Physical Kinetics

the theory of nonequilibrium macroscopic processes—that is, processes arising in systems that have been removed from a state of thermal (thermodynamic) equilibrium. Physical kinetics includes the thermodynamics of nonequilibrium processes, the kinetic theory of gases (including plasma), and the theory of transfer processes in solids, as well as the general statistical theory of nonequilibrium processes, which began to develop in the 1950’s.

All nonequilibrium processes in adiabatically isolated systems (systems that do not experience heat exchange with surrounding bodies) are irreversible processes, which occur with an increase of entropy. Entropy attains a maximun in the state of equilibrium.

As in the case of equilibrium states, two alternate methods— phenomenological, or thermodynamic (thermodynamics of nonequilibrium processes), and statistical—are used in physical kinetics for the description of systems.

Thermodynamic description of nonequilibrium processes. In a thermodynamic description of nonequilibrium processes, changes in space and time of such macroscopic parameters of the state of the system as the mass density of the i-th component p_i(r, t), the density of the momentum ρu(r, t), the local temperature T(r, t) the mass flux of the i-th component j_i (r, t), and the flux density of the internal energy q(r, t) are considered; here r is the coordinate, t is time, u is the mean mass velocity, and ρ is the mass density. At equilibrium the quantities ρ, ρ_i, and T are constant, and the fluxes are equal to zero.

A thermodynamic description of nonequilibrium processes is possible only for states that are close to equilibrium when the parameters of state change sufficiently slowly in space and time. In the case of gases this means that all the thermodynamic parameters characterizing the state of the system change little over the free path length and during a time equal to the mean free time of the molecules (the average time between two successive collisions of molecules). Slow processes are very often encountered in practice, since the establishment of equilibrium occurs only after a very large number of collisions. Such processes include diffusion, heat conduction, and electrical conduction. Deviations from the state of thermodynamic equilibrium are characterized by the gradients of temperature, concentration (ρ_i/ρ), and mass velocity (thermodynamic forces), whereas the fluxes of energy, mass of the i-th component, and momentum are linearly related to the thermodynamic forces. The coefficients in these relationships are called kinetic coefficients.

Let us take as an example diffusion in binary mixtures, which is the process of equalization of the concentration of components resulting from the random thermal motion of the molecules. The phenomenological equation that describes the process of diffusion is derived using the law of conservation of matter and the experimentally observed direct proportionality of the material flux of one of the components resulting from diffusion to its concentration gradient (with the opposite sign). The proportionality constant is called the coefficient of diffusion. According to the equation of diffusion, the rate of change in the concentration of matter is proportional to the divergence of the concentration gradient, with the proportionality constant equal to the coefficient of diffusion.

Solution of the diffusion equation makes possible determination of the time required for equalization of the concentration of the molecules in the system (for example, in a vessel containing a gas) resulting from diffusion (the relaxation time). The relaxation time T_r is of the order of T_r ~ L²/D, where L is the linear dimensions of the vessel and D is the diffusion coefficient. The relaxation time increases with increasing vessel size and with decreasing diffusion coefficient. The diffusion coefficient is proportional to the mean free path of the molecules λ and their mean thermal velocity ν. Therefore, relaxation time is proportional to T_r ~ L²/λν, = (L/λ)² · λ/ν, where λ/ν = T is the mean free time. It is evident that T_r » T when L » λ. Thus, the condition L » λ (the dimensions of the system are large compared to the mean free path of the molecules) is necessary for the process leading to the establishment of equilibrium to be considered a slow process.

The equations describing heat conduction, internal friction, electrical conduction, and other phenomena are derived similarly. In phenomenological theory the coefficient of diffusion, the thermal conductivity, and the viscosity, as well as the electrical conductivity, must be determined experimentally.

The processes mentioned above are called direct. This emphasizes the fact that during diffusion the concentration gradient of a given substance generates a flux of the substance, a temperature gradient generates a flux of internal energy that varies only with temperature with constant concentration of the molecules, and electric current is generated by a potential gradient.

