Transport phenomena in non-uniform gas subjected to laser radiation

The paper discusses the theory of transport processes in one-component gas located in capillary subjected to resonant laser radiation and both temperature and pressure gradients. The equations for the kinetic coefficients determining heat- and mass transport in the gas are derived on the basis of modified Boltzmann equations for the excited and unexcited particles. The cross kinetic coefficients satisfy the Onsager reciprocity for all Knudsen numbers and laws of gas particles interaction with each other and with boundary surface of the capillary. Analysis of possible non-equilibrium stationary states of first and second order for the one-component gas in the capillary has been developed on the basis of the Prigogine theorem of stationary states. Equations describing the stationary states in Knudsen limit (Kn >> 1) and slip-flow regime (Kn << 1) were derived.


Introduction
Thermodynamic description of transport phenomena is based on the laws of conservation of mass, momentum and energy and the principle of increasing entropy. According to the thermodynamics of irreversible processes 1 the entropy production in the system is determined as a sum of products of generalized fluxes and generalized forces bound to those fluxes. These forces are generally associated with spatial inhomogeneity of thