viriaal

Viriaal is a term used in physics to refer to two related concepts: the virial theorem and the virial expansion. The term is widely employed in classical mechanics, astrophysics, and statistical mechanics, and its Dutch spelling is viriaal.

The virial theorem relates time-averaged quantities in a bound mechanical system. For a system of N particles with potential energy V that is homogeneous of degree n, the theorem states that 2⟨T⟩ = n⟨V⟩, where T is the total kinetic energy. In gravitational systems, where the potential scales as r^(-1) and n = -1, this becomes 2⟨T⟩ = -⟨V⟩. The virial theorem provides a practical link between kinetic and potential energy components and is used to estimate masses, energy budgets, and dynamical states of systems such as star clusters, galaxies, and molecular ensembles. It is derived from the time evolution of the virial quantity G = ∑ r_i · p_i and applies to stationary or quasi-stationary configurations where time averages exist.

The virial expansion is a framework in statistical mechanics for expressing the equation of state of a many-particle system as a series in density. In the form p = ρ k_B T [1 + B2(T)ρ + B3(T)ρ^2 + ...], the coefficients Bi(T) are virial coefficients that encode interactions between particles. The second virial coefficient B2(T) reflects pairwise interactions and can be computed from the pair potential u(r). Virial coefficients quantify deviations from ideal gas behavior and are central to understanding real gases and molecular fluids via the Mayer cluster expansion.

See also: virial coefficient, virial expansion, Mayer expansion, virial theorem.