In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability. Liouville's theorem applies to conservative systems, that is, systems in which the effects of friction are absent or can be ignored. The general mathematical formulation for such systems is the measure-preserving dynamical system. Liouville's theorem applies when there are degrees of freedom that can be interpreted as positions and momenta; not all measure-preserving dynamical systems have these, but Hamiltonian systems do. The general setting for conjugate position and momentum coordinates is available in the mathematical setting of symplectic geometry. Liouville's theorem ignores the possibility of chemical reactions, where the total number of particles may change over time, or where energy may be transferred to internal degrees of freedom. The non-squeezing theorem, which applies to all symplectic maps (the Hamiltonian is a symplectic map) implies further restrictions on phase-space flows beyond volume/density/measure conservation. There are extensions of Liouville's theorem to cover these various generalized settings, including stochastic systems.
Liouville equation
The Liouville equation describes the time evolution of the phase space distribution function. Josiah Willard Gibbs was the first to recognize this as the fundamental equation of statistical mechanics. It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838. Consider a Hamiltonian dynamical system with canonical coordinates q i {\displaystyle q_{i}} and conjugate momenta p i {\displaystyle p_{i}} , where i = 1 , … , n {\displaystyle i=1,\dots ,n} . Then the phase space distribution ρ ( p , q , t ) {\displaystyle \rho (p,q,t)} determines the probability ρ ( p , q , t ) d n q d n p {\displaystyle \rho (p,q,t)\;\mathrm {d} ^{n}q\,\mathrm {d} ^{n}p} that the system will be found in the infinitesimal phase space volume d n q d n p {\displaystyle \mathrm {d} ^{n}q\,\mathrm {d} ^{n}p} at time t . {\displaystyle t.} The Liouville equation is
∂ ρ ∂ t + ∑ i = 1 n ( ∂ ρ ∂ q i q ˙ i + ∂ ρ ∂ p i p ˙ i ) = 0. {\displaystyle {\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{n}\left({\frac {\partial \rho }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \rho }{\partial p_{i}}}{\dot {p}}_{i}\right)=0.}
Time derivatives are denoted by dots, and are evaluated according to Hamilton's equations for the system. This equation demonstrates the conservation of density in phase space (which was Gibbs's name for the theorem). Liouville's theorem states that:
A proof of Liouville's theorem uses the n-dimensional divergence theorem. The proof is based on the fact that the evolution of ρ {\displaystyle \rho } obeys an 2n-dimensional version of the continuity equation:
∂ ρ ∂ t + ∇ → ⋅ ( ρ u → ) = 0 {\displaystyle {\frac {\partial \rho }{\partial t}}+{\vec {\nabla }}\cdot (\rho {\vec {u}})=0}
with u → = ( q ˙ 1 , q ˙ 2 , … , q ˙ n , p ˙ 1 , p ˙ 2 , . . . , p ˙ n ) {\displaystyle {\vec {u}}=({\dot {q}}_{1},{\dot {q}}_{2},\dots ,{\dot {q}}_{n},{\dot {p}}_{1},{\dot {p}}_{2},...,{\dot {p}}_{n})} being the "velocity" vector of p i {\displaystyle p_{i}} and q i {\displaystyle q_{i}} . The above equation means that change of the total probability within a small volume in phase space is equal to the net flux of probability density into or out of the volume. After inserting u → {\displaystyle {\vec {u}}} in the above equation, we reach
∂ ρ ∂ t + ∑ i = 1 n ( ∂ ( ρ q ˙ i ) ∂ q i + ∂ ( ρ p ˙ i ) ∂ p i ) = 0. {\displaystyle {\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{n}\left({\frac {\partial (\rho {\dot {q}}_{i})}{\partial q_{i}}}+{\frac {\partial (\rho {\dot {p}}_{i})}{\partial p_{i}}}\right)=0.}
That is, the 3-tuple ( ρ , ρ q ˙ i , ρ p ˙ i ) {\displaystyle (\rho ,\rho {\dot {q}}_{i},\rho {\dot {p}}_{i})} is a conserved current. The above equation can be reduced to the Liouville equation based on the following identity
ρ ∑ i = 1 n ( ∂ q ˙ i ∂ q i + ∂ p ˙ i ∂ p i ) = ρ ∑ i = 1 n ( ∂ 2 H ∂ q i ∂ p i − ∂ 2 H ∂ p i ∂ q i ) = 0 , {\displaystyle \rho \sum _{i=1}^{n}\left({\frac {\partial {\dot {q}}_{i}}{\partial q_{i}}}+{\frac {\partial {\dot {p}}_{i}}{\partial p_{i}}}\right)=\rho \sum _{i=1}^{n}\left({\frac {\partial ^{2}H}{\partial q_{i}\,\partial p_{i}}}-{\frac {\partial ^{2}H}{\partial p_{i}\partial q_{i}}}\right)=0,}
where H {\displaystyle H} is the Hamiltonian, and we have used the relationships q ˙ i = ∂ H / ∂ p i {\displaystyle {\dot {q}}_{i}=\partial H/\partial p_{i}} and p ˙ i = − ∂ H / ∂ q i {\displaystyle {\dot {p}}_{i}=-\partial H/\partial {q_{i}}} . The derivation of the Liouville equation can be viewed as the motion through phase space as a 'fluid flow' of system points. The theorem that the convective derivative of the density, d ρ / d t {\displaystyle d\rho /dt} , is zero follows from the equation of continuity by noting that the 'velocity field' ( p ˙ , q ˙ ) {\displaystyle ({\dot {p}},{\dot {q}})} in phase space has zero divergence (which follows from Hamilton's relations).
