Relaxation (physics)

In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be characterized by a relaxation time τ. The simplest theoretical description of relaxation as function of time t is an exponential law exp(-t/τ).

Relaxation in simple linear systems

Mechanics: 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) = Ae ^{− t / T}cos(μt − δ). The constant T is called the relaxation time of the system and the constant μ is the quasi-frequency.

Electronics: The RC circuit

In an RC circuit containing a charged capacitor and a resistor, the voltage decays exponentially:

$V(t)=V_0 e^{-\frac{t}{RC}} \ ,$

The constant $\tau = RC\$ is called the characteristic/relaxation time of the circuit.

Relaxation in condensed matter physics

In condensed matter physics, relaxation is usually studied as a linear response to a small external perturbation. Since the underlying microscopic processes are active even in the absence of external perturbations, one can also study "relaxation in equilibrium" instead of the usual "relaxation into equilibrium" (see fluctuation-dissipation theorem).

Dielectric relaxation time

In dielectric materials, the dielectric polarization P depends on the electric field E. If E changes, P(t) reacts: the polarization relaxes towards a new equilibrium.

The dielectric relaxation time is closely related to the electrical conductivity. In a semiconductor it is a measure how long it takes to become neutralized by conduction process. This relaxation time is small in metals and can be large in semiconductors and insulators.

Liquids and amorphous solids

Main article: Structural relaxation

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.

Spin relaxation in NMR

Main article: Relaxation (NMR)

In nuclear magnetic resonance, relaxation is of prime importance. See Relaxation (NMR).

Relaxation in atmospheric sciences

Desaturation of clouds

Consider a supersaturated portion of a cloud. Then shut off the updrafts, entrainment, or any other vapor sources/sinks and things that would induce the growth of the particles (ice or water). Then wait for this supersaturation to reduce and become just saturation (relative humidity = 100%), which is the equilibrium state. The time it took for this to happen is called relaxation time. It will happen as ice crystals or liquid water content grow within the cloud and thus eat up the moisture that's in it. It is very important in cloud physics modeling because if models do not consider a relaxation time, things will grow very exaggerated.

In water clouds where the concentrations are larger (hundreds per cm³) and the temperatures are warmer (thus allowing for much lower supersaturation rates as compared to ice clouds), the relaxation times will be very low (seconds to minutes).

In ice clouds the concentrations are lower (just a few per liter) and the temperatures are colder (very high supersaturation rates) and so the relaxation times can be hours and hours.

τ=(4πDNRK)^-1

where

• D = diffusion coefficient [m²/s]
• N = concentration (of ice crystals or water droplets) [m^-3]
• R = mean radius of particles [m]
• K = capacitance [unitless]

Relaxation in 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 the velocity of a star to be changed by gravitational perturbations from other stars. Various events occur on timescales relating to the relaxation time, including core collapse and energy equipartition.

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 acceleration resulting from the perturbation.^[1]

See also

• Relaxation oscillator

References

1. ^ Sparke, L. & Gallagher, J. (2000). Galaxies in the Universe: An Introduction, 1st ed., Sec. 3.2. Cambridge University Press. ISBN 0-521-59241-0