The Virial Theorem

On the day of 1870 Jun 13 Rudolf Clausius delivered a lecture titled AOn a Mechanical Theorem Applicable to Heat@ to a gathering of members of the Association for Natural and Medical Sciences of the Lower Rhine. In the course of that lecture he laid out what we call the virial theorem, which he summed up in the statement, AThe mean vis viva of the system is equal to its virial@. In that statement he used the Latin vis (force, strength) viva (living) to denote what we now call kinetic energy and virial (the adjectival form of the plural vires) to denote what we now call potential energy.

Consider a set of particles, each particle
pondering a mass m_{i}, bound into a single system by suitable forces
and isolated from any forces emanating from any other systems of particles. We
claim to have a complete mathematical description of that system and from that
description we draw one aspect for further examination
B the moment of mass,

(Eq=n 1)

in which M represents the total mass of the system and the
vector **x**_{i} represents the location of the i-th particle
relative to the origin of our coordinate grid. The vector
+ **x, **thus represents the location of
the system=s center of mass. On the
assumption that dm_{i}/dt=0 (that is, that the particles=
masses don=t change, such as in
collisions in which two particles trade some net mass from one to the other) for
all values of the index, we differentiate Equation 1 with respect to time and
get

(Eq=n 2)

in which the velocity of the i-th particle comes out as **v**_{i}=d**x**_{i}/dt.
In that way we calculate the net linear momentum of the system. If we
differentiate that equation with respect to time, we get

(Eq=n 3)

In that equation the equality to zero comes from Newton=s third law of motion and the fact that our system does not interact with any other systems.

Next we draw from our description of the
system our calculation of its moment of inertia, by multiplying each term in
Equation 1 by **x**_{i}, for further examination;

(Eq=n 4)

As we did with the moment of mass, we differentiate that equation with respect to time;

(Eq=n 5)

in which **p**_{i}=m_{i}**v**_{i}
represents the i-th particle=s linear
momentum. The sum has the mathematical form of an action and each term looks
like the action that we would have to assert to move a particle with its
momentum from the origin of our grid to the point represented by **x**_{i}.
If we put the center of mass of our system at the origin of the grid, then that
sum represents the action that comes into being when we create the system in its
given configuration from an undifferentiated mass M sitting motionless on the
origin. Of course, in accordance with the principle of least action, we figure
that the number that we calculate with that sum represents the minimum action
required to transform an undifferentiated motionless mass into an extended bound
system whose parts move.

If we divide Equation 5 by two and differentiate it with respect to time, we get

(Eq=n 6)

in which I have exploited the twin facts that **p**_{i}=m_{i}**v**_{i}
and that Newton=s second law of motion
equates the rate at which a body=s
linear momentum changes to the force exerted upon that body by other bodies. The
first term on the right side of that equation represents twice the total kinetic
energy contained in the system due to the motions of its parts and the second
term represents Clausius= virial.

One way of calculating the virial term begins with the superposition principle, which tells us that for i…j we have

(Eq=n 7)

in which **f**_{ij} equals the force that the j-th
particle exerts upon the i-th particle. We then have the virial as

(Eq=n 8)

in which I have exploited Newton=s
third law of motion in the form **f**_{ji}=-**f**_{ij}.
But **x**_{i}-**x**_{j}=**r**_{ij} and

(Eq=n 9)

in which U_{ij} represents the potential energy of the
i-th particle vis-a-vis the j-th particle. Because the potential energy usually
conforms to the description

(Eq=n 10)

in which K_{ij} represents some combination of constants
and the properties of the interacting particles (such as gravitational mass or
electric charge) and n usually represents a negative number (n=-1 for
inverse-square forces, such as gravity or electricity), we have

(Eq=n 11)

So now we have Equation 6 in the form

(Eq=n 12)

in which T represents the total kinetic energy manifested in the system and U represents the total potential energy manifested in the system. If we let n=-1 (gravity, electric force), we get

(Eq=n 13)

which astrophysicists call Lagrange=s Identity, the virial theorem as it applies to astrophysical systems.

