In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by a conservative force (where the work done is independent of path), with that of the total potential energy of the system.
[1] The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics; this average total kinetic energy is related to the temperature of the system by the equipartition theorem.
However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in thermal equilibrium.
If the force between any two particles of the system results from a potential energy V(r) = αrn that is proportional to some power n of the interparticle distance r, the virial theorem takes the simple form
In 1870, Rudolf Clausius delivered the lecture "On a Mechanical Theorem Applicable to Heat" to the Association for Natural and Medical Sciences of the Lower Rhine, following a 20-year study of thermodynamics.
as applied in classical gravitational dynamics, the original form of which was included in Lagrange's "Essay on the Problem of Three Bodies" published in 1772.
[2] The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell, Lord Rayleigh, Henri Poincaré, Subrahmanyan Chandrasekhar, Enrico Fermi, Paul Ledoux, Richard Bader and Eugene Parker.
Richard Bader showed that the charge distribution of a total system can be partitioned into its kinetic and potential energies that obey the virial theorem.
[3] As another example of its many applications, the virial theorem has been used to derive the Chandrasekhar limit for the stability of white dwarf stars.
Consider N = 2 particles with equal mass m, acted upon by mutually attractive forces.
Suppose the particles are at diametrically opposite points of a circular orbit with radius r. The velocities are v1(t) and v2(t) = −v1(t), which are normal to forces F1(t) and F2(t) = −F1(t).
The respective magnitudes are fixed at v and F. The average kinetic energy of the system in an interval of time from t1 to t2 is
This is because the dot product of the displacement with equal and opposite forces F1(t), F2(t) results in net cancellation.
[4] Assuming that the masses are constant, G is one-half the time derivative of this moment of inertia:
Since no particle acts on itself (i.e., Fjj = 0 for 1 ≤ j ≤ N), we split the sum in terms below and above this diagonal and add them together in pairs:
In a common special case, the potential energy V between two particles is proportional to a power n of their distance rij:
If a region of space is unusually full of galaxies, it is safe to assume that they have been together for a long time, and the virial theorem can be applied.
Doppler effect measurements give lower bounds for their relative velocities, and the virial theorem gives a lower bound for the total mass of the cluster, including any dark matter.
The expectation value ⟨dQ/dt⟩ of this time derivative vanishes in a stationary state, leading to the quantum virial theorem:
[13] Henri Poincaré proved and applied a form of the virial theorem in 1911 to the problem of formation of the Solar System from a proto-stellar cloud (then known as cosmogony).
[18] The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law:[19][20][failed verification]
Finally, pik is the fluid-pressure tensor expressed in the local moving coordinate system
In a finite configuration without pressure-bearing walls or magnetic coils, the surface integral will vanish.
If a total mass M is confined within a radius R, then the moment of inertia is roughly MR2, and the left hand side of the virial theorem is MR2/τ2.
The difference from the classical case arises due to considering the pressure field and the field of particles’ acceleration inside the system, while the derivative of the scalar G is not equal to zero and should be considered as the material derivative.
An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature:[24]
Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models, such as an isothermal sphere), it can be hard to define specific, finite measures of their mass and size.
The virial theorem, and related concepts, provide an often convenient means by which to quantify these properties.
The core temperature increases even as energy is lost, effectively a negative specific heat.
[26] This continues beyond the main sequence, unless the core becomes degenerate since that causes the pressure to become independent of temperature and the virial relation with n equals −1 no longer holds.