{\partial\dot{q_i}}{q_i} + \frac{\partial\dot{p_i}}{p_i} \right) = 0 \]

Answer 2

The incompressibility of the phase space fluid is due to the conservation of phase space volume. As particles move through phase space, their density does not change, leading to the preservation of phase space volume. This results in the incompressibility of the fluid in phase space.