And now for a little sleight of mind. We want
to calculate the average value of Equation 12 over a long span of time. To that
end we integrate Equation 12 with respect to time and divide the result by t_{v},
which we can increase endlessly. We get

(Eq=n 14)

the zero coming about as a result of the limit based on the fact
that the rate at which the system=s
moment of inertia changes is, like the system itself, bounded while we can
extend t_{v} toward infinity. In the case n=-1, which is the most common
case, we have the sum of the system=s
total energy and its kinetic energy on the right side of that equation. The
system=s total energy remains
constant, so its average energy must do the same. Adding kinetic energy, which
never takes a negative value, to the total energy must zero out the average, so
we infer, as we should have expected for a bound system, that the potential
energy takes a negative value that is, on average, twice that of the average
kinetic energy in the system.

Now, to see whether we might gain some additional insight into this theorem, let=s approach the problem from a different direction. Let=s look at our system of particles as it manifests itself in phase space.

If we have a number of bodies bound by mutually exerted forces into a single system, then that system fills an unchanging volume in phase space. That statement gives us the basic content of Liouville=s theorem as it applies to physics. We can use that theorem to prove and verify the virial theorem by calculating the volume of a region of phase space occupied by a mechanical system and then calculating the rate at which that volume changes with the elapse of time. For the phase volume of an N-dimensional space we have

(Eq=n 15)

Note that the 2N on the upper right side of the elogated ess of integration does not represent the limit of the integration, but denotes the fact that we want to carry out 2N separate integrations, one for each dimension of the phase space. Differentiating that equation with respect to time gives us

(Eq=n 16)

In this calculation I have left the summation over the set of particles comprising the system implicit in order to reduce the number of subscripts in the formulae. In going from the first line to the second line I have exploited the fact that the operations of differentiation and integration commute with each other. In going from the second line to the third line I have exploited the definition of force and made the first term equal to Clausius= virial and then exploited the fact that multiplying a body=s linear momentum by its velocity calculates twice the body=s kinetic energy. Liouville=s theorem requires that the time derivative of a phase space volume equal zero. The integral in the third line represents a phase volume, which must equal a nonzero constant, so we must infer that we must have

(Eq=n 17)

standing true to mathematical physics. But that equation simply gives us the virial theorem again. Note that this version does not have the statistical nature of the previous version: Equation 17 stands true to Reality at every instant, not merely on average over a long elapse of instants.

On first impression the virial theorem seems trivial. It doesn=t seem to offer us any insight into the structure of Reality and we can=t use it to solve problems. At best we can use it to analyze data to see whether something might be missing from it.

Consider the gravitational force as it applies to bodies revolving on orbits about each other. For simplicity let=s consider a small body revolving about a much more massive body so that the small body carries almost all of the system=s kinetic energy. In the gravitational forcefield the small body=s potential energy, vis-a-vis the large body, stands proportional to the negative first power of the distance between the bodies, so the virial theorem takes the form

(Eq=n 18)

In this case we must use the averages to cover the possibility
that the bodies follow elliptical orbits. We describe the dynamics of a
perfectly circular orbit by balancing the small body=s
inertial reaction force against the gravitational force that the larger body
exerts upon it. In this case the inertial reaction corresponds to the
centrifugal force, m**v**^{2}/r, so we have

(Eq=n 19)

If the small body follows an elliptical orbit, that equation won=t quite balance at any given instant, but averaged over one or more orbital periods it will. Multiplying that equation by the radial distance gives us the more generally correct

(Eq=n 20)

as the virial theorem leads us to expect. If we then multiply that equation by r/m and represent the orbital speed as v=ωr, we get

(Eq=n 21)

which equals a constant. Because the angular velocity ω stands in inverse proportion to the orbital period, that equation encodes Johannes Kepler=s third law of planetary motion.

