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).

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.

Exponential decay probability density

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

Let the homogeneous differential equation:

model damped unforced oscillations of a weight on a spring.

Oscillation repetitive variation of some measure about a central value

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.

Electronics: The RC circuit

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.

Electronic oscillator electronic circuit that produces a repetitive, oscillating electronic signal, often a sine wave or a square wave

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.

Relaxation oscillator

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.

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).

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.

Stress relaxation

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.

Stress relaxation

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.

Stress (mechanics) physical quantity that expresses internal forces in a continuous material

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).

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. 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.

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

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

Chemical relaxation methods

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]

Relaxation 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 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 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). [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 = (4πDNRK )−1 seconds, where:

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 [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.

See also

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Dielectric electrically poorly conducting or non-conducting, non-metallic substance of which charge carriers are generally not free to move

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Exponential distribution probability distribution

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Multivariate normal distribution

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Equipartition theorem theorem

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Toroidal coordinates

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Stretched exponential function

The stretched exponential function

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Köhler theory

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.

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  1. Kittel, Rep. Prog. Phys. 1947; Hall, Phys. Rev. 1948; Wintner Phys. Rev. 1948.
  2. Atkins P. and de Paula J. Atkins' Physical Chemistry (8th ed., W.H.Freeman 2006) p.805-7, ISBN   0-7167-8759-8
  3. 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.
  4. Spitzer, Lyman (1987). Dynamical evolution of globular clusters. Princeton, NJ: Princeton University Press. p. 191. ISBN   0691083096.