Phase Space, Trajectories, and Liouville's Theorem

The previous lesson took the Boltzmann distribution as given and read the properties of matter off it. The rest of the course derives that distribution, and every derivation rests on one geometric object: the space of all mechanical states of the system. A microstate is a complete specification of the positions and momenta of every particle, and the collection of all such specifications is phase space. The dynamics is a flow on this space, and the central fact about the flow — that it preserves volume — is Liouville's theorem. It is the foundation on which the postulate of equal a priori probabilities, and therefore all of equilibrium statistical mechanics, is built.¹

The microstate as a point in phase space

A system of $N$ point particles in three dimensions is described classically by $3 N$ generalized coordinates $q = (q_{1}, \dots, q_{3 N})$ and their $3 N$ conjugate momenta $p = (p_{1}, \dots, p_{3 N})$ . A single point

X = (q_{1}, \dots, q_{3 N}, p_{1}, \dots, p_{3 N})

in the $6 N$ -dimensional space fixes the instantaneous mechanical state of the entire system.

The distinction between two spaces is worth fixing at the outset, because both appear throughout the subject.

$Γ$ -space (the full phase space) has $6 N$ dimensions. One point is the state of the whole $N$ -particle system.
$μ$ -space (the single-particle phase space) has six dimensions, the position and momentum of one particle. The state of the whole system is then a cloud of $N$ points in $μ$ -space, one per particle.

The $μ$ -space picture is the natural home of the Maxwell-Boltzmann distribution of the last lesson — each molecule is a point, and the distribution is the density of the cloud. The $Γ$ -space picture is the natural home of ensemble theory, developed in the next two lessons, where a whole system is one point and the ensemble is a cloud of systems. Liouville's theorem is a statement in $Γ$ -space.

Hamiltonian flow

The evolution of the microstate is fixed by Hamilton's equations. For a Hamiltonian $H (q, p, t)$ giving the total energy as a function of the coordinates and momenta,

\overset{q}{˙}_{i} = \frac{\partial H}{\partial p _{i}}, \overset{p}{˙}_{i} = - \frac{\partial H}{\partial q _{i}}, i = 1, \dots, 3 N .

These $6 N$ first-order equations define a velocity field on $Γ$ : at each point $X$ they give the phase-space velocity

\dot{X} = (\overset{q}{˙}, \overset{p}{˙}) = (\frac{\partial H}{\partial p}, - \frac{\partial H}{\partial q}) .

The microstate moves along the integral curve of this field, tracing a phase-space trajectory. Because the field is single-valued, exactly one trajectory passes through each point, and trajectories never cross: a deterministic system has a unique future and a unique past from any state.

For a time-independent Hamiltonian the energy is conserved, $\dot{H} = 0$ along the motion, so a trajectory is confined to the constant-energy surface $H (q, p) = E$ , a $(6 N - 1)$ -dimensional hypersurface in $Γ$ . Additional conserved quantities — total momentum, total angular momentum for an isolated system — confine it further. The trajectory winds on the intersection of these surfaces, and for a generic (non-integrable) system it explores that intersection densely, a property taken up in the next lesson as ergodicity.

A phase-space trajectory is confined to the constant-energy surface H = E and winds over it; the energy surface is one level set of the Hamiltonian in the 6N-dimensional phase space.

The phase-space density of an ensemble

Statistical mechanics does not follow one trajectory; it considers a large collection of identically prepared systems — an ensemble — occupying many points of $Γ$ at once. When the number of systems is large the cloud is described by a smooth density.

The ensemble average of any observable $A (q, p)$ is its mean over the density,

⟨ A ⟩ (t) = Γ \int A (q, p) ρ (q, p, t) d^{3 N} q d^{3 N} p .

Equilibrium statistical mechanics is the search for the stationary density $ρ_{eq}$ , the one for which $⟨ A ⟩$ does not change in time. Finding it requires knowing how $ρ$ evolves, which is what Liouville's theorem supplies.

Liouville's theorem

The ensemble members are neither created nor destroyed as they move: each system follows its own trajectory. The density therefore obeys a continuity equation in $Γ$ , exactly as a conserved fluid does in ordinary space. Writing the $6 N$ -dimensional phase-space velocity as $\dot{X} = (\overset{q}{˙}, \overset{p}{˙})$ , conservation of ensemble members reads

\frac{\partial ρ}{\partial t} + \nabla_{X} \cdot (ρ \dot{X}) = 0,

where $\nabla_{X} \cdot$ is the divergence over all $6 N$ phase-space coordinates. Expanding the divergence,

\nabla_{X} \cdot (ρ \dot{X}) = i = 1 \sum 3 N [\frac{\partial ( ρ q ˙ _{i} )}{\partial q _{i}} + \frac{\partial ( ρ p ˙ _{i} )}{\partial p _{i}}] = i \sum (\overset{q}{˙}_{i} \frac{\partial ρ}{\partial q _{i}} + \overset{p}{˙}_{i} \frac{\partial ρ}{\partial p _{i}}) + ρ i \sum (\frac{\partial q ˙ _{i}}{\partial q _{i}} + \frac{\partial p ˙ _{i}}{\partial p _{i}}) .