In addition to the direct processes, there are also the cross processes. An example of a cross process is thermodiffusion, which is the transfer of matter caused by a temperature gradient rather than a concentration gradient (that would be ordinary diffusion). Thermodiffusion generates a concentration gradient, which leads to ordinary diffusion. If a constant temperature difference is maintained within the system, a steady state is established, in which the material fluxes caused by the temperature and concentration gradients become equalized. Under these conditions, mixtures of gases exhibit increased concentration of molecules with lower molecular weight in areas of higher temperature (this phenomenon is used in the separation of isotopes).

Concentration gradients, in turn, generate fluxes of internal energy. This is the basis of diffusional heat conduction. Temperature gradients cause orderly motion of the charged particles that may be present in a solid, giving rise to thermoelectric current.

The principle of symmetry of kinetic coefficients, established by L. Onsager, is of great importance in physical kinetics. At equilibrium, the thermodynamic parameters a_i, (pressure, temperature, and others) that characterize the state of a macroscopic system are constant over time: da_i/dt = 0. The most important state function of the system, the entropy S, which depends on a_i, reaches a maximum at equilibrium, and therefore its partial derivatives are ∂S/∂a_i = 0. For small deviations of the system from equilibrium, the derivatives ∂S/∂a_i and da_i/dt are small but nonzero and are related by approximate linear relations. The proportionality constants in these relationships are the kinetic coefficients. If the coefficient γik denotes the rate of change of the parameter of the system a_i as a function of ∂S/∂a_k, then the equality γik = γki is valid in accordance with the Onsager principle (in the absence of a magnetic field and rotation of the system as a whole). The Onsager principle follows from the property of microscopic reversibility, which is expressed in terms of the invariance of the equations of motion of the system’s particles with respect to reversal of the sign of time: t → — t. In particular, it follows from this principle that there is a relation between the coefficient defining the evolution of heat by a current because of uneven heating of the conductor (the Thomson effect) and the coefficient defining the evolution of heat by a current at the junctions of different conductors and semiconductors (the Peltier effect).

Statistical description of nonequilibrium processes. The statistical theory of nonequilibrium processes is more detailed and profound than the thermodynamic theory. In contrast to the thermodynamic method, statistical theory makes possible the calculation, on the basis of definite concepts concerning the structure of matter and intermolecular forces, of the kinetic coefficients that define the rates of diffusion, internal friction (viscosity), electrical conduction, and so on. However, this theory is very complex.

The statistical method for describing systems in the equilibrium, as well as in the nonequilibrium, state is based on the computation of distribution functions. Universal distribution functions for coordinates and momentums (or velocities) of all particles exist for equilibrium states; the functions determine the probability that the quantities will assume fixed values. For systems in thermal contact with the surrounding medium of constant temperature, this is the canonical Gibbs distribution, and for isolated systems, it is the microcanonical Gibbs distribution, both of which are completely defined by the energy of the system.

Nonequilibrium states depend to a much larger extent than equilibrium states on the microscopic properties of systems (the properties of atoms and molecules and the forces of interaction between them). The general methods for deriving distribution functions (in terms of the coordinates and momentums of all particles within the system), which are analogous to the canonical Gibbs distribution but describe nonequilibrium processes, were developed only during the 1950’s and 1960’s.

