Relaxation time

Relaxation time is a general concept in physics for the characteristic time in which a system changes to an equilibrium condition from a non-equilibrium condition. It can measure the time-dependent response of a system to well-defined external stimuli.

Dielectric relaxation time
For instance, the properties of a dielectric change on a time scale determined by the relaxation time when an external electric field is changed. This so-called dielectric relaxation time is a property of a solid that is closely related to its conductivity. The dielectric relaxation time is a measure of the time it takes for charge in a semiconductor to become neutralized by conduction process. It is small in metals and can be large in semiconductors and insulators.

Amorphous solids
An amorphous solid, such as amorphous indomethacin displays a temperature dependence of molecular motion, which can be quantified as the average relaxation time for the solid in a metastable supercooled liquid or glass to approach the molecular motion characteristic of a crystal. Differential scanning calorimetry can be used to quantify enthalpy change due to molecular structural relaxation.

Mathematical example: Damped unforced oscillator
Let the homogenous differential equation: $$m\frac{d^2 y}{d t^2}+\gamma\frac{d y}{d t}+ky=0$$ model damped unforced oscillations of a weight on a spring.

The displacement will then be of the form $$y(t) = A e^{-t/T} \cos(\mu t - \delta)$$. The constant T is called the relaxation time of the system and the constant μ is the quasi-frequency.

Astronomy
In astronomy, relaxation time relates to clusters of gravitationally-interacting bodies (star clusters, galaxy clusters, globular clusters). The relaxation time is a measure of the time it takes for one object in a system to be significantly perturbed by other objects in the system. In the case of stars in a galaxy, the relaxation time measures the time for one star to undergo a strong encounter with another star. Various events occur on timescales relating to the relaxation time, including core collapse and energy exchange between stars (minimization of the total energy in a cluster).

The relaxation time is related to the velocity of a body (typically a star) and the perturbation rate. In the example of a star cluster, a particular star will have an orbit with a velocity v. As the star passes by other stars, the orbit will be perturbed by the gravitational field of nearby stars. The relaxation time is similar to the ratio of the velocity to the time derivative of the perturbation.