Relaxation (physics)

Relaxation (from lat. Relaxatio “attenuation, reduction”) is the process of establishing thermodynamic , and therefore statistical equilibrium in a physical system consisting of a large number of particles .

Relaxation is a multi-stage process, since not all physical parameters of the system (particle distribution by coordinates and momenta, temperature , pressure , concentration in small volumes and throughout the system and others) tend to equilibrium at the same speed. Usually, an equilibrium is first established for some parameter (partial equilibrium), which is also called relaxation . All relaxation processes are nonequilibrium processes in which energy dissipation occurs in the system, that is, entropy is produced (entropy does not decrease in a closed system). In various systems, relaxation has its own characteristics, depending on the nature of the interaction between the particles of the system; therefore, relaxation processes are very diverse. The time to establish equilibrium (partial or complete) in the system is called the relaxation time.

The process of establishing equilibrium in gases is determined by the mean free path of particles $l$ ${\ displaystyle l}$ $l$ and free time $t$ ${\ displaystyle t}$ $t$ (average distance and average time between two consecutive collisions of molecules). Attitude $l/t$ ${\ displaystyle l / t}$ ${\ displaystyle l / t}$ has the order of magnitude of the particle velocity. Quantities $l$ ${\ displaystyle l}$ $l$ and $t$ ${\ displaystyle t}$ $t$ very small compared to macroscopic scales of length and time. On the other hand, for gases, the mean free path is much longer than the collision time $t_{0}$ ${\ displaystyle t_ {0}}$ $t_0$ $(t\gg t_{0})$ ${\ displaystyle (t \ gg t_ {0})}$ ${\ displaystyle (t \ gg t_ {0})}$ . Only under this condition is relaxation determined only by pairwise collisions of molecules .

Content

Description of the relaxation process

For monatomic gases

In monatomic gases (without internal degrees of freedom, that is, with only translational degrees of freedom), relaxation occurs in two stages.

At the first stage, in a short period of time, on the order of the time of collision of molecules, the initial, even highly nonequilibrium, state is randomized in such a way that details of the initial state become irrelevant and the so-called “abbreviated description” of the nonequilibrium state of the system becomes possible when knowledge of the distribution probability is not required all particles of the system in coordinates and momenta, and it is enough to know the distribution of one particle in coordinates and momenta depending on time, that is, one hour ary molecular distribution function. (All other distribution functions of a higher order, describing the distribution over the states of two, three, etc. particles, depend on time only through a single-particle function).

The single-particle function satisfies the Boltzmann kinetic equation , which describes the relaxation process. This stage is called kinetic and is a very fast relaxation process.

At the second stage, during a time of the order of the mean free path of molecules and as a result of only a few collisions in macroscopically small volumes of the system, local equilibrium is established; it corresponds to a locally equilibrium, or quasi-equilibrium, distribution, which is characterized by the same parameters as in the case of complete equilibrium of the system, but depending on spatial coordinates and time. These small volumes still contain a lot of molecules, and since they interact with the environment only on their surface, they can be considered approximately isolated. The parameters of the local-equilibrium distribution during relaxation slowly tend to equilibrium, and the state of the system usually differs little from the equilibrium. Relaxation time for local equilibrium $t_{p}\gg t_{0}$ ${\ displaystyle t_ {p} \ gg t_ {0}}$ . After the establishment of local equilibrium, hydrodynamic equations ( Navier - Stokes equations , equations of heat conduction , diffusion , etc.) are used to describe the relaxation of the nonequilibrium state of the system. It is assumed that the thermodynamic parameters of the system ( density , temperature, etc.) and mass velocity (average mass transfer rate) change little over time $t$ ${\ displaystyle t}$ and in the distance $l$ ${\ displaystyle l}$ . This stage of relaxation is called hydrodynamic. Further relaxation of the system to the state of complete statistical equilibrium, in which the average particle velocities, average temperature, average concentration , etc. are equalized, occurs slowly as a result of a very large number of collisions.

Such processes ( viscosity , thermal conductivity , diffusion , electrical conductivity , etc.) are called slow. Appropriate relaxation time $t_{p}$ ${\ displaystyle t_ {p}}$ depends on size $L$ ${\ displaystyle L}$ system and great compared to $t$ ${\ displaystyle t}$ : $t_{0}\approx t(L/l)2\gg t$ ${\ displaystyle t_ {0} \ approx t (L / l) 2 \ gg t}$ that takes place at $l\ll L$ ${\ displaystyle l \ ll L}$ , i.e. for not very rarefied gases.

For polyatomic gases

In polyatomic gases (with internal degrees of freedom), the energy exchange between translational and internal degrees of freedom can be slowed down, and a relaxation process associated with this phenomenon occurs. The fastest way - in a time of the order of the time between collisions - equilibrium is established in the translational degrees of freedom; such an equilibrium state can be characterized by the corresponding temperature. The equilibrium between translational and rotational degrees of freedom is established much more slowly. Excitation of vibrational degrees of freedom can occur only at high temperatures. Therefore, in polyatomic gases, multistage processes of energy relaxation of vibrational and rotational degrees of freedom are possible.

