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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).

**Thermodynamic equilibrium** is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermodynamic equilibrium there are no net macroscopic flows of matter or of energy, either within a system or between systems. In a system in its own state of internal thermodynamic equilibrium, no macroscopic change occurs. Systems in mutual thermodynamic equilibrium are simultaneously in mutual thermal, mechanical, chemical, and radiative equilibria. Systems can be in one kind of mutual equilibrium, though not in others. In thermodynamic equilibrium, all kinds of equilibrium hold at once and indefinitely, until disturbed by a thermodynamic operation. In a macroscopic equilibrium, almost or perfectly exactly balanced microscopic exchanges occur; this is the physical explanation of the notion of macroscopic equilibrium.

**Tau** is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300.

A quantity is subject to **exponential decay** if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where *N* is the quantity and λ (lambda) is a positive rate called the **exponential decay constant**:

- Relaxation in simple linear systems
- Mechanics: Damped unforced oscillator
- Electronics: The RC circuit
- Relaxation in condensed matter physics
- Stress relaxation
- Dielectric relaxation time
- Liquids and amorphous solids
- Spin relaxation in NMR
- Chemical relaxation methods
- Relaxation in atmospheric sciences
- Desaturation of clouds
- Relaxation in astronomy
- See also
- References

Let the homogeneous differential equation:

model damped unforced oscillations of a weight on a spring.

**Oscillation** is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The term *vibration* is precisely used to describe mechanical oscillation. Familiar examples of oscillation include a swinging pendulum and alternating current.

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.

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

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. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

The constant 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 *.

The **RC time constant**, also called tau, the time constant of an RC circuit, is equal to the product of the circuit resistance and the circuit capacitance, i.e.

An **electronic oscillator** is an electronic circuit that produces a periodic, oscillating electronic signal, often a sine wave or a square wave. Oscillators convert direct current (DC) from a power supply to an alternating current (AC) signal. They are widely used in many electronic devices. Common examples of signals generated by oscillators include signals broadcast by radio and television transmitters, clock signals that regulate computers and quartz clocks, and the sounds produced by electronic beepers and video games.

In electronics a **relaxation oscillator** is a nonlinear electronic oscillator circuit that produces a nonsinusoidal repetitive output signal, such as a triangle wave or square wave. The circuit consists of a feedback loop containing a switching device such as a transistor, comparator, relay, op amp, or a negative resistance device like a tunnel diode, that repetitively charges a capacitor or inductor through a resistance until it reaches a threshold level, then discharges it again. The period of the oscillator depends on the time constant of the capacitor or inductor circuit. The active device switches abruptly between charging and discharging modes, and thus produces a discontinuously changing repetitive waveform. This contrasts with the other type of electronic oscillator, the harmonic or linear oscillator, which uses an amplifier with feedback to excite resonant oscillations in a resonator, producing a sine wave. Relaxation oscillators are used to produce low frequency signals for applications such as blinking lights and electronic beepers and in voltage controlled oscillators (VCOs), inverters and switching power supplies, dual-slope analog to digital converters, and function generators.

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).

A **linear response function** describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as susceptibility, impulse response or impedance, see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related.

The **fluctuation–dissipation theorem** (**FDT**) or **fluctuation–dissipation relation** (**FDR**) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a general proof that thermal fluctuations in a physical variable predict the response quantified by the admittance or impedance of the same physical variable, and vice versa. The fluctuation–dissipation theorem applies both to classical and quantum mechanical systems.

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

**Continuum mechanics** is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

In materials science, **stress relaxation** is the observed decrease in stress in response to strain generated in the structure. This is primarily due to keeping the structure in a strained condition for some finite interval of time and hence causing some amount of plastic strain. This should not be confused with creep, which is a constant state of stress with an increasing amount of strain.

In continuum mechanics, **stress** is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure, each particle gets pushed against by all the surrounding particles. The container walls and the pressure-inducing surface push against them in (Newtonian) reaction. These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules. Stress is frequently represented by a lowercase Greek letter sigma (σ).

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. It is important in dielectric spectroscopy. Very long relaxation times are responsible for dielectric absorption.

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.

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] }

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

In chemical kinetics, relaxation methods are used for the measurement of very fast reaction rates. A system initially at equilibrium is perturbed by a rapid change in a parameter such as the temperature (most commonly), the pressure, the electric field or the pH of the solvent. The return to equilibrium is then observed, usually by spectroscopic means, and the relaxation time measured. In combination with the chemical equilibrium constant of the system, this enables the determination of the rate constants for the forward and reverse reactions.^{ [2] }

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 supersaturation to reduce and become just saturation (relative humidity = 100%), which is the equilibrium state. The time it takes for the supersaturation to dissipate 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 for accurate mathematical modelling.

In water clouds where the concentrations are larger (hundreds per cm^{3}) 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).^{ [3] }

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

*T*= (4π*DNRK*)^{−1}seconds, where:

*D*= diffusion coefficient [m^{2}/s]*N*= concentration (of ice crystals or water droplets) [m^{−3}]*R*= mean radius of particles [m]*K*= capacitance [unitless].

