Liouville Equation

In this section we will build a bridge from Classical Mechanics to StatisticalPhysics. The bridge is Liouville equation.

We start with the Hamiltonian formalism of the Classical Mechanics,where the state of a system with m degrees of freedom is described by mpairs of conjugated variables called (generalized) coordinates and momentaqs, ps, s = 1, 2, . . . , m. The equations of motion are generated with theHamiltonian function, H(qs, ps), by the following rule

qs =∂H

∂ps, (1)

ps = −∂H

∂qs. (2)

Fore example, if we have N three-dimensional particles of mass M interact-ing with each other via a pair potential U , and also interacting with someexternal potential V , then the Hamiltonian for this system reads:

H =N∑

j=1

p2j

2M+

N∑

j=1

V (rj) +∑

i<j

U(ri − rj) , (3)

where rj and pj are the radius-vector and momentum of the j-th particle,respectively. In this example, m = 3N : each component of each radius-vector represents a separate degree of freedom.

The following property of Eqs. (1)-(2) will be crucial for us. If we needto describe time evolution of some function A(qs, ps) due to the evolutionof coordinates and momenta, then the following relation takes place

A = H,A , (4)

where the symbol in the r.h.s. is a shorthand notation—called Poissonbracket—for the following expression

H, A =∑s

∂H

∂ps

∂A

∂qs− ∂H

∂qs

∂A

∂ps. (5)

[The proof is straightforward. The chain rule for dA(qs(t), ps(t))/dt andthen Eqs. (1)-(2) for qs and ps.]

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Hence, any quantity A(qs, ps) is a constant of motion if, and only if, itsPoisson bracket with the Hamiltonian is zero. In particular, the Hamiltonianitself is a constant of motion, since H, H = 0, and this is nothing else thanthe conservation of energy, because the physical meaning of the Hamiltonianfunction is energy expressed in terms of coordinates and momenta.

Definition: The phase space is a 2m-dimensional space of points, or,equivalently, vectors of the following form:

X = (q1, q2, . . . , qm, p1, p2, . . . , pm) . (6)

Each point/vector in the phase space represents a state of the mechanicalsystem. If we know X at some time moment, say, t = 0, then the furtherevolution of X—the trajectory X(t) in the phase space—is unambiguouslygiven by Eqs. (1)-(2), since these are the first-order differential equationswith respect to the vector function X(t). (For the same reason differenttrajectories cannot intersect!)

The phase space is convenient for statistical description of mechanicalsystem. Suppose that the initial state for a system is known only with acertain finite accuracy. This means that actually we know only the proba-bility density W0(X) of having the point X somewhere in the phase space.If the initial condition is specified in terms of probability density, then thesubsequent evolution should be also described probabilistically, that is wehave to work with the distribution W (X, t), which should be somehow re-lated to the initial condition W (X, 0) = W0(X). Our goal is to establishthis relation.

We introduce a notion of a statistical ensemble. Instead of dealing withprobability density, we will work with a quantity which is proportional to it,and is much transparent. Namely, we simultaneously take some large numberNens of identical and independent systems distributed in accordance withW (X, t). We call this set of systems statistical ensemble. The j-th memberof the ensemble is represented by its point Xj in the phase space. The crucialobservation is that the quantity Nens W (X, t) gives the concentration of thepoints Xj. Hence, to find the evolution of W we just need to describe theevolution of the concentration of the points Xj , which is intuitively easier,since each Xj obeys the Hamiltonian equation of motion.

A toy model. To get used to the ensemble description, and also to obtainsome important insights, consider the following dynamical model with justone degree of freedom:

H = (1/4)(p2 + q2)2 . (7)

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The equations of motion are:

q = (p2 + q2) p , (8)

p = −(p2 + q2) q . (9)

The quantityω = p2 + q2 (10)

is a constant of motion, since, up to a numeric factor, it is a square root ofenergy. We thus have a linear system of equations

q = ωp , (11)

p = −ωq , (12)

which is easily solved:

q(t) = q0 cosωt + p0 sinωt , (13)

p(t) = p0 cosωt− q0 sinωt , (14)

where q0 ≡ q(0), p0 ≡ p(0), and ω = p20 + q2

0. We see that our system isa non-linear harmonic oscillator. It performs harmonic oscillations, but incontrast to a linear harmonic oscillator, the frequency of oscillations is afunction of energy.

Now we take Nens = 1000 replicas of our system and uniformly dis-tribute them within the square 0.75 ≤ q ≤ 1.25, −0.25 ≤ p ≤ 0.25 of thetwo-dimensional phase space. Then we apply the equations of motion (13)-(14) to each points and trace the evolution. Some characteristic snapshotsare presented in Fig. 1. In accordance with the equations of motion, eachpoint rotates along corresponding circle of the radius

√p20 + q2

0. Since ouroscillators are non-linear, points with larger radii rotate faster, and this leadsto the formation of the spiral structure. The number of the spiral windingsincreases with time. With a fixed number of points in the ensemble, at somelarge enough time it becomes simply impossible to resolve the spiral struc-ture. For all practical purposes, it means that instead of dealing with theactual distribution W (X, t), which is beyond our “experimental” resolution,we can work with an effective distribution Weff(X, t) obtained by slightlysmearing W (X, t). [Actually, this or that sort of “smearing” (either ex-plicit or implicit) is an unavoidable ingredient of any Statistical-Mechanicaldescription!] In contrast to the genuine distribution W (X, t) that keeps in-creasing the number of spiral windings, the smeared distribution Weff(X, t)

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saturates to a certain equilibrium (=time-independent) function, perfectlydescribing our ensemble at large times (see the plot for t = 1000). With ourequations of motion, we see that the generic structure of our equilibriumWeff(X) (no matter what is the initial distribution) is Weff(X) = f(p2 + q2),the particular form of the function f coming from the initial distribution.Indeed, with respect to an individual member of the ensemble, the evolu-tion is a kind of roulette that randomizes the position of corresponding phasespace point Xj along the circle of the radius

√p2 + q2. Below we will see

how this property is generalized to any equilibrium ensemble of Hamiltoniansystems.

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Figure 1: Evolution of the ensemble of 1000 systems described by the Hamil-tonian (7).

After playing with a toy model, we are ready to consider a general case.From now on we normalize the function W (X, t) to the number of the en-semble members. Correspondingly, the number of points in the phase spacevolume Ω0 at the time t is given by the integral

NΩ0(t) =∫

Ω0

W (X, t) dΩ , (15)

where dΩ = dq1 . . . dqm dp1 . . . dpm is the element of the phase space volume;the integration is over the volume Ω0. To characterize the rate of variationof the number of points within the volume Ω0, we use the following timederivative

NΩ0 =∫

Ω0

∂

∂tW (X, t) dΩ . (16)

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By the definition of the function W (X, t), its variable X does not dependon time, so that the time derivative deals only with the variable t.

There is an alternative way of calculating NΩ0 . We may count the num-ber of points that cross the surface of the volume Ω0 per unit time:

NΩ0 = −∫

surface of Ω0

J · dS . (17)

Here J is the flux of the points [number of points per unit (and perpendicularto velocity) surface per unit time]; dS = n dS, where n is the unit normalvector at a surface point, and dS is the surface element. We assume that nis directed outwards and thus write the sign minus in the right-hand side of(17).

In accordance with the known theorem of calculus, the surface integral(17) can be converted into the bulk integral

∫

surface of Ω0

J · dS =∫

Ω0

∇ · J dΩ , (18)

where ∇ is the vector differential operator

∇ =(

∂

∂q1, . . . ,

∂

∂qm,

∂

∂p1, . . . ,

∂

∂pm

). (19)

We arrive at the equality∫

Ω0

∂

∂tW (X, t) dΩ = −

∫

Ω0

∇ · J dΩ . (20)

Since Eq. (20) is true for an arbitrary Ω0, including an infinitesimally smallone, we actually have

∂

∂tW (X, t) = −∇ · J . (21)

This is a quite general relation, known as the continuity equation. It arisesin theories describing flows of conserving quantities (say, particles of fluidsand gases). The dimensionality of the problem does not matter.

Now we are going to independently relate the flux J to W (X, t) and thusend up with a closed equation in terms of W (X, t). By the definition of Jwe have

J = W (X, t) X , (22)

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because the flux of particles is always equal to their concentration timesvelocity. In our case, the velocity X is just a function of X following fromthe Hamiltonian (we utilize the equations of motion):

X = (q1, . . . qm, p1, . . . pm) =(

∂H

∂p1, . . . ,

∂H

∂pm, −∂H

∂q1, . . . , − ∂H

∂qm

).

(23)Plugging this into the continuity equation and doing some algebra leadingto the cancellation of terms ∂2H/∂qs∂ps by the terms -∂2H/∂ps∂qs, weultimately arrive at an elegant formula (we take advantage of the previouslyintroduced Poisson bracket)

∂

∂tW (X, t) = W,H . (24)

This is the Liouville equation—the equation of motion for the distributionfunction W (X, t). Since it is the first-order differential equation with re-spect to time, it unambiguously defines the evolution of any given initialdistribution.

While the form of the Liouville equation definitely has something incommon with Eq. (4), the physical meaning of the two is radically different.In the l.h.s. of Eq. (4) we are dealing with the full derivative with respectto time, A ≡ A(qs(t), ps(t)), while the variable X in Eq. (24) is essentiallytime-independent; it just labels a fixed point in the phase space. Note alsothe different sign: W,H = −H,W.

Nevertheless, the relation Eq. (4) becomes crucially important for under-standing the structure of the equilibrium solutions of the Liouville equation.Indeed for any equilibrium (=time-independent) solution W (X) we haveH, W = 0. Thus, if we formally—the procedure has no direct physicalmeaning!—plug X = X(t) into W (X), where X(t) is any trajectory satisfy-ing the equations of motions, then the result will be time-independent. Thatis any equilibrium W is formally equal to some constant of motion, and viceversa! We have already seen an example of this when playing with our toymodel. Now we see that this is a general theorem (known as Liouville’stheorem).

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