Relaxation time

# Relaxation time

Discussion

Encyclopedia
In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

.
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).

### Mechanics: Damped unforced oscillator

Let the homogenous differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

:
model damped unforced oscillation
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

s of a weight on a spring.

The displacement will then be of the form . 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
RC circuit
A resistor–capacitor circuit ', or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source...

containing a charged capacitor and a resistor, the voltage decays exponentially:
The constant 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
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

materials, the dielectric polarization
Polarization density
In classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is...

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
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...

it is a measure how long it takes to become neutralized by conduction process. This relaxation time is small in metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...

s and can be large in semiconductors and insulator
Electrical insulation
thumb|250px|[[Coaxial Cable]] with dielectric insulator supporting a central coreThis article refers to electrical insulation. For insulation of heat, see Thermal insulation...

s.

### Liquids and amorphous solids

An amorphous solid
Amorphous solid
In condensed matter physics, an amorphous or non-crystalline solid is a solid that lacks the long-range order characteristic of a crystal....

, 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
Supercooling
Supercooling, also known as undercooling, is the process of lowering the temperature of a liquid or a gas below its freezing point without it becoming a solid....

liquid or glass
Glass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...

to approach the molecular motion characteristic of a crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

. Differential scanning calorimetry
Differential scanning calorimetry
Differential scanning calorimetry or DSC is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature of a sample and reference is measured as a function of temperature. Both the sample and reference are maintained at nearly the same temperature...

can be used to quantify enthalpy
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

change due to molecular structural relaxation.

### Spin relaxation in NMR

In nuclear magnetic resonance
Nuclear magnetic resonance
Nuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...

, relaxation is of prime importance. See Relaxation (NMR)
Relaxation (NMR)
In nuclear magnetic resonance spectroscopy and magnetic resonance imaging the term relaxation describes several processes by which nuclear magnetization prepared in a non-equilibrium state return to the equilibrium distribution. In other words, relaxation describes how fast spins "forget" the...

.

### 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
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, relaxation time relates to clusters of gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

ally-interacting bodies (star cluster
Star cluster
Star clusters or star clouds are groups of stars. Two types of star clusters can be distinguished: globular clusters are tight groups of hundreds of thousands of very old stars which are gravitationally bound, while open clusters, more loosely clustered groups of stars, generally contain less than...

s, galaxy cluster
Galaxy cluster
A galaxy cluster is a compact cluster of galaxies. Basic difference between a galaxy group and a galaxy cluster is that there are many more galaxies in a cluster than in a group. Also, galaxies in a cluster are more compact and have higher velocity dispersion. One of the key features of cluster is...

s, globular cluster
Globular cluster
A globular cluster is a spherical collection of stars that orbits a galactic core as a satellite. Globular clusters are very tightly bound by gravity, which gives them their spherical shapes and relatively high stellar densities toward their centers. The name of this category of star cluster is...

s). 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
Core collapse
Core collapse can refer to:* The collapse of the stellar core of a massive star, such as the core collapse that produces a Type II supernova.* The dynamic process that leads to a concentration of stars at the core of a Globular cluster....

and energy equipartition.

The relaxation time is related to the velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

of a body (typically a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

) and the perturbation
Perturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

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
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

of nearby stars. The relaxation time is similar to the ratio of the velocity to the acceleration resulting from the perturbation.