Question bank — Liouville's theorem — phase space volume conservation

False. $\rho$ is only constant along a trajectory ($d\rho /dt=0$); at a fixed spot $\partial \rho /\partial t$ can be huge as a dense blob drifts in or out.

False. Volume (measure) is conserved, but the blob generically stretches into thin filaments while thinning in another direction — that stretch-and-fold is exactly how chaos and mixing arise.

False. It needs the flow to be divergence-free, which is guaranteed only by Hamilton's equations via the mixed-partial cancellation. A damped oscillator ($\dot{p}=\cdots -\gamma p$) violates it.

False. The blob does change — it shears from a rectangle into a parallelogram. The area survives because shear leaves the perpendicular height ($\Delta p$, since $\dot{p}=0$) untouched.

False. It rigidly rotates about the origin (circles in rescaled $Q,P$). Rotation has Jacobian determinant $1$, so no shrinking. Shrinking would require dissipation.

False. The opposite: bounded, volume-preserving flow forces near-returns — that is the Poincaré Recurrence Theorem. Conserved volume is precisely what makes recurrence unavoidable.

False. A single point has zero volume trivially. The content is about a cloud (an ensemble) of neighbouring states flowing together — that's why it underpins Statistical Mechanics — Ensembles.

False. Only Canonical Transformations preserve it (their Jacobian is symplectic, determinant $1$). A generic coordinate change rescales volume.

The error: continuity holds for any conserved fluid, dissipative or not. Liouville appears only after you additionally prove $\nabla \cdot \dot{\mathbf{x}}=0$, which kills the $\rho (\nabla \cdot \dot{\mathbf{x}})$ term.

The sign error: ${\dot{p}}_i=-\partial H/\partial q_i$, so the second term is $-{\partial }^{2}H/\partial p_i\partial q_i$. The two mixed partials then cancel to $0$, not add.

Two errors. Friction gives $\nabla \cdot \dot{\mathbf{x}}=-\gamma \ne 0$, so it's not a Liouville system, and while $\rho$ rises the theorem $d\rho /dt=0$ is violated — points pile up at the attractor.

The error: $d\rho /dt$ is the material derivative (following the flow). $\rho$ still varies from point to point in space and can vary in time at a fixed point; it is only constant to an observer riding along with a phase point.

The error: the mixed-partial cancellation ${\partial }^{2}H/\partial q\partial p={\partial }^{2}H/\partial p\partial q$ holds even for $H(q,p,t)$. The flow is still divergence-free; a time-dependent $H$ just makes $\partial \rho /\partial t\ne 0$ possible, which the theorem allows.

The error: $\partial \rho /\partial t=-\{\rho ,H\}$ rearranges to $d\rho /dt={\partial }_t\rho +\{\rho ,H\}=0$. A nonzero bracket is exactly balanced by ${\partial }_t\rho$; the total derivative is zero. See Poisson Brackets.

The volume of a flowing blob changes at rate $dV/dt=\int \nabla \cdot \dot{\mathbf{x}}dV$. If the integrand is zero everywhere, the integral is zero, so $V$ is frozen — the same reason incompressible fluids keep their volume.

Because we never know the exact state — only that it lies somewhere in a small region. Conserved volume means probability is conserved, which is the foundation that lets Statistical Mechanics — Ensembles treat $\rho$ as a genuine probability fluid.

It says the very structure of Hamilton's Equations — one $+\partial H/\partial p$ and one $-\partial H/\partial q$ with a shared $H$ — is what pairs the $q$ and $p$ expansions to cancel. This antisymmetric pairing is the seed of Symplectic Geometry.

Shear slides layers parallel to one axis without moving them apart perpendicular to it. Area = base × perpendicular height, and the height ($\Delta p$) is untouched, so area is invariant while the shape skews.

Liouville guarantees no volume is lost, forcing trajectories to crowd and come arbitrarily close — that's Poincaré Recurrence Theorem — but "close" is not "exact"; measure conservation gives near-return, not exact periodicity.

It licenses treating $\rho$ as an incompressible conserved fluid, so a stationary ensemble (${\partial }_t\rho =0$) must satisfy $\{\rho ,H\}=0$ — i.e. depend only on conserved quantities like energy. That is where equilibrium ensembles come from.

Trivially its zero volume stays zero. The theorem's real content only bites for a region of positive volume, i.e. an ensemble with spread.

Neither. It stays exactly $\Delta q\Delta p$ for all time; it just becomes an ever-more-slanted, ever-thinner-looking parallelogram whose true area is unchanged.

Yes. Divergence-free is a local statement everywhere, including fixed points; a point sitting still contributes no expansion. Volume conservation of any surrounding blob is unaffected.

Volume decays as $e^{-\gamma t}\to 0$; the blob collapses onto the origin attractor. This is the clean failure case that proves Liouville genuinely requires the Hamiltonian (divergence-free) structure.

Yes. The divergence still vanishes at every instant (mixed partials cancel regardless of explicit $t$), so volume is preserved even as the flow field itself changes in time.

Volume equality is necessary but not sufficient. The flow is a specific volume-preserving (symplectic) map, so many equal-volume shapes are still unreachable from each other by the dynamics.