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 homogeneous 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.

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 relaxation time of the circuit. A nonlinear oscillator circuit which generates a repeating waveform by the repetitive discharge of a capacitor through a resistance is called a relaxation oscillator.

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).

Stress relaxation

In continuum mechanics, stress relaxation is the gradual disappearance of stresses from a viscoelastic medium after it has been deformed.

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 of 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

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.

The term "structural relaxation" was introduced in the scientific literature in 1947/48 without any explanation, applied to NMR, and meaning the same as "thermal relaxation".^[1]

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 will thus consume the contained moisture. The dynamics of relaxation are very important in cloud physics modeling because if models do not take relaxation time into account, then it is highly probable that error will creep into the system.

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, for instance, stars in a galaxy. The relaxation time is a measure of the time it takes for one object in the system (the "test star") to be significantly perturbed by other objects in the system (the "field stars"). It is most commonly defined as the time for the test star's velocity to change by of order itself.

Suppose that the test star has velocity v. As the star moves along its orbit, its motion will be randomly perturbed by the gravitational field of nearby stars. The relaxation time can be shown to be ^[2]

T_r = {0.34\sigma^3\over G^2 m\rho\ln\Lambda}

\approx 0.95\times 10^{10} \!\left({\sigma\over 200\,\mathrm{km\,s}^{-1}}\right)^{\!3} \!\!\left({\rho\over 10^6\,M_\odot\,\mathrm{pc}^{-3}}\right)^{\!-1} \!\!\left({m_\star\over M_\odot}\right)^{\!-1} \!\!\left({\ln\Lambda\over 15}\right)^{\!-1}\!\mathrm{yr}

where ρ is the mean density, m is the test-star mass, σ is the 1d velocity dispersion of the field stars, and ln Λ is the Coulomb logarithm.

Various events occur on timescales relating to the relaxation time, including core collapse, energy equipartition, and formation of a Bahcall-Wolf cusp around a supermassive black hole.

References

↑ Kittel, Rep. Progr. Phys. 1947; Hall, Phys. Rev. 1948; Wintner Phys. Rev. 1948.
↑ Spitzer, Lyman (1987). Dynamical evolution of globular clusters. Princeton, NJ: Princeton University Press. p. 191. ISBN 0691083096.