Liouville's Theorem — Phase Space

Phase-space volume is conserved under Hamiltonian flow — ensemble density stays constant

Particles

ω (frequency)

Damping γ (dissipative)

Phase-space diagnostics

ModeHamiltonian

Blob area (est.)—

t—

Hamiltonian:
H = p²/2m + ω²q²/2
⇒ circles in phase space

Dissipative:
q̈ + γq̇ + ω²q = 0
⇒ spiral inward (volume shrinks)

Liouville's theorem (1838): the phase-space volume of any region of initial conditions is preserved under Hamiltonian evolution. Equivalently, the phase-space density ρ(q,p,t) satisfies ∂ρ/∂t + {ρ,H} = 0 (Poisson bracket = 0 along trajectories). This is why a harmonic oscillator ensemble shears but never contracts or expands. In contrast, dissipative systems (γ>0) violate Liouville's theorem — the phase-space blob spirals inward and shrinks. This distinction underlies all of statistical mechanics: Liouville → ergodic hypothesis → entropy, and the arrow of time.