Virial Theorem

7/28/2019 Virial Theorem

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Lemma 1: Given a system governed by the canonical ensemble, with the properties

limzjj

H(z) = ,

and

H(z) > C0, z ,

for some global constant C0, the Virial Theorem holds.

Proof: Applying the definition of the probability density function for the canonical ensemble to the average in

the Virial theorem (1) results in the following:

ziH

zj :=

1

C

ziH

zjexp

1

kB TH(z)

dz. (2)

Here C is a normalization factor determined by the relation

C =

exp

1

kB TH(z)

dz.

Regrouping terms in (2) results in

ziH

zj =

kB T

C

zi

zj

exp

1

kB TH(z)

dz,

or rather,

ziH

zj =

kB T

C

zj

zi exp

H(z)

kB T

dz +

kB T

C

zizj

exp

H(z)

kB T

dz.

Using the definition of C and zi/zj = ij reduces the relation to:

ziH

zj =

kB T

C

zj

zi exp

H(z)

kB T

dz + kB T ij.

Applying the assumption that H(z) when zj j, we find that

zj

zi exp

H(z)

kB T

dz = 0.

Hence, the Virial Theorem holds:

ziH

zj = kB T ij ,

. We should note that in the case of classic periodic boundary conditions this proof only applies to components

of the momenta. A different proof is required for components of the position, since the energy remains bounded

as particles approach the edges of the simulation cell.

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3 Microcanonical Ensemble

In this section we will consider systems with bounded or unbounded domains, with the distribution of states

governed by the microcanonical ensemble. The direct proof of the Virial Theorem in this ensemble is very similarto the proof for the canonical ensemble. However, the assumptions are slightly more complicated, and we will

require some additional relations from thermodynamics. In particular, we will assume that the temperature is

defined through the entropy, S:

S := kB ln

1

h3N N!

(E H(z)) dz

, (3)

1

T:=

S

E. (4)

Here, the function is the unit (Heaviside) step-function, and the (h3N N!) is a normalization factor required

to get the correct units and additivity properties for entropy.

Recall that the probability of finding the system in a state with configuration (q, p) = z is given by theprobability density function:

mc (z) [E H(z)] .

The constant of proportionality is determined so that the integral of over the entire domain is one.

We will assume that each component zj is defined on the domain j with domain boundary j . We will

also assume that the Hamiltonian is bounded below, which is the case in classical Molecular Dynamics. Some

arguments can be made in the unbounded case (e.g., pure gravitation), but we will not consider that case here.

Lemma 2: Given a system governed by the microcanonical ensemble, with the property

limzjj

H(z) E, with zi i, i = j,

and

H(z) > C0, z ,

for some global constant C0, the Virial Theorem holds.

Proof: Applying the definition of the probability density function for the microcanonical ensemble to the

average in the Virial theorem (1) results in the following:

ziH

zj :=

1

C

ziH

zj[E H(z)] dz. (5)

Here C is a normalization factor determined by the relation

C =

[E H(z)] dz.

Using the property that the delta-function is the derivative of the step-function yields:

ziH

zj =

1

C

E

ziH

zj [E H(z)] dz.

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Rearranging terms (as in the previous section) results in

ziH

zj =

1

C

E

zj{ [E H(z)] zi (E H(z))} dz

+ ij1

C

E

[E H(z)] (E H(z)) dz

1

C

E

[E H(z)] (E H(z))H

zj(z) dz.

In the expression above the last term is zero, since the delta function is only nonzero when H(z) = E, but

at that point the integrand is zero. Using the assumption that the boundary of integration is inside of the

boundary of the set H(z) E,

zj{ [E H(z)] zi (E H(z))} dz = 0,

since [E H(z)] = 0 on the boundary of j . Hence, we are left with the following expression:

ziH

zj = ij

1

C

E

(E H(z)) [E H(z)] dz.

Differentiating with respect to E results in:

ziH

zj = ij

1

C

[E H(z)] dz + ij1

C

(E H(z)) [E H(z)] dz.

The second term is zero, and the first term can be rewritten as:

ziH

zj = ij

Eln

[E H(z)] dz

1

.

Applying the definition of entropy (3) yields:

ziH

zj = ij kB

S

E

1

.

Finally, using the relationship between temperature and entropy we arrive at the desired conclusion:

ziH

zj = ij kB T,

.

Once again, in the case of periodic boundary conditions this proof only applies to components of the mo-

menta. The reason is that the domain associated with a component of the coordinates is typically strictly

contained within the energy shell, hence H(z) is not equal to E on the boundary of the domain of integration.