Other formulations
Poisson bracket The theorem above is often restated in terms of the Poisson bracket as
∂ ρ ∂ t = { H , ρ } {\displaystyle {\frac {\partial \rho }{\partial t}}=\{H,\rho \}}
or, in terms of the linear Liouville operator or Liouvillian,
i L ^ = ∑ i = 1 n [ ∂ H ∂ p i ∂ ∂ q i − ∂ H ∂ q i ∂ ∂ p i ] = − { H , ∙ } {\displaystyle \mathrm {i} {\widehat {\mathbf {L} }}=\sum _{i=1}^{n}\left[{\frac {\partial H}{\partial p_{i}}}{\frac {\partial }{\partial q^{i}}}-{\frac {\partial H}{\partial q^{i}}}{\frac {\partial }{\partial p_{i}}}\right]=-\{H,\bullet \}}
as
∂ ρ ∂ t + i L ^ ρ = 0. {\displaystyle {\frac {\partial \rho }{\partial t}}+{\mathrm {i} {\widehat {\mathbf {L} }}}\rho =0.}
Ergodic theory In ergodic theory and dynamical systems, motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem. In Hamiltonian mechanics, the phase space is a smooth manifold that comes naturally equipped with a smooth measure (locally, this measure is the 6n-dimensional Lebesgue measure). The theorem says this smooth measure is invariant under the Hamiltonian flow. More generally, one can describe the necessary and sufficient condition under which a smooth measure is invariant under a flow. The Hamiltonian case then becomes a corollary.
Symplectic geometry We can also formulate Liouville's Theorem in terms of symplectic geometry. For a given system, we can consider the phase space ( q μ , p μ ) {\displaystyle (q^{\mu },p_{\mu })} of a particular Hamiltonian H {\displaystyle H} as a manifold ( M , ω ) {\displaystyle (M,\omega )} endowed with a symplectic 2-form
ω = d p μ ∧ d q μ . {\displaystyle \omega =dp_{\mu }\wedge dq^{\mu }.}
The volume form of our manifold is the top exterior power of the symplectic 2-form, and is just another representation of the measure on the phase space described above. On our phase space symplectic manifold we can define a Hamiltonian vector field generated by a function f ( q , p ) {\displaystyle f(q,p)} as
X f = ∂ f ∂ p μ ∂ ∂ q μ − ∂ f ∂ q μ ∂ ∂ p μ . {\displaystyle X_{f}={\frac {\partial f}{\partial p_{\mu }}}{\frac {\partial }{\partial q^{\mu }}}-{\frac {\partial f}{\partial q^{\mu }}}{\frac {\partial }{\partial p_{\mu }}}.}
Specifically, when the generating function is the Hamiltonian itself, f ( q , p ) = H {\displaystyle f(q,p)=H} , we get
X H = ∂ H ∂ p μ ∂ ∂ q μ − ∂ H ∂ q μ ∂ ∂ p μ = d q μ d t ∂ ∂ q μ + d p μ d t ∂ ∂ p μ = d d t {\displaystyle X_{H}={\frac {\partial H}{\partial p_{\mu }}}{\frac {\partial }{\partial q^{\mu }}}-{\frac {\partial H}{\partial q^{\mu }}}{\frac {\partial }{\partial p_{\mu }}}={\frac {dq^{\mu }}{dt}}{\frac {\partial }{\partial q^{\mu }}}+{\frac {dp^{\mu }}{dt}}{\frac {\partial }{\partial p_{\mu }}}={\frac {d}{dt}}}
where we utilized Hamilton's equations of motion and the definition of the chain rule. In this formalism, Liouville's Theorem states that the Lie derivative of the volume form is zero along the flow generated by X H {\displaystyle X_{H}} . That is, for ( M , ω ) {\displaystyle (M,\omega )} a 2n-dimensional symplectic manifold,
L X H ( ω n ) = 0. {\displaystyle {\mathcal {L}}_{X_{H}}(\omega ^{n})=0.}
In fact, the symplectic structure ω {\displaystyle \omega } itself is preserved, not only its top exterior power. That is, Liouville's Theorem also gives
L X H ( ω ) = 0. {\displaystyle {\mathcal {L}}_{X_{H}}(\omega )=0.}
Quantum Liouville equation The analog of Liouville equation in quantum mechanics describes the time evolution of a mixed state. Canonical quantization yields a quantum-mechanical version of this theorem, the von Neumann equation. This procedure, often used to devise quantum analogues of classical systems, involves describing a classical system using Hamiltonian mechanics. Classical variables are then re-interpreted as quantum operators, while Poisson brackets are replaced by commutators. In this case, the resulting equation is
