# 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.

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

Let the homogeneous differential equation:

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

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 ${\displaystyle y(t)=Ae^{-t/T}\cos(\mu t-\delta )}$. 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:

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.

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

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 .

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 ranging from simplest clock generators to digital instruments and complex computers and peripherals etc. 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.

## 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 thermodynamic 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.

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 which is not a physical quantity. 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]

### Monomolecular First-Order Reversible Reaction

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

${\displaystyle {\ce {A ->[k] B ->[k'] A}}}$

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

In other words, reactant A and product B are forming into one another based off of 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, ${\displaystyle {d[A] \over dt}=-k[A]+k'[B]}$, where brackets around A and B indicate concentrations.

If we say that at ${\displaystyle t=0,[A(t)]=[A]_{0}}$, 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:

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

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

${\displaystyle {d[A] \over dt}=-k[A]+k'[B]=-k[A]+k'([A]_{0}-[A])=-(k+k')[A]+k'[A]_{0}}$, which becomes the separable differential equation ${\displaystyle {1 \over -(k+k')[A]+k'[A]_{0}}d[A]=dt}$

This equation can be solved by substitution to yield ${\displaystyle [A]={k'-ke^{-(k+k')t} \over k+k'}[A]_{0}}$

## 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:
• 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 [4]

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

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.

## Related Research Articles

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.

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 electrochemistry, the Nernst equation is an equation that relates the reduction potential of an electrochemical reaction to the standard electrode potential, temperature, and activities of the chemical species undergoing reduction and oxidation. It was named after Walther Nernst, a German physical chemist who formulated the equation.

In physics, Langevin equation is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation.

Exponential growth is a specific way that a quantity may increase over time. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent.

The reaction rate or rate of reaction is the speed at which reactants are converted into products. For example, the oxidative rusting of iron under Earth's atmosphere is a slow reaction that can take many years, but the combustion of cellulose in a fire is a reaction that takes place in fractions of a second. For most reactions, the rate decreases as the reaction proceeds.

A single chemical reaction is said to be autocatalytic if one of the reaction products is also a catalyst for the same or a coupled reaction. Such a reaction is called an autocatalytic reaction.

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.

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.

The reaction quotient (Q) measures the relative amounts of products and reactants present during a reaction at a particular point in time. 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 rate law or rate equation for a chemical reaction is an equation that links the reaction rate with the concentrations or pressures of the reactants and constant parameters. For many reactions the rate is given by a power law such as

In chemistry, a steady state is a situation in which all state variables are constant in spite of ongoing processes that strive to change them. For an entire system to be at steady state, i.e. for all state variables of a system to be constant, there must be a flow through the system. A simple example of such a system is the case of a bathtub with the tap running but with the drain unplugged: after a certain time, the water flows in and out at the same rate, so the water level stabilizes and the system is in a steady state.

In science, e-folding is the time interval in which an exponentially growing quantity increases by a factor of e; it is the base-e analog of doubling time. This term is often used in many areas of science, such as in atmospheric chemistry, medicine and theoretical physics, especially when cosmic inflation is investigated. Physicists and chemists often talk about the e-folding time scale that is determined by the proper time in which the length of a patch of space or spacetime increases by the factor e mentioned above.

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.

The Cole–Cole equation is a relaxation model that is often used to describe dielectric relaxation in polymers.

The stretched exponential function

Equilibrium chemistry is concerned with systems in chemical equilibrium. The unifying principle is that the free energy of a system at equilibrium is the minimum possible, so that the slope of the free energy with respect to the reaction coordinate is zero. This principle, applied to mixtures at equilibrium provides a definition of an equilibrium constant. Applications include acid–base, host–guest, metal–complex, solubility, partition, chromatography and redox equilibria.

Pressure jump is a technique used in the study of chemical kinetics. It involves making rapid changes to the pressure of an experimental system and observing the return to equilibrium or steady state. This allows the study of the shift in equilibrium of reactions that equilibrate in periods between milliseconds to hours, these changes often being observed using absorption spectroscopy, or fluorescence spectroscopy though other spectroscopic techniques such as CD, FTIR or NMR can also be used.

## References

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.