However, for the components of the momenta (which are unbounded), the inclusion is reversed, and we reach

the energy shell before reaching the edge of the domain (since the domain of the momenta is the entire real

line). Hence, a different proof is required if we want to show the Virial Theorem is true for the coordinates

under periodic boundary conditions.

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4 Clausius Virial Theorem

In this section we consider an alternative approach for proving the Virial Theorem, using the Equipartition

Theorem with the Clausius Virial Theorem.

Theorem (Equipartition): Given a system governed by the canonical or microcanonical ensemble, and a

component of phase-space, zi, which appears quadratically in the energy function, and is defined on the entire

real line, then

ziH

zi = kB T

Proof: Clearly the energy is unbounded as zi approaches the boundary of its domain (infinity), hence both

Lemmas 1 and 2 apply, .

Note that for classical mechanical systems, components of the momenta always satisfy the assumptions of

the Equipartition Theorem.

Theorem (Classical Clausius): Given a system governed by the microcanonical ensemble, with a bounded

and differentiable trajectory in phase space, then

piH

pi = qi

H

qi.

Proof: Using Hamiltons equations of motion, we have the following identity:

d

dt(qipi) = piqi + qi pi = pi

H

pi qi

H

qi

Hence we need only to prove that d (qipi) /dt is zero. By the ergodic hypothesis, the ensemble average is thesame as the time average. Using the time average, we have

d

dt(qipi) := lim

1

0

d

dt(qipi) dt = lim

qi()pi() qi(0)pi(0)

Since both the coordinates and momenta are bounded for all time, the expression must be zero. Hence,

piH

pi = qi

H

qi,

.

Note that given Lemma 2, the Classical Clausius theorem does not provide any new information. In par-

ticular, it cannot be applied in this form to systems with periodic boundary conditions for one of two possible

reasons. Suppose we enforce the coordinate path to remain within the unit periodic cell, translating each coor-

dinate when it reaches the edge of the cell. In this case the trajectory is clearly not globally differentiable, with

discontinuities at each point that the trajectory reaches the edge of the cell. On the other hand, we could take

the view that the coordinate path can wander freely in space, without boundaries, and compute forces based on

the minimum (or multiple) image convention. Now we have the problem that the trajectory is not uniformly

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bounded, and hence the Classical Clausius theorem still does not apply. For the remainder of this section,

we will adopt the convention of an unbounded, but differentiable coordinate path, and introduce a Modified

Clausius theorem.

Theorem (Modified Clausius): Given a system governed by the microcanonical ensemble, with unbounded

coordinates, the Hamiltonian invariant under periodic translation, globally bounded momenta, and a differen-

tiable a trajectory in phase space, then

piH

pi qi

H

qi = Li lim

t

1

t

k

|pi (k)| .

Here, Li is the length of the simulation cell in the direction of the coordinate qi. The index k is over all crossings

of a cell boundary with time of each crossing given by k. The variable qi is the periodic version of qi, where qiis contained inside of a single unit cell.

Proof: We start by decomposing qi into a bounded periodic component, qi, and a sum of step-functions, qi.

We define the later term as follows:

qi (t) := qi (t) qi (t) =k

Lik (t k) .

The summation is over all crossings of a periodic cell, with Li and k defined as in the assumptions of the

Theorem. The symbol k represents the sign of the crossing, depending on if it crosses the left or right

boundary of the cell. Expanding the expression for ddt

(piqi) as before results in

d

dt(piqi) +

d

dt(piqi) =

d

dt(piqi) = piqi + qi pi,

or after subtracting qi pi from both sides

ddt

(piqi) + ddt

(piqi) qi pi = piqi + qi pi.

For the first term we can use the fact that qi and pi are bounded to conclude that

d

dt(qipi) := lim

1

0

d

dt(qipi) dt = lim

qi()pi() qi(0)pi(0)

is zero. After expanding the second term, we are left with

pid

dtqi = piqi + qi pi = pi

H

pi qi

H

qi.

Applying the definition of q, we can expand the left-hand side to obtain

pid

dtqi := lim

1

0

pi (t) Lik

k(t k) dt,

= lim

1

Lik

kpi (k) .

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L L L

qi

qi

qi

= +

Figure 1: Example of a decomposition of an unbounded qi into the sum of a periodic, qi, part and a series of

step functions, qi.

If we recognize that k and pi(k) always have the same sign (i.e. the crossing direction and the direction of

the momenta are always the same), we can replace kpi(k) with the absolute value of the momenta, yieldingin the desired result:

lim

1

Lik

|pi (k)| = piH

pi qi

H

qi,

.

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Virial Theorem

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