∂ ρ ∂ t = 1 i ℏ [ H , ρ ] , {\displaystyle {\frac {\partial \rho }{\partial t}}={\frac {1}{i\hbar }}[H,\rho ],}
where ρ is the density matrix. When applied to the expectation value of an observable, the corresponding equation is given by Ehrenfest's theorem, and takes the form
d d t ⟨ A ⟩ = − 1 i ℏ ⟨ [ H , A ] ⟩ , {\displaystyle {\frac {d}{dt}}\langle A\rangle =-{\frac {1}{i\hbar }}\langle [H,A]\rangle ,}
where A {\displaystyle A} is an observable. Note the sign difference, which follows from the assumption that the operator is stationary and the state is time-dependent. In the phase-space formulation of quantum mechanics, substituting the Moyal brackets for Poisson brackets in the phase-space analog of the von Neumann equation results in compressibility of the probability fluid, and thus violations of Liouville's theorem incompressibility. This, then, leads to concomitant difficulties in defining meaningful quantum trajectories.
Examples
Simple harmonic oscillator phase-space volume
Consider an N {\displaystyle N} -particle system in three dimensions, and focus on only the evolution of d N {\displaystyle \mathrm {d} {\mathcal {N}}} particles. Within phase space, these d N {\displaystyle \mathrm {d} {\mathcal {N}}} particles occupy an infinitesimal volume given by
d Γ = ∏ i = 1 N d 3 p i d 3 q i . {\displaystyle \mathrm {d} \Gamma =\displaystyle \prod _{i=1}^{N}d^{3}p_{i}d^{3}q_{i}.}
We want d N d Γ {\displaystyle {\frac {\mathrm {d} {\mathcal {N}}}{\mathrm {d} \Gamma }}} to remain the same throughout time, so that ρ ( Γ , t ) {\displaystyle \rho (\Gamma ,t)} is constant along the trajectories of the system. If we allow our particles to evolve by an infinitesimal time step δ t {\displaystyle \delta t} , we see that each particle phase space location changes as
{ q i ′ = q i + q i ˙ δ t , p i ′ = p i + p i ˙ δ t , {\displaystyle {\begin{cases}q_{i}'=q_{i}+{\dot {q_{i}}}\delta t,\\p_{i}'=p_{i}+{\dot {p_{i}}}\delta t,\end{cases}}}
where q i ˙ {\displaystyle {\dot {q_{i}}}} and p i ˙ {\displaystyle {\dot {p_{i}}}} denote d q i d t {\displaystyle {\frac {dq_{i}}{dt}}} and d p i d t {\displaystyle {\frac {dp_{i}}{dt}}} respectively, and we have only kept terms linear in δ t {\displaystyle \delta t} . Extending this to our infinitesimal hypercube d Γ {\displaystyle \mathrm {d} \Gamma } , the side lengths change as
d q i ′ = d q i + ∂ q i ˙ ∂ q i d q i δ t , d p i ′ = d p i + ∂ p i ˙ ∂ p i d p i δ t . {\displaystyle {\begin{aligned}dq_{i}'=dq_{i}+{\tfrac {\partial {\dot {q_{i}}}}{\partial q_{i}}}dq_{i}\delta t,\\[2pt]dp_{i}'=dp_{i}+{\tfrac {\partial {\dot {p_{i}}}}{\partial p_{i}}}dp_{i}\delta t.\end{aligned}}}
To find the new infinitesimal phase-space volume d Γ ′ {\displaystyle \mathrm {d} \Gamma '} , we need the product of the above quantities. To first order in δ t {\displaystyle \delta t} , we get the following:
d q i ′ d p i ′ = d q i d p i [ 1 + ( ∂ q i ˙ ∂ q i + ∂ p i ˙ ∂ p i ) δ t ] . {\displaystyle dq_{i}'dp_{i}'=dq_{i}dp_{i}\left[1+\left({\frac {\partial {\dot {q_{i}}}}{\partial q_{i}}}+{\frac {\partial {\dot {p_{i}}}}{\partial p_{i}}}\right)\delta t\right].}
So far, we have yet to make any specifications about our system. Let us now specialize to the case of N {\displaystyle N} 3 {\displaystyle 3} -dimensional isotropic harmonic oscillators. That is, each particle in our ensemble can be treated as a simple harmonic oscillator. The Hamiltonian for this system is given by
H = ∑ i = 1 3 N ( 1 2 m p i 2 + m ω