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 ^{ [4] }

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.

- Characteristic time, a.k.a. relaxation time
- Relaxation oscillator
- Time constant

The **Beer–Lambert law**, also known as **Beer's law**, the **Lambert–Beer law**, or the **Beer–Lambert–Bouguer law** relates the attenuation of light to the properties of the material through which the light is travelling. The law is commonly applied to chemical analysis measurements and used in understanding attenuation in physical optics, for photons, neutrons or rarefied gases. In mathematical physics, this law arises as a solution of the BGK equation.

In a chemical reaction, **chemical equilibrium** is the state in which both reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the system. Usually, this state results when the forward reaction proceeds at the same rate as the reverse reaction. The reaction rates of the forward and backward reactions are generally not zero, but equal. Thus, there are no net changes in the concentrations of the reactant(s) and product(s). Such a state is known as dynamic equilibrium.

In classical mechanics, a **harmonic oscillator** is a system that, when displaced from its equilibrium position, experiences a restoring force *F* proportional to the displacement *x*:

In physics, **optical depth** or **optical thickness**, is the *natural logarithm* of the ratio of incident to *transmitted* radiant power through a material, and **spectral optical depth** or **spectral optical thickness** is the natural logarithm of the ratio of incident to *transmitted* spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.

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 they do in an electrical conductor but only slightly shift from their average equilibrium positions causing **dielectric polarization**. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

In probability theory and statistics, the **exponential distribution** is the probability distribution that describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

In probability theory and statistics, the **multivariate normal distribution**, **multivariate Gaussian distribution**, or **joint normal distribution** is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be *k*-variate normally distributed if every linear combination of its *k* components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

In classical statistical mechanics, the **equipartition theorem** relates the temperature of a system to its average energies. The equipartition theorem is also known as the **law of equipartition**, **equipartition of energy**, or simply **equipartition**. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in rotational motion.

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

The **equilibrium constant** of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

An **activity coefficient** is a factor used in thermodynamics to account for deviations from ideal behaviour in a mixture of chemical substances. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same and, as a result, properties of the mixtures can be expressed directly in terms of simple concentrations or partial pressures of the substances present e.g. Raoult's law. Deviations from ideality are accommodated by modifying the concentration by an *activity coefficient*. Analogously, expressions involving gases can be adjusted for non-ideality by scaling partial pressures by a fugacity coefficient.

In chemistry, a **reaction quotient** is a function of the activities or concentrations of the chemical species involved in a chemical reaction. In the special case that the reaction is at equilibrium the reaction quotient is constant and equal to the equilibrium constant that appears in the expression of the law of mass action.

The **temperature jump method** is a technique used in chemical kinetics for the measurement of very rapid reaction rates. It is one of a class of chemical relaxation methods pioneered by the German physical chemist Manfred Eigen in the 1950s. In these methods, a reacting system initially at equilibrium is perturbed rapidly and then observed as it *relaxes* back to equilibrium. In the case of temperature jump, the perturbation involves rapid heating which changes the value of the equilibrium constant, followed by relaxation to equilibrium at the new temperature.

**Toroidal coordinates** are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.

The **stretched exponential function**

In mathematics — specifically, in stochastic analysis — an **Itô diffusion** is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion of a particle subjected to a potential in a viscous fluid. Itô diffusions are named after the Japanese mathematician Kiyosi Itô.

**Viscoplasticity** is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.

**Köhler theory** describes the process in which water vapor condenses and forms liquid cloud drops, and is based on equilibrium thermodynamics. It combines the Kelvin effect, which describes the change in saturation vapor pressure due to a curved surface, and Raoult's Law, which relates the saturation vapor pressure to the solute. It is an important process in the field of cloud physics. It was initially published in 1936 by Hilding Köhler, Professor of Meteorology in the Uppsala University.

**Nuclear magnetic resonance (NMR) in porous materials** covers the application of using NMR as a tool to study the structure of porous media and various processes occurring in them. This technique allows the determination of characteristics such as the porosity and pore size distribution, the permeability, the water saturation, the wettability, etc.

- ↑ Kittel, Rep. Prog. Phys. 1947; Hall, Phys. Rev. 1948; Wintner Phys. Rev. 1948.
- ↑ Atkins P. and de Paula J.
*Atkins' Physical Chemistry*(8th ed., W.H.Freeman 2006) p.805-7, ISBN 0-7167-8759-8 - ↑ Rogers, R.R.; Yau, M.K. (1989).
*A Short Course in Cloud Physics*. International Series in Natural Philosophy.**113**(3rd ed.). Elsevier Science. ISBN 0750632151. - ↑ Spitzer, Lyman (1987).
*Dynamical evolution of globular clusters*. Princeton, NJ: Princeton University Press. p. 191. ISBN 0691083096.

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