Chemistry - Liouville's theorem (Hamiltonian)

In the physical sciences, Liouville's theorem (also sometimes known as the Liouville equation) is a key theorem in statistical mechanics of classical systems. It is also important in the mathematical study of Hamiltonian mechanics and symplectic topology. The two disciplines make rather different expressions of the key result, and so both are given here. The quantum mechanical analogue of the key result is also given.

Contents

1 History

2 Introductory remarks - 1. Phase space distributions

3 Introductory remarks - 2. Ensembles

4 Physical sciences expression

5 Informal demonstration, (and symplectic space analogue)

6 'Fluids' 'proof'

7 Mathematical expression

8 Quantum-mechanical expression

9 Uses

10 References

History

Liouville's theorem is named after the French mathematician Joseph Liouville (1809-1882). It is one of two such-named theorems, the other being in the field of complex analysis (see Liouville's theorem (complex analysis).

The present theorem is of fundamental importance in statistical mechanics of classical systems, where it is also known (after J. Willard Gibbs) as the conservation of density in phase space.

Introductory remarks - 1. Phase space distributions

The Liouville equation describes the time evolution of phase space density. Consider a set of systems with a large number of degrees of freedom, N. In statistical mechanics such systems typically involve large numbers of particles, (typically of order Avogadro's number). The phase space vector $\Gamma \equiv (q_i ,p_i ;1 \le i \le N)$ is defined by all the coordinates $q i$ and all the momenta, $p i$ of all the particles, i, in the system.

Introductory remarks - 2. Ensembles

Macroscopically we may consider an ensemble of such systems such that the energy (or alternatively temperature), the volume (or alternatively pressure) and the number (or alternatively chemical potential) of particles are fixed. Obviously there are very many microstates (represented by phase vectors) which are consistent with the very small set of specified macroscopic state.

Different equilibrium statistical mechanical ensembles correspond to systems with different macroscopic/thermodynamic constraints. The microcanonical ensemble has fixed energy, volume and N. The canonical ensemble is appropriate when the temperature volume and N are fixed. The isothermal isobaric ensemble is used when temperature pressure and N are fixed.

The equations of motion for the microscopic systems mean that from an initial phase vector $Γ(0)$ at time t = 0, the subsequent time evolution $Γ(t)$ is completely determined. If one imagines an initial ensemble composed of infinitely many t = 0 microstates, then one can describe this initial ensemble by an initial phase space density.

The Liouville equation describes how this phase space density $D (Γ, t)$ evolves in time.

Physical sciences expression

This theorem is of fundamental importance in statistical mechanics of classical systems, where it is also known (after J. Willard Gibbs) as the conservation of density in phase space.

For arbitrary deterministic dynamics, it can be shown that the phase space density D obeys,

$\frac{\partial D }{\partial t}=-\sum_{i=1}^{N}\left[ \frac{\partial D\dot{q}_{i} }{\partial q_{i}}+\frac{\partial D\dot{p}_{i} }{\partial p_{i}}\right].$

(Physically, this equation, which is sometimes itself called the Liouville equation, states the 'continuity of flow' of the system-points through phase-space - see section 4 'fluids-proof' below).

For symplectic systems (which includes all Hamiltonian systems, see Hamilton's relations)

$\sum_{i=1}^{N}\left[ \frac{\partial \dot{q}_{i} }{\partial q_{i}}+\frac{\partial \dot{p}_{i} }{\partial p_{i}}\right]=0.$

(It is important to remember that for systems which lose heat to their surroundings the above equation is not true).

The density D of systems in the neighbourhood of a particular system-point as it moves through phase-space (i.e., the 'convective derivative' of D) is given by

$\frac{dD }{dt} = \frac{\partial D }{\partial t}+\sum_{i=1}^{N}\left[ \frac{\partial D }{\partial q_{i}}\dot{q}_{i}+\frac{\partial D }{\partial p_{i}}\dot{p}_{i}\right].$

Thus, for isolated symplectic (Hamiltonian) systems the convective derivative is zero,

$\frac{dD }{dt} = 0.$

This is Liouville's Theorem (as usually stated in the physics and chemistry of classical systems).

If the ensemble is statistically stationary (the partial time derivative of D is zero) then this equation is satisfied by D = D(H) where H is the Hamiltonian function of the system.

In the microcanonical ensemble

D = D 0 δ(H - H 0)

where δ is a Dirac delta function, and in the canonical ensemble

D = D 0 exp( - H / θ)

where θ is some constant.

Informal demonstration, (and symplectic space analogue)

The key result (local constancy of D in the neighbourhood of a given system-point) can be demonstrated (to the satisfaction of a physicist) by considering motion of a 'cloud' of points through phase space. The local density of points D is given by N/V where N is the number of points in the cloud, of volume V.

Constancy of N - in a deterministic system, phase-space trajectories can never cross. Were two trajectories to intersect, it would imply that some configuration of the system would have two possible futures. Assuming that the system is deterministic (given perfect knowledge of its condition), then such intersections are impossible. Thus systems neither enter nor leave V.

Constancy of V - this follows because any expansion of the volume along a co-ordinate q_i is exactly balanced by the shrinking of the volume in the direction of the conjugate momentum p_i. This balance follows from Hamilton's relations between p_i, q_i and their rates of change.

In more detail:

Consider the time rate-of-change (taken with the flow, ie convective derivative) of a small phase-space volume, made up of a 'cloud' of points:-

$\Delta V = \Delta q_1 \Delta p_1 \cdots \Delta q_i \Delta p_i \cdots$

Now the rate of separation of a line element (made up of system points) $Δ q i$ say is given by the difference in 'velocity' between its two ends, i.e.

$\frac {d} {dt} {\Delta q_i} = \left(\frac {\partial}{\partial q_i} {\dot q_i}\right) \cdot {\Delta q_i}$

and similarly for other q and p.

Thus

$\frac {d} {dt} ({\Delta p_i}{\Delta q_i}) = \Delta p_i \Delta q_i \cdot \left(\frac{\partial}{\partial p_i}{\dot p_i}+\frac {\partial}{\partial q_i}{\dot q_i}\right)$

and on substituting the Hamilton's relations for ${\dot p_i} , {\dot q _i}$ , this last bracket is seen to be zero. Thus, overall, V is constant, and so, since N is, so is D.

(The equivalent result in the terms of symplectic spaces is that the 2-form S, formed from the wedge product of $Δ p i$ and $Δ q i$ has a Lie derivative for its Hamiltonian evolution, (given by the Poisson bracket with respect to the vector field {H, } which vanishes. - See say p 484 of Ref 1 for a 'popular maths' exposition)

'Fluids' 'proof'

Alternatively, we note that the phase-space velocity field $v= ({\dot p} , {\dot q } )$ has zero divergence, and that the local 'flux' of system-points through phase-space is $J = D v$

Then applying conservation of system points to the flux J into a fixed control volume gives:

$- \frac {\partial D} {\partial t} = \operatorname{div} (D v) = v\cdot\operatorname{grad}(D) + D \operatorname{div}(v).\,$

From Hamilton's relations $\operatorname{div}(v) = 0$ , and so

$\frac {d D}{dt}= \frac {\partial D} {\partial t} +v\cdot\operatorname{grad}(D) = 0.\,$

Mathematical expression

Typically, in an appropriately normalised system, ρ is the probability that a physical system will be found in an infinitesimal volume $d τ$ of phase space, τ standing for both position and momentum coordinates. In a system of N particles, τ is a convenient shorthand for the set of variables

$\{\,\vec{q}_1,\vec{q}_2,\ldots,\vec{q}_N,;\vec{p}_{1},\vec{p}_{2},\ldots,\vec{p}_N\,\}.$

In a system with Hamiltonian H and distribution function ρ, the theorem states that

$\frac{\partial}{\partial t}\rho(\tau,t)=-\{\,\rho(\tau,t) ,H\,\}$

where the curly braces denote a Poisson bracket.

A closely similar expression can be written by using the Liouvillian operator

${\hat{L}}=\sum_{i=1}^{N}\left[ \frac{\partial H}{\partial p_{i}} \frac{\partial }{\partial q_{i}}-\frac{\partial H}{\partial q_{i}}\frac{\partial }{\partial p_{i}}\right].$

so that

$\frac{\partial \rho }{\partial t}+{\hat{L}}\rho =0.$

Quantum-mechanical expression

Canonical quantization yields a quantum-mechanical version of this theorem. 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

$\frac{\partial}{\partial t}\rho=-\frac{i}{\hbar}[\rho,H]$

where ρ is the density matrix.

Uses

The Liouville equation is valid for both equilibrium and nonequilibrium systems. It is the fundamental equation of nonequilibrium statistical mechanics. It integral to the proof of the fluctuation theorem from which the second law of thermodynamics can be derived. It is also the key component of the derivation of Green-Kubo relations for linear transport coefficients such as shear viscosity, thermal conductivity or electrical conductivity. The Liouville equation is a mathematically exact consequence of the equations of motion for the particles comprising a system.

References

1 Penrose R (2004) The Road to Reality, Jonathan Cape, London

Categories: Hamiltonian mechanics | Theorems | Statistical mechanics