The virial theorem, in the form of Equation 18, also applies to systems of many bodies revolving about a common center of mass, such as the stars and nebulae comprising a galaxy. Astronomers can apply appropriate statistical techniques to their measurements of stellar velocities and locations in a galaxy to determine the amounts of kinetic energy and potential energy manifested in the galaxy=s structure. When they do so, they typically get results that look more like

(Eq=n 22)

For that equation to stand true to Reality, either the gravitational force in a galaxy must stand in inverse proportion to the twelfth power of distance from the galactic center or the galaxy must contain almost ten times the mass of its visible part in something that contributes very little to the galaxy=s store of kinetic energy. We have no evidence to support any hypothesis that the law of gravitational force differs from inverse-square, so we take Equation 22 as an indication that the galaxy (and the Universe by extension) contains roughly ten times the mass of its visible components in the form of some dark matter that does not form luminous bodies and does not revolve in elliptical orbits about the galactic center.

Or consider a very massive body with a cylindrical hole bored through it. We have a small body floating at the massive body=s center of mass and attached to the larger body through a spring that runs along the hole. When the system rotates about an axis perpendicular to the orientation of the spring, the small body will move away from the larger body=s center until the centrifugal force appearing to push it balances the force that the spring exerts between the bodies in accordance with Hooke=s law, f=-kr. In this case the small body=s potential energy, vis-a-vis the larger body, stands in direct proportion to the square of the radial distance between the small body and the center of the large body, so the virial theorem takes the form

(Eq=n 23)

Balancing inertial reaction force against the restoring force of the spring gives us

(Eq=n 24)

so multiplying that equation by radial distance gives us

(Eq=n 25)

because

(Eq=n 26)

Again, we see how the virial theorem conforms to classical dynamics.

Finally, because it has a statistical nature, the virial theorem must also conform to the ergodic theorem. To paraphrase James C. Maxwell, we say that a system is ergodic if it passes through all of the points in phase space that correspond to the system=s total energy. A one-dimensional harmonic oscillator, which consists of a body on a spring attached to a much more massive body, gives us a good example: it has a two-dimensional phase space, in which it traces out an ellipse with the elapse of time. If we have N identical oscillators (with N representing a very large number), all started at random times, then their locations in space at any given instant mimic the distribution of locations in one oscillator in one period. If we measure that one oscillator over an interval with randomly chosen beginning and ending times, the we must use as the distribution function the limit as Δt grows endlessly. Here we enter the realm of the ergodic theorem.

If a dynamic system passes through every point in phase space corresponding to its total energy, thereby fulfilling Maxwell=s criterion, then for the average of a system parameter Q we must have standing true to mathematical physics the statement that

(Eq=n 27)

in which +Q,_{s}
represents the statistical average of Q over the system at a single instant.
Although its content seems intuitive enough, we need this theorem to justify a
tacit assumption that physicists make in devising statistical thermodynamics and
other fields involving stochastic elements. In statistical dynamics theory gives
us +Q,_{s}
while experiments, because of the ultra-short times between particle collisions,
give us +Q,_{t}.
The ergodic theorem tells us that we can legitimately equate those two averages
and thus use experiments to prove and verify theories.

We connect the virial theorem to the ergodic theorem by way of the fact that Clausius obtained his version of the virial theorem by using phase averages while we obtain the virial theorem by taking the time average of Lagrange=s Identity,

(Eq=n 28)

and noting that the left side equals zero for stable systems.
That fact stands true to Reality under the proviso that
**τ**
greatly exceeds the longer of the system=s
relaxation time or system crossing time.

Finally, we note a fact that must lead to additional study. We have a hint that the virial theorem and the concept of ergodicity relate to the connection between fully reversible classical dynamics and the irreversible processes of thermodynamics. In pursuing that hint we may find a connection between ergodicity and the nature of time itself.

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