Liouville's theorem is commonly stated as $$\frac{d\rho}{dt} = \frac{\partial{\rho}}{\partial{t}} + [\rho, H]$$

Where, $H$ is Hamiltonian of the system, $\rho$ is the density, $[...]$ is the usual Poisson bracket.

My Textbook (name given below) interprets the theorem as

"The "local" density of the representative points, as viewed by an observer moving with a representative point, stays constant in time."

My question is that if an observer is moving with the representative point $(q_i, p_i)$ then wouldn't the density he will observe for the representative points in the whole phase space be basically looking constant in time to him? (If there's some reason for the localisation condition to be introduced in the statement, then what is the extent of this locality?)


  1. Statistical Mechanics - R.K. Pathria, Paul D. Beale
  • $\begingroup$ More on the definition of local. $\endgroup$
    – Qmechanic
    Mar 26, 2021 at 14:20
  • $\begingroup$ Are you completely current on the Lagrangian description of fluid flow, in sharp contrast to the Eulerian picture? $\endgroup$ Mar 26, 2021 at 14:50
  • $\begingroup$ WP. $\endgroup$ Mar 26, 2021 at 14:54
  • 2
    $\begingroup$ This might be useful. $\endgroup$ Mar 26, 2021 at 14:59
  • 2
    $\begingroup$ $\rho$ represents an ensemble, which essentially is a measure in the phase space. It's not a dynamical variable, but a weight in phase space (Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley & Sons, 1975, Sec. 2.2.) How an "observer" can sit somewhere and "view" a piece of the phase space evolve with its abstract statistical measure I cannot figure out. And the density changes from point to point. Otherwise it would be singular. This is clearly stated in last reference provided by @CosmasZachos. $\endgroup$
    – joigus
    Mar 27, 2021 at 7:50

1 Answer 1


This is just a pedagogic way of "understanding" Liouville's theorem. Whereas the global "shape" of the probability distribution changes with time, if you place a walker in $q_0,p_0$ with a probability $\rho(q_0,p_0)dp_0dp_0$ in its backpack, this walker will always carry the same probability $\rho(q_t,p_t)dp_tdp_t=\rho(q_0,p_0)dp_0dp_0$ through its Hamiltonian evolution.


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