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3.2: Phase Flows in Classical Mechanics

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Hamiltonian evolution

The master equation provides us with a semi-phenomenological description of a dynamical system’s relaxation to equilibrium. It explicitly breaks time reversal symmetry. Yet the microscopic laws of Nature are (approximately) time-reversal symmetric. How can a system which obeys Hamilton’s equations of motion come to equilibrium?

Let’s start our investigation by reviewing the basics of Hamiltonian dynamics. Recall the Lagrangian L=L(q,˙q,t)=TV. The Euler-Lagrange equations of motion for the action S[q(t)]=dtL are

˙pσ=ddt(L˙qσ)=Lqσ ,

where pσ is the canonical momentum conjugate to the generalized coordinate qσ:

pσ=L˙qσ .

The Hamiltonian, H(q,p) is obtained by a Legendre transformation,

H(q,p)=rσ=1pσ˙qσL .

Note that

dH=rσ=1(pσd˙qσ+˙qσdpσLqσdqσL˙qσd˙qσ)Ltdt =rσ=1(˙qσdpσLqσdqσ)Ltdt .

Thus, we obtain Hamilton’s equations of motion,

Hpσ=˙qσ,Hqσ=Lqσ=˙pσ

and

dHdt=Ht=Lt .

Define the rank 2r vector φ by its components,

φi={qiif\ 1irpirif\ ri2r .

Then we may write Hamilton’s equations compactly as

˙φi=JijHφj ,

where

J=(0r×r1r×r1r×r0r×r)

is a rank 2r matrix. Note that Jt=J, J is antisymmetric, and that J2=12r×2r.

Dynamical systems and the evolution of phase space volumes

Consider a general dynamical system,

dφdt=V(φ) ,

where φ(t) is a point in an n-dimensional phase space. Consider now a compact2 region R0 in phase space, and consider its evolution under the dynamics. That is, R0 consists of a set of points {φ|φR0}, and if we regard each φR0 as an initial condition, we can define the time-dependent set R(t) as the set of points φ(t) that were in R0 at time t=0:

R(t)={φ(t)|φ(0)R0} .

Now consider the volume Ω(t) of the set R(t). We have

Ω(t)=R(t)dμ

where

dμ=dφ1dφ2dφn ,

for an n-dimensional phase space. We then have

Ω(t+dt)=R(t+dt)dμ=R(t)dμ|φi(t+dt)φj(t)| ,

where

|φi(t+dt)φj(t)|(φ1,,φn)(φ1,,φn)

is a determinant, which is the Jacobean of the transformation from the set of coordinates {φi=φi(t)} to the coordinates {φi=φi(t+dt)}. But according to the dynamics, we have

φi(t+dt)=φi(t)+Vi(φ(t))dt+O(dt2)

and therefore

φi(t+dt)φj(t)=δij+Viφjdt+O(dt2) .

We now make use of the equality

lndetM=TrlnM ,

for any matrix M, which gives us3, for small ε,

det(1+εA)=expTrln(1+εA)=1+εTrA+12ε2((TrA)2Tr(A2))+

Thus,

Ω(t+dt)=Ω(t)+R(t)dμVdt+O(dt2) ,

which says

dΩdt=R(t)dμV=R(t)dSˆnV

Here, the divergence is the phase space divergence,

V=ni=1Viφi ,

and we have used the divergence theorem to convert the volume integral of the divergence to a surface integral of ˆnV, where ˆn is the surface normal and dS is the differential element of surface area, and R denotes the boundary of the region R. We see that if V=0 everywhere in phase space, then Ω(t) is a constant, and phase space volumes are preserved by the evolution of the system.

For an alternative derivation, consider a function ϱ(φ,t) which is defined to be the density of some collection of points in phase space at phase space position φ and time t. This must satisfy the continuity equation,

ϱt+(ϱV)=0 .

This is called the continuity equation. It says that ‘nobody gets lost’. If we integrate it over a region of phase space R, we have

ddtRdμϱ=Rdμ(ϱV)=RdSˆn(ϱV) .

It is perhaps helpful to think of ϱ as a charge density, in which case J=ϱV is the current density. The above equation then says

dQRdt=RdSˆnJ ,

where QR is the total charge contained inside the region R. In other words, the rate of increase or decrease of the charge within the region R is equal to the total integrated current flowing in or out of R at its boundary.

clipboard_e7321fd35a4ca7363b128a8d296ffb6ba.png
Figure 3.2.1: Time evolution of two immiscible fluids. The local density remains constant.

The Leibniz rule lets us write the continuity equation as

ϱt+Vϱ+ϱV=0 .

But now suppose that the phase flow is divergenceless, V=0. Then we have

DϱDt(t+V)ϱ=0 .

The combination inside the brackets above is known as the convective derivative. It tells us the total rate of change of ϱ for an observer moving with the phase flow. That is

ddtϱ(φ(t),t)=ϱφidφidt+ϱt=ni=1Viρφi+ϱt=DϱDt .

If Dϱ/Dt=0, the local density remains the same during the evolution of the system. If we consider the ‘characteristic function’

ϱ(φ,t=0)={1if φR00otherwise

then the vanishing of the convective derivative means that the image of the set R0 under time evolution will always have the same volume.

Hamiltonian evolution in classical mechanics is volume preserving. The equations of motion are

˙qi=+Hpi,˙pi=Hqi

A point in phase space is specified by r positions qi and r momenta pi, hence the dimension of phase space is n=2r:

φ=(qp),V=(˙q˙p)=( H/pH/q) .

Hamilton’s equations of motion guarantee that the phase space flow is divergenceless:

V=ri=1{˙qiqi+˙pipi}=ri=1{qi(Hpi)+pi(Hqi)}=0 .

Thus, we have that the convective derivative vanishes, viz.

DϱDtϱt+Vϱ=0 ,

for any distribution ϱ(φ,t) on phase space. Thus, the value of the density ϱ(φ(t),t) is constant, which tells us that the phase flow is incompressible. In particular, phase space volumes are preserved.

Liouville’s equation and the microcanonical distribution

Let ϱ(φ)=ϱ(q,p) be a distribution on phase space. Assuming the evolution is Hamiltonian, we can write

ϱt=˙φϱ=rk=1(˙qkqk+˙pkpk)ϱ=iˆLϱ ,

where ˆL is a differential operator known as the Liouvillian:

ˆL=irk=1{HpkqkHqkpk} .

Equation ???, known as Liouville’s equation, bears an obvious resemblance to the Schrödinger equation from quantum mechanics.

Suppose that Λa(φ) is conserved by the dynamics of the system. Typical conserved quantities include the components of the total linear momentum (if there is translational invariance), the components of the total angular momentum (if there is rotational invariance), and the Hamiltonian itself (if the Lagrangian is not explicitly time-dependent). Now consider a distribution ϱ(φ,t)=ϱ(Λ1,Λ2,,Λk) which is a function only of these various conserved quantities. Then from the chain rule, we have

˙φϱ=aϱΛa˙φΛa=0 ,

since for each a we have

dΛadt=rσ=1(Λaqσ˙qσ+Λapσ˙pσ)=˙φΛa=0 .

We conclude that any distribution ϱ(φ,t)=ϱ(Λ1,Λ2,,Λk) which is a function solely of conserved dynamical quantities is a stationary solution to Liouville’s equation.

Clearly the microcanonical distribution,

ϱE(φ)=δ(EH(φ))D(E)=δ(EH(φ))dμδ(EH(φ)) ,

is a fixed point solution of Liouville’s equation.


This page titled 3.2: Phase Flows in Classical Mechanics is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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