# How is virial theorem for point particles related to the virial theorem for an ideal gas?

Some presentations of the virial theorem are mechanical (see this page by John Baez for an example). They assume that there is a system of point particles interacting only via Newtonian gravity (along with other assumptions, e.g. that the particles don't fly away to infinity), and show $\langle T \rangle = -\frac12\langle V\rangle$. The physics that goes in is just Newton's laws.

Other presentations are based on thermodynamics (see pp 81 of these notes by Mike Guidry, for example). They imagine a gas in hydrostatic equilibrium, find the pressure, and use the ideal gas law to derive the same result, $T = -\frac12 V$.

The physical assumptions that go into these seem pretty different. In the mechanical case, we have only gravitational interactions. In the thermodynamic case, the interactions aren't even specified. Presumably the gas particles are bouncing off each other according to some sort of force law, but we only actually need to know that the ideal gas law holds (and use the condition for hydrostatic equilibrium).

Although the theorems seem physically different, they have the same name and come to the same conclusion (except that the thermodynamic one doesn't need the time-averaging). How are these two versions of the virial theorem related to each other? Other than crunching through each proof separately, how can one see that they ought to give the same result?

note: I'm asking about the special case of the virial theorem described above, not the general virial theorem for more general force laws, for example

• Hmmm ... I think it is relevant that Guidry examines a gas in hydrostatic equilibrium in which pressure is balancing the self-gravitation. In other words the long-range force in at play is the same one treated in conventional discussions such as the one by Baez. Isn't it? If we ask about an ideal gas confined in a box we would expect to get different results. Jun 27 '18 at 5:49
• This question held my attention through my shower. There are a whole series of ways you might argue it, but I think writing a Virial in terms of central potentials in multiple powers would lead to something like $\langle T \rangle = f_1 \langle U_\text{dominate} \rangle + f_2 \langle U_\text{subdominate} \rangle + \dots$ and that with an ideal gas assumption even the subdominate term is small. Alas, my idea to write a mini-dictionary of models that might reasonable be applied to stars got derailed by the non-central magnetic forces in a plasma. MHD strikes again. Jun 27 '18 at 6:19

1. Generally speaking, while the virial theorems in classical mechanics (CM) and classical statistical mechanics (CSM) have the same form, it is important to realize that the average procedure $\langle\cdot \rangle$ is a long-time average in CM and a statistical average in CSM. See also my related Phys.SE answer here.

2. Note that in fluid dynamical models, mechanical and statistical notions become mixed together, thereby leading to hybrid virial theorems. E.g. the energy of a fluid is typically a mixture of thermodynamical internal energy and mechanical (kinetic & potential) energies. See e.g. my related Phys.SE answer here.

How are these two versions of the virial theorem related to each other?

Short answer, Guidry's derivation does not really derive another "version of virial theorem" of general validity; his result

$$2U+\Omega =0 ~~~(1)$$

involving internal energy of gas $U$ and potential energy of gas $\Omega$ implies more restricted behaviour than the behaviour that the usual general virial theorem result for gravitationally bound systems

$$2\langle T \rangle + \langle E_p\rangle =0 ~~~(2),$$ where $E_p$ is gravitational potential energy, does. This is because Guidry's derivation assumes some special assumptions to get (1).

Long answer, his argument is based on some very special assumptions:

• the gas has defined pressure $p$ at every point of space;

• all different spherical layers of gas are at rest;

• the particles interact according to Newton's law of gravitation (see the formula for gravitational P.E. he uses);

• the gas particles interact also via short-range repulsive interaction of negligible interaction energy (so the layers experience pressure but no contribution to internal energy due to this repulsion needs to be included, internal energy density is $3/2p$ as for particles that do not repel each other);

• there is $R$ where pressure $P=0$ or at least $\lim_{r\to\infty} r^3 P \to 0$.

The last assumption seems unwarranted for a system of particles that are attracted to each other only gravitationally, but let us suspend this disbelief.*

His derivation uses these assumptions to arrive at the result that internal energy of the gas is equal to minus half the potential energy of the mass, and both energies are constant in time, since the spherical layers are assumed not to move.

On the other hand, the general virial theorem, when applied to gravitationally bound system of particles, uses much less restrictive assumptions: just that the system does not lose particles too violently, so the assumption behind derivation of virial theorem's is valid, but there is no requirement of hydrostatic equilibrium. Then the time averages obey the relation (2). This result means that both kinetic energy and potential energy can change in time, unlike Guidry's quantities $U$ and $\Omega$. Less special assumptions, less restrictive condition that the system obeys.

* Particles that interact purely gravitationally tend to form systems with no apparent self-delimited boundary; usually, density decays with distance but there is no closed surface where it falls down to 0; some particles can get arbitrarily far from the system.