The distribution functions may be used to determine any microscopic quantities that characterize the state of the system and to follow their changes in space as a function of time. This is accomplished by the calculation of statistical means. The determination of the distribution function, which depends on the coordinates and momentums of all of the particles, is an insoluble problem, since it is equivalent to a solution of the equations of motion for all particles of the system. However, an exact determination of the distribution function is not required for practical purposes, because it contains excessively detailed information on the motion of the separate particles that is not essential for determining the behavior of the system as a whole. Hence an approximate statistical description in terms of simpler distribution functions is used. The singlet distribution function f(P, r, t), which gives the average number of particles with fixed values of the momentums P (or velocities v) and coordinates r, is adequate for the description of gases of medium density. Gases of higher density require the knowledge of pair distribution functions. A general method for the derivation of singlet and more complex distribution functions (depending on the coordinates and momentums of two or more particles) was developed by N. N. Bogoliubov, M. Born, M. Green, and others. These equations are called kinetic equations. They include the Boltzmann kinetic equation for rarefied gases, derived by L. Boltzmann on the basis of the balance of particles with velocities in the intervals Δv_x, Δv_y, Δv_z within the volume ΔxΔyΔz (v_x, v_y, and v_z are projections of the velocity ν onto the coordinate axes x, y, and z). The kinetic equations of L. D. Landau and A. A. Vlasov are varieties of the Boltzmann equation for ionized gases (plasma).

Kinetic equations may be derived not only for gases but also for small excitations in condensed systems. The thermal motion of the system is characterized by excitations of various types. In a gas these excitations are the translational motion of the constituent particles, as well as the internal excitation of atoms and molecules. The thermal motion is usually characterized by excitations of a more complex nature. Thus, thermal excitation in crystalline materials may be represented as elastic waves propagating along the crystal or, more precisely, as normal lattice vibrations. The collective excitations in plasma are vibrations of the electric charge density caused by long-range Coulomb forces. Electron excitations (electron transitions from states below in the Fermi surface to states above the surface) are possible in metals, whereas in semiconductors hole excitations (the appearance of electron-free states in the valence band on transition of electrons into the conduction band) are also possible. At low temperatures, in the weakly excited state, the excitation energy may always be represented as a sum of certain elementary excitations or, in quantum terms, quasiparticles. The concept of quasi-particles is applicable not only to crystals but also to liquids, gases, and amorphous materials, if the temperature is not too high. Distribution functions for the quasiparticles of a system in a nonequilibrium state satisfy the kinetic equation.

In quantum systems the distribution function depends on the spin of particles (or quasiparticles). In particular, for particles with half-integral spin the equilibrium distribution function is the Fermi-Dirac distribution function, whereas for particles (quasiparticles) with either integral or zero spin it is the Bose-Einstein distribution function. In addition to external interactions, kinetic equations also take into account interactions between particles or quasiparticles, in which case the interactions are regarded as pair collisions. It is these interactions that lead to the establishment of equilibrium states. In many cases, the distribution function does not depend explicitly on time. Such functions are called steady-state functions, and they describe processes that are time-independent. Changes in the distribution function caused by external forces during steady-state processes are compensated by corresponding changes caused by collisions.

In simple cases, the change of the distribution function f of the system as the result of collisions may be estimated roughly, assuming that the change is proportional to the magnitude of the deviation from the equilibrium function, since collisions change the distribution function only when there are deviations from the equilibrium state. The quantity that is the inverse of the proportionality constant in this relationship is called the relaxation time. It is impossible to take the interaction into account in a general case using such simple means, and the kinetic equation contains the “collision integral,” which more precisely takes into account the result of changes in the distribution function caused by the interaction of particles (quasiparticles).

The nonequilibrium distribution function is found and the energy, mass, and momentum fluxes are calculated by solving the kinetic equation, which makes possible derivation of equations for the heat conduction, diffusion, and momentum transfer (the Navier-Stokes equation), with kinetic coefficients expressed by molecular constants. (However, the kinetic equation may be formulated only for gases consisting of particles or quasiparticles.)

The fundamental principles of the theory of nonequilibrium processes have been reliably established. Methods for the derivation of equations for the transfer of energy, mass, and momentum in various systems have been developed not only for gases but also for liquids. In such cases, expressions are obtained for the kinetic coefficients that are part of these equations by means of correlation functions (functions describing the correlation in space and time) of the fluxes of these physical quantities—that is, eventually, through the molecular constants. These expressions are very complex and may be calculated by advanced numerical methods.