# Relaxation

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## relaxation

**1.**

*Physics*the return of a system to equilibrium after a displacement from this state

**2.**

*Maths*a method by which errors resulting from an approximation are reduced by using new approximations

## Relaxation

the process of the achievement of thermodynamic and, consequently, statistical equilibrium in a physical system consisting of a large number of particles. Relaxation is a multistage process, since the physical parameters of the system—such as the distribution of the particles with respect to coordinates and momenta and the temperature, pressure, and concentration in small volumes and throughout the entire system—do not all approach equilibrium at the same rate. Usually, a state of partial equilibrium is first reached, that is, equilibrium with respect to some parameter. The term “relaxation” is also applied to this case. All relaxation processes are nonequi-librium processes in which energy is dissipated in the system—that is, entropy is produced (in a closed system the entropy increases). The characteristics of relaxation are different in different systems and depend on the nature of the interaction between the system’s particles; relaxation processes are therefore extremely varied. The characteristic time required for a system to approach partial or complete equilibrium is called the relaxation time.

The process of the achievement of equilibrium in a gas is determined by the mean free path *l* of the particles and by the mean free time τ, which are the average distance and average time, respectively, between two successive collisions of molecules. The ratio *l*/τ is of the order of the speed of the particles. The quantities *l* and τ are very small compared with macroscopic scales of length and time. On the other hand, for gases the mean free time is much greater than the collision time τ_{0}: τ ≫ τ_{0}. Relaxation is determined solely by binary collisions of molecules only when this condition is satisfied.

* Monatomic gases*. In monatomic gases, which lack internal degrees of freedom and have only translational degrees of freedom, relaxation occurs in two stages. In the first stage, the initial state, which may be even a markedly nonequilibrium state, is randomized in a short period of time of the order of the collision time τ

_{0}of the molecules. The randomization occurs in such a way that the details of the initial state become insignificant, and an abbreviated description of the nonequilibrium state of the system is possible. This means that knowledge of the probability of the distribution of all particles in the system with respect to coordinates and momenta is not required. It is sufficient to know the single-particle distribution function, that is the distribution of one particle with respect to coordinates and momenta as a function of time. (Higher-order distribution functions, which describe distributions with respect to the states of two, three, or more particles, depend on time only through the single-particle function.) The one-particle function satisfies the Boltzmann equation, which describes the process of relaxation. This stage is called the kinetic stage and is a very rapid relaxation process.

In the second stage, local equilibrium is achieved in macro-scopically small volumes of the system in a time of the order of the mean free time τ of the molecules as a result of only a few collisions. To this local equilibrium there corresponds a local-equilibrium, or quasi-equilibrium, distribution, which is characterized by the same parameters as in the case of complete equilibrium of the system but is dependent on the spatial coordinates and time. These small volumes still contain numerous molecules, and since they interact with the surroundings only at the surface, they may be considered approximately isolated. The parameters of a local-equilibrium distribution in a relaxation process slowly tend toward the equilibrium parameters, and the state of the system usually differs little from the equilibrium state. The relaxation time for local equilibrium is t_{e} ≈ τ. After local equilibrium is reached, the equations of hydrodynamics, such as the Navier-Stokes equations and the heat-flow and diffusion equations, are used to describe the relaxation of the nonequilibrium state of the system. Here it is assumed that the thermodynamic parameters of the system, such as density and temperature, and the mass velocity (the average rate of transfer of mass) vary little during a time τ and over a distance *l*. This stage of relaxation is called the hydrodynamic stage.

The further relaxation of the system to a state of complete statistical equilibrium, in which the average speeds of the particles, the average, temperature, the average concentration, and other parameters are equalized, proceeds slowly because of the large number of collisions. Such slow processes include viscosity, heat conduction, diffusion, and electrical conduction. The corresponding relaxation time t_{e} depends on the dimensions *L* of the system and is great compared with τ: *t _{e}* ~

*τ*(

*L*/

*l*)

^{2}≫ τ. This relation holds when

*l*≪

*L*—that is, for gases that are not extremely rarefied.

* Polyatomic gases*. In polyatomic gases, which have internal degrees of freedom, the exchange of energy between transla-tional and internal degrees of freedom may be slow, and a relaxation process associated with this phenomenon arises. Equilibrium with respect to translational degrees of freedom is achieved rather quickly—in a time of the order of the time between collisions; such an equilibrium state can be characterized by the corresponding temperature. Equilibrium between translational and rotational degrees of freedom is achieved much more slowly. The excitation of vibrational degrees of freedom can occur only at high temperatures. Multistage processes of relaxation of vibrational and rotational degrees of freedom are therefore possible in polyatomic gases.

