Spotlight: the Virial Theorem
The Virial Theorem (Clausius, 1870) governs equilibria involving a large number of bodies; it has been successfully applied in the last century to a number of relevant physics problems; in astrophysics, cosmology, molecular physics, quantum mechanics, and in statistical mechanics. In itself, the theorem provides a general equation that relates the time average of the kinetic energy of a stable system consisting of N particles with the time average of the potential energy that binds these particles.
For the careful reader, it is worth noting that the Virial theorem is not really about equilibria at all. Rather, it is systems that remain bounded in phase space for extensive periods of time by some central potential. This condition of being bound for long periods of time is sometimes called “virial equilibrium”, which is distinct from thermodynamic equilibrium. The two are often conflated however since most systems that remain bounded for long times naturally settle down to some sort of steady state. This allows for the hand-wavy equation of steady-state with equilibrium (although the steady-state property is not strictly required, only boundedness). It then becomes important to remember that the virial theorem ultimately stems from a mechanical and not a thermodynamic perspective.
The equations of motion for a two-body problem can be solved analytically. It is therefore possible to directly obtain expressions that describe all future motions of the bodies. Adding just one more body to the problem renders the situation considerably more difficult. There is no general analytic solution to the three-body problem. In fact, Poincaré proved that for any N > 2, there is no closed-form analytic solution for the N-body problem. Often, however, it is not uncommon to be interested in systems of millions or billions of bodies. For instance, a galaxy may have more than 5 × 10¹¹ stars. To describe exactly the motion of stars in galaxies we would need to solve the 5× 10¹¹-body problem. Equally, a volume of interacting gas particles may contain over 10²³ molecules, requiring a 10²³-body problem. This is, of course, impossible, but we can still make some simple considerations about the general properties of such a system by invoking the Virial theorem.
The virial theorem is incredibly powerful; although some will recognize it and some can state it correctly, only a few appreciate its significance. This may largely be a result of its obscure origin, for the virial theorem did not arrive in the form we see it today but rather evolved from the triumphs made in the kinetic theory of gases. In order to place the theorem in its proper perspective, it is worth recounting some of that history.
Inspired by Carnot’s work on heat engines, Claussius began working on mechanical heat in the mid-19th Century. His twenty years of work then culminated in the formulation of what we now see as the earliest presentation of the virial theorem. On June 13, 1870, Claussius delivered a lecture before the Association for Natural and Medical Sciences of the Lower Rhine “On a Mechanical Theorem Applicable to Heat.” [Ref. Claussius, R. J. E. 1870. Phil. Mag. S. 4, Vol. 40, p. 122]. In giving this lecture, Claussius stated the theorem as “The mean vis viva of the system is equal to its virial.” In the 19th century, scientists often assigned Latin names to systems and their characteristics. So, as is known to all students of celestial mechanics the vis viva (Latin for “living force”) is in reality the total kinetic energy of the system. Virial is then derived from Latin as well, stemming from the word virias (plural of vis) meaning forces. Although the characteristic of the system Claussius called the virial is no longer given much significance as a physical concept, the name has become attached to the theorem and its evolved forms.
After the turn of the century, the applications of the theorem became more varied and widespread. Lord Rayleigh formulated a generalization of the theorem in 1903 in which the tensor virial theorem was formulated and later so extensively developed by Chandrasekhar during the 1960s. Poincare too used a form of the virial theorem in 1911 to investigate the dynamic stability of cosmological structures during the 1940s. Paul Ledoux developed a variational form of the theorem which was used to obtain the pulsation period for stars. Chandrasekhar and Fermi extended the virial theorem in 1953 to include the presence of magnetic fields.
Below is the usual treatment of the virial theorem, following from Goldstein’s Third Edition of Classical Mechanics. It is worth noting that there are many ways to achieve the same result. While reading, I also came across using Lagrange’s identity to arrive at virial theorem (though I must confess that this beats me). The original form of Lagrange’s identity was included in his “Essay on the Problem of Three Bodies” published in 1772, predating Claussius by a century. Nonetheless, it is widely regarded that the credit for the virial theorem belongs to Claussius.
The General Case
We begin by considering an isolated system of N bodies bounded by some potential. The moment of inertia for our system is defined as follows:
Taking the time derivative yields
The mass terms combine with the velocity components to yield momentum, p. The new sum of the momenta dotted with the positions of the bodies results in our desired quantity called the virial, eta. Taking the time derivative of the virial
The time average over a really long period of time is then
where the left-hand side equals zero since 𝜏 → ∞ = 0 if the constraints and velocities of all the particles remain finite. So, in other words, as long as there is an upper-bound to η, we obtain the following
The right-hand side of this equation is called the virial of the system. For a system of particles subject to a conservative central force F = -∇U, the virial theorem equals
For a single particle acting under a central potential, this reduces to
Now, consider the general potential of the following form
whose derivative then follows
Substituting back,
For various potentials then, the kinetic energy is defined differently. Consider harmonic motion. For a linear restoring force when n = 1, the time-averaged kinetic energy equals the time-averaged potential energy. Consider inverse-square forces: for a force where n = -2, the time-averaged kinetic energy equals -1/2 the time-averaged potential energy.
A Remark
The reason that the moment of inertia shows up is in some ways a little accidental. It’s a compact way to remember and to derive the virial theorem, but it is not a priori the “correct” physical quantity to look at since summing the moments of inertia does not even seem to be an intuitive place, to begin with. In my opinion, a more appropriate way to approach the virial theorem is in terms of symmetry (following Noether’s theorem). Below is an article briefly discussing Noether’s theorem:
In reading the above, or otherwise, you may come across the fact that continuous symmetries are associated with a conserved quantity: rotational symmetry is associated with angular momentum conservation, linear momentum conservation follows from translational symmetry and time symmetry yields conservation of energy. It turns out that for a system that is in virial equilibrium, there is an approximate dilatation symmetry that emerges in the long time limit (that is, as 𝜏 → ∞). Linked here are some articles that discuss this: Zhang et al., Milgrom 1994, StackExchange. Dilatations are the symmetries associated with a global rescaling of all positions r → rq for sufficiently small q. The quantity η = dI/dt is the conserved quantity associated with dilation but it is equally important to note that η is not exactly conserved since it is only bounded above and below a time-dependent constant, and so appears approximately constant in large 𝜏 limit. One often-cited reason applies to systems in virial equilibrium; velocities and coordinates of the particles of the system have upper and lower limits so that ηbound, is bounded between two extremes, ηmin, and ηmax, and the average goes to zero in the limit of infinite τ:
Another Remark
Averaging a system over a long time period may be equal to averaging the system over the ensemble. This is the ergodic hypothesis. Mathematically it can be written as the following
Essentially, this is the equivalent of stating if a bound system has a large number of bodies (N → ∞), it is equivalent to seeing the system’s evolution over a large time (𝜏 → ∞). We can then apply the virial theorem to a galaxy by determining the kinetic and potential energies of all the stars at a given moment instead of measuring the energies over a long period of time. Since the time scales for changes for such huge systems are very long, it is much easier to simply take the average over all stars. The ergodic theorem then says that we can replace the mean value from being a time average to being an average over all bodies in the system, making matters far simpler.
With that, I will end this article here. The virial theorem is generally a difficult concept to tackle so I apologize if there were any mistakes made here. Feel free to rip me apart in the comments! And as always, thank you for reading!