For gas mixtures

In mixtures of gases with very different masses of molecules, the energy exchange between the components is slowed down, as a result of which a state with different component temperatures and the processes of their temperature relaxation can occur. For example, in plasma the masses of ions and electrons are very different. The equilibrium of the electronic component is most quickly established, then the ionic component comes into equilibrium, and a much longer time is required to establish equilibrium between the electrons and ions; therefore, states in a plasma can exist for a long time in which the ionic and electronic temperatures are different and, therefore, the processes of component temperature relaxation occur.

For liquids

In liquids, the concept of time and the mean free path of particles (and, consequently, the kinetic equation for a single-particle distribution function) loses its meaning. A similar role for the liquid is played by the quantities $t_{1}$ ${\ displaystyle t_ {1}}$ and $l_{1}$ ${\ displaystyle l_ {1}}$ - the time and length of the correlation of dynamic variables describing the flow of energy or momentum; $t_{1}$ ${\ displaystyle t_ {1}}$ and $l_{1}$ ${\ displaystyle l_ {1}}$ characterize the attenuation in time and space of the mutual influence of molecules, that is, correlation. In this case, the concept of the hydrodynamic relaxation stage and the local-equilibrium state is fully valid. In macroscopically small volumes of liquid, but still large enough in comparison with the correlation length $l_{1}$ ${\ displaystyle l_ {1}}$ , the locally equilibrium distribution is established over a time of the order of the correlation time $t_{1}$ ${\ displaystyle t_ {1}}$ $(t_{p}\gg t_{1})$ ${\ displaystyle (t_ {p} \ gg t_ {1})}$ as a result of intense interaction between molecules (rather than pair collisions, as in a gas), but these volumes can still be considered approximately isolated. At the hydrodynamic stage, relaxation in a liquid, thermodynamic parameters and mass velocity satisfy the same hydrodynamic equations as for gases (provided that the changes in the thermodynamic parameters and mass velocity over time $t_{1}$ ${\ displaystyle t_ {1}}$ and in the distance $l_{1}$ {\ displaystyle l_ {1}} ) Relaxation time to complete thermodynamic equilibrium $t_{p}\gg t_{1}(L/l_{1})^{2}$ ${\ displaystyle t_ {p} \ gg t_ {1} (L / l_ {1}) ^ {2}}$ (just like in gas and solid) can be estimated using kinetic coefficients . For example, the relaxation time of concentration in a binary mixture in volume $L^{3}$ ${\ displaystyle L ^ {3}}$ of order $t_{p}\gg L^{2}/D$ ${\ displaystyle t_ {p} \ gg L ^ {2} / D}$ where $D$ ${\ displaystyle D}$ - diffusion coefficient, temperature relaxation time $t_{p}\gg L^{2}/c$ ${\ displaystyle t_ {p} \ gg L ^ {2} / c}$ where $c$ ${\ displaystyle c}$ Is the coefficient of thermal diffusivity , etc. For a liquid with internal degrees of freedom of molecules, it is possible to combine a hydrodynamic description of translational degrees of freedom with additional equations to describe the relaxation of internal degrees of freedom ( relaxation hydrodynamics ).

For solids and quantum liquids

In solids , as in quantum liquids , relaxation can be described as relaxation in a gas of quasiparticles. In this case, one can introduce the time and mean free path of the corresponding quasiparticles (provided that the excitation of the system is small).

For example, in a crystal lattice at low temperatures, elastic vibrations can be interpreted as phonon gas. The interaction between phonons leads to quantum transitions, that is, to collisions between them. Energy relaxation in the crystal lattice is described by the kinetic equation for phonons. In the system of spin magnetic moments of a ferromagnet, magnons are quasiparticles. Relaxation (for example, magnetization) can be described by the kinetic equation for magnons. The relaxation of the magnetic moment in a ferromagnet takes place in two stages: at the first stage, due to the relatively strong exchange interaction, the equilibrium value of the absolute value of the magnetic moment is established.

At the second stage, due to the weak spin-orbit interaction, the magnetic moment is slowly oriented along the axis of easy magnetization; this stage is similar to the hydrodynamic stage of relaxation in gases.

Note

Literature

Uhlenbeck D., Ford J. Lectures on statistical mechanics. - Per. from English - M.: Mir, 1965.
Bondarevsky S.I., Ablesimov N.E. Relaxation effects in nonequilibrium condensed systems. Self-exposure as a result of radioactive decay. - Vladivostok: Dalnauka, 2002 .-- 232 p.
Ablesimov N.E., Zemtsov A.N. Relaxation effects in nonequilibrium condensed systems. Basalts: from eruption to fiber. - M.: ITiG FEB RAS, 2010 .-- 400 p.
Osipov A. I. Thermodynamics yesterday, today, tomorrow. Part 1. Equilibrium thermodynamics (inaccessible link) // Coolant. - 1999. - No. 4. - p. 79-85.