* Gaseous mixtures*. In mixtures of gases whose molecules are of markedly differing mass, the energy exchange between the components is slow. Consequently, a state wherein the components have different temperatures may occur, and temperature relaxation processes are possible. For example, in a plasma the masses of the ions and electrons are very different. The electronic component reaches equilibrium before the ionic component. A much longer time is required to achieve equilibrium between electrons and ions. For this reason, in a plasma there can exist for a long time states in which the ion and electron temperatures are different and in which, consequently, the processes of relaxation of the temperatures of the components occur.

* Liquids*. In liquids, the concept of the mean free time and path of particles and, consequently, the kinetics equations for the single-particle distribution function are not applicable. For liquids, a role analogous to that of the mean free time and mean free path is played by the quantities τ

_{1}and l

_{1}, the correlation time and correlation length, respectively, of the dynamic variables describing the fluxes of energy or momentum. The quantities τ

_{1}and

*l*

_{1}, characterize the attenuation in time and space of the mutual influence of the molecules—that is, they characterize the attenuation of a correlation. Here the concepts of the hydrodynamic stage of relaxation and of the local-equilibrium state remain fully valid. In macroscopically small volumes of a liquid that are still quite large compared with the correlation length

*l*a local-equilibrium distribution is achieved in a time of the order of the correlation time τ

_{l}_{1}(

*t*≈ τ

_{e}_{1}) as a result of intensive interaction between the molecules. Although the local-equilibrium distribution is not a result of binary collisions, as in a gas, these volumes may still be assumed to be approximately isolated. In the hydrodynamic stage of relaxation in a liquid, the thermodynamic parameters and mass velocity satisfy the same equations of hydrodynamics as for gases, if the change in the thermodynamic parameters and mass velocity during a time τ

_{1}and over a distance

*l*

_{1}, is small. As in a gas or solid, the relaxation time to complete thermodynamic equilibrium is

*t*≈ τ

_{e}_{1}(

*L*/

*l*

_{1})

^{2}and can be estimated by using the kinetic coefficients (

*see*KINETICS, PHYSICAL). For example, the concentration relaxation time in a binary mixture in the volume L

^{3}is of the order of

*t*≈

_{e}*L*

^{2}/

*D*, where

*D*is the diffusion coefficient. Another example is the temperature relaxation time, which is

*t*≈

_{e}*L*

^{2}/

*x*, where

*x*is the thermal diffusivity. For a liquid where the molecules have internal degrees of freedom, the hydrodynamic description of translational degrees of freedom can be combined with additional equations-to describe the relaxation of the internal degrees of freedom.

* Solids*. In solids, as in quantum fluids, relaxation can be described as relaxation in a gas of quasiparticles. In this case, we can speak of the mean free time and path of the corresponding quasiparticles, provided that the excitation of the system is small. For example, elastic vibrations in a crystal lattice at low temperatures can be interpreted as a phonon gas. The interaction between the phonons results in quantum transitions, that is, in collisions between the phonons. In the system of spin magnetic moments of a ferromagnetic substance, magnons are the quasiparticles; relaxation of, for example, magnetization can be described by the kinetic equation for magnons. The relaxation of the magnetic moment in a ferromagnetic material occurs in two stages. In the first stage, the absolute value of the magnetic moment reaches its equilibrium value through a relatively strong exchange interaction. In the second stage, the magnetic moment is slowly oriented along the direction of easy magnetization as a result of a weak spin-orbit interaction. This stage is analogous to the hydrodynamic stage of relaxation in gases.

### REFERENCE

Uhlenbeck, G., and J. Ford.*Lektsiipo statisticheskoi mekhanike*. Moscow, 1965. (Translated from English.)

D. N. ZUBAREV

## Relaxation

in physiology and medicine, the decrease in the tonus of the skeletal musculature, caused, in particular, by various chemical substances and manifested in a decrease of motor activity or complete immobilization, or paralysis.

The extensiveness, degree, and duration of relaxation depend on the site of disruption of conduction of the nerve impulse and on the chemical substance used. Narcotic substances act on the central sections of the nervous system and produce a widespread but incomplete relaxation. Substances used for local anesthesia act on peripheral nerves, causing local incomplete relaxation. The most extensive and complete relaxation is observed upon injection of special preparations called muscle relaxants.