The final sum vanishes identically. Substituting Hamilton's equations,

i \sum (\frac{\partial q ˙ _{i}}{\partial q _{i}} + \frac{\partial p ˙ _{i}}{\partial p _{i}}) = i \sum (\frac{\partial}{\partial q _{i}} \frac{\partial H}{\partial p _{i}} - \frac{\partial}{\partial p _{i}} \frac{\partial H}{\partial q _{i}}) = 0,

because mixed second partials of $H$ are equal. The Hamiltonian flow is divergence-free. The continuity equation collapses to

\frac{\partial ρ}{\partial t} + i \sum (\overset{q}{˙}_{i} \frac{\partial ρ}{\partial q _{i}} + \overset{p}{˙}_{i} \frac{\partial ρ}{\partial p _{i}}) = 0.

The left side is the total time derivative of $ρ$ along a trajectory — the rate of change seen by an observer riding with the flow.

The two readings of the theorem must be kept apart. The comoving (Lagrangian) derivative $d ρ / d t$ is zero: a bundle of systems keeps the same density as it moves. The local (Eulerian) derivative $\partial ρ / \partial t$ at a fixed point need not be zero, because the flow can carry a denser bundle past that point. Only when the two agree — when $\partial ρ / \partial t = 0$ everywhere — is the ensemble stationary.

Incompressibility and volume conservation

The divergence-free property has a direct geometric statement. Consider a small region $R_{0}$ of phase space at time $t_{0}$ , of volume

Ω (t_{0}) = R_{0} \int d^{3 N} q d^{3 N} p .

Let each point of $R_{0}$ flow forward under the dynamics to time $t$ , carrying the region to $R_{t}$ . The map $X (t_{0}) \mapsto X (t)$ has a Jacobian, and the divergence-free condition makes that Jacobian unity: the transported region has the same volume,

Ω (t) = R_{t} \int d^{3 N} q d^{3 N} p = Ω (t_{0}) .

The shape of the region can distort without bound — a compact blob is stretched into long thin filaments that thread through phase space — but its total volume is fixed. The simple harmonic oscillator makes the mechanism concrete. Its phase space is the $(x, p)$ plane, its trajectories are ellipses, and a cell of initial conditions is rotated rigidly around the origin, conserving area exactly. A more generic system shears the cell, as sketched below.

A phase-space cell is carried by the flow and sheared out of shape, yet its area is unchanged (Liouville). Left: an initial square of initial conditions. Right: the same set of systems later, stretched but equal in area.

Volume conservation is why phase-space volume is the right measure of the number of states. If the dynamics could compress a region, an even distribution over states would not stay even, and the counting on which entropy rests would be frame-dependent. Liouville's theorem guarantees it is not.

Stationary ensembles and equilibrium

A stationary ensemble is one whose local density does not change, $\partial ρ / \partial t = 0$ . Liouville's theorem then forces

i \sum (\overset{q}{˙}_{i} \frac{\partial ρ}{\partial q _{i}} + \overset{p}{˙}_{i} \frac{\partial ρ}{\partial p _{i}}) = {ρ, H} = 0,

where the Poisson bracket of two phase functions is

{A, B} = i \sum (\frac{\partial A}{\partial q _{i}} \frac{\partial B}{\partial p _{i}} - \frac{\partial A}{\partial p _{i}} \frac{\partial B}{\partial q _{i}}) .

The compact form of Liouville's theorem is therefore

\frac{\partial ρ}{\partial t} = - {ρ, H} = {H, ρ} .

A density is stationary when it Poisson-commutes with the Hamiltonian. The cleanest way to guarantee that is to make $ρ$ a function of the coordinates and momenta only through conserved quantities. If $ρ = ρ (H)$ depends on $(q, p)$ solely through the energy, then ${ρ, H} = ρ^{'} (H) {H, H} = 0$ automatically, and the ensemble is stationary.

This single conclusion organizes the whole ensemble program. It does not yet say which function of the energy to take. Two natural choices generate the ensembles of the following modules:

$ρ$ constant on an energy surface, zero off it. All accessible microstates of fixed energy are equally weighted. This is the microcanonical ensemble, and its justification is the postulate of equal a priori probabilities in the next lesson.
$ρ \propto e^{- β H}$ . An exponential of the energy, stationary because it is a function of $H$ . This is the canonical ensemble, derived for a system in contact with a heat bath.

Liouville's theorem narrows the candidates for equilibrium to functions of the conserved quantities; a physical postulate selects the one nature uses.

The energy shell

An isolated system has a fixed energy, but no real system is prepared with the energy known to infinite precision, and the constant-energy surface $H = E$ has zero $6 N$ -dimensional volume. The construction that carries finite measure is a thin energy shell: the set of microstates whose energy lies between $E$ and $E + δ E$ , for a small tolerance $δ E$ .

The energy shell is the thin region between the surfaces H = E and H = E plus delta-E; it carries the finite phase-space volume that the microcanonical ensemble spreads its probability over.

The shell volume is the surface area of the energy surface times its thickness in energy. Writing $Γ (E)$ for the phase-space volume enclosed by $H \leq E$ , the shell of width $δ E$ has volume

Ω (E) δ E = \frac{\partial Γ}{\partial E} δ E,

so $Ω (E) = \partial Γ/ \partial E$ is the density of states in phase space — the area of the energy surface measured with the right normal thickness. For $N$ of order $1 0^{22}$ the enclosed volume $Γ (E)$ grows so violently with $E$ that almost all of it sits within a hair of the surface: the shell and the whole enclosed volume have logarithms that agree to relative order $1/ N$ . The microcanonical entropy $S = k ln Ω$ is therefore insensitive to the arbitrary choice of $δ E$ , a point the microcanonical module makes quantitative for the ideal gas.

Coarse-graining and the μ-space description

Liouville's theorem creates a tension with the second law. The fine-grained density $ρ$ is carried rigidly by the flow, so any functional of it that the flow preserves — in particular the Gibbs entropy $- k \int ρ ln ρ$ , built in the next lessons — cannot increase. Yet observed entropy does increase. The resolution is coarse-graining: partition $Γ$ into small but finite cells and replace $ρ$ by its average $\overset{ρ}{ˉ}$ over each cell.

The filamentation guaranteed by Liouville's theorem stretches an initial blob into ever-finer threads that wind through many cells. The fine-grained density stays constant on those threads, but the cell average spreads out as the threads distribute themselves, and the coarse-grained entropy rises. The exact volume is conserved; the volume that a finite-resolution measurement can resolve grows. This is the mechanical seed of irreversibility, developed when the Gibbs entropy is introduced.

For a dilute gas of weakly interacting particles a second reduction is available. When correlations between particles are negligible the $Γ$ -space density factorizes into single-particle densities, and the whole state is captured by the occupation of the six-dimensional $μ$ -space. The number of particles in a $μ$ -space cell is a histogram — the coarse-grained one-particle distribution — and its evolution under collisions is the subject of kinetic theory. At equilibrium this $μ$ -space density is the Maxwell speed distribution of the previous lesson.

The mu-space occupation histogram: each particle of the gas is one point in the six-dimensional single-particle phase space, and the coarse number in each cell is the one-particle distribution whose equilibrium form is Maxwell-Boltzmann.

From dynamics to probability

The mechanics of this lesson is exact and reversible: Hamilton's equations run equally well backward, and Liouville's theorem holds in both directions of time. Nothing here selects a preferred state or a direction of change. What it provides is the stage. Phase-space volume is the invariant measure of the number of microstates; the energy shell is the finite region an isolated system lives on; stationarity restricts equilibrium densities to functions of the conserved quantities. The physics — which stationary density is realized, and why entropy grows — enters through a probabilistic postulate about accessible microstates, taken up next.

Reif, Fundamentals of Statistical and Thermal Physics, Ch. 2 — the statistical description of a many-particle system through its accessible microstates, and the role of phase space as the arena for that description. ↩
Reif, Fundamentals of Statistical and Thermal Physics, §2.1–2.2 — specification of the microstate by generalized coordinates and momenta, and the subdivision of phase space into cells. ↩
Reif, Fundamentals of Statistical and Thermal Physics, §2.3, and Kardar, Statistical Physics of Particles, §3.2 — the continuity equation for the ensemble density, the divergence-free Hamiltonian flow, and Liouville's theorem $d ρ / d t = 0$ . ↩
Kardar, Statistical Physics of Particles, §3.3, and Pathria & Beale, Statistical Mechanics, §2.2 — the Poisson-bracket form $\partial ρ / \partial t = {H, ρ}$ and the conclusion that stationary densities depend on phase-space coordinates only through conserved quantities. ↩