Relaxation (physics)

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

In simple linear systems

Mechanics: Damped unforced oscillator

Let the homogeneous differential equation:

${\displaystyle m{\frac {d^{2}y}{dt^{2}}}+\gamma {\frac {dy}{dt}}+ky=0}$

model

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

${\displaystyle V(t)=V_{0}e^{-{\frac {t}{RC}}}\ ,}$

The constant ${\displaystyle \tau =RC\ }$ is called the relaxation time or RC time constant 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.

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

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, i.e., the surface charges equalize. It is important in dielectric spectroscopy. Very long relaxation times are responsible for dielectric absorption.

The dielectric relaxation time is closely related to the

Liquids and amorphous solids

An

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]^[2][3]

Spin relaxation in NMR

In nuclear magnetic resonance (NMR), various relaxations are the properties that it measures.

Chemical relaxation methods

In

Monomolecular first-order reversible reaction

A monomolecular, first order reversible reaction which is close to equilibrium can be visualized by the following symbolic structure:

$${\displaystyle {\ce {A}}~{\overset {k}{\rightarrow }}~{\ce {B}}~{\overset {k'}{\rightarrow }}~{\ce {A}}}$$

$${\displaystyle {\ce {A <=> B}}}$$

In other words, reactant A and product B are forming into one another based on reaction rate constants k and k'.

To solve for the concentration of A, recognize that the forward reaction (${\displaystyle {\ce {A ->[{k}] B}}}$) causes the concentration of A to decrease over time, whereas the reverse reaction (${\displaystyle {\ce {B ->[{k'}] A}}}$) causes the concentration of A to increase over time.

Therefore, ${\displaystyle {d{\ce {[A]}} \over dt}=-k{\ce {[A]}}+k'{\ce {[B]}}}$, where brackets around A and B indicate concentrations.

If we say that at ${\displaystyle t=0,{\ce {[A]}}(t)={\ce {[A]}}_{0}}$, and applying the law of conservation of mass, we can say that at any time, the sum of the concentrations of A and B must be equal to the concentration of ${\displaystyle A_{0}}$, assuming the volume into which A and B are dissolved does not change:

$${\displaystyle {\ce {[A]}}+{\ce {[B]}}={\ce {[A]}}_{0}\Rightarrow {\ce {[B]}}={\ce {[A]}}_{0}-{\ce {[A]}}}$$

Substituting this value for [B] in terms of [A]₀ and [A](t) yields

$${\displaystyle {d{\ce {[A]}} \over dt}=-k{\ce {[A]}}+k'{\ce {[B]}}=-k{\ce {[A]}}+k'({\ce {[A]}}_{0}-{\ce {[A]}})=-(k+k'){\ce {[A]}}+k'{\ce {[A]}}_{0},}$$

which becomes the separable differential equation

$${\displaystyle {\frac {d{\ce {[A]}}}{-(k+k'){\ce {[A]}}+k'{\ce {[A]}}_{0}}}=dt}$$

This equation can be solved by substitution to yield

$${\displaystyle {\ce {[A]}}={k'-ke^{-(k+k')t} \over k+k'}{\ce {[A]}}_{0}}$$

In atmospheric sciences

Desaturation of clouds

Consider a supersaturated portion of a cloud. Then shut off the updrafts, entrainment, and any other vapor sources/sinks and things that would induce the growth of the particles (ice or water). Then wait for this

mathematical modelling

.

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).[5]

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 as long as several hours. Relaxation time is given as

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].

In astronomy

In

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^[6]

${\displaystyle {\begin{aligned}T_{r}&={0.34\sigma ^{3} \over G^{2}m\rho \ln \Lambda }\\&\approx 0.95\times 10^{10}\!\left({\sigma \over 200\,\mathrm {km\,s} ^{-1}}\right)^{\!3}\!\!\left({\rho \over 10^{6}\,M_{\odot }\,\mathrm {pc} ^{-3}}\right)^{\!-1}\!\!\left({m \over M_{\odot }}\right)^{\!-1}\!\!\left({\ln \Lambda \over 15}\right)^{\!-1}\!\mathrm {yr} \end{aligned}}}$

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

• Relaxation oscillator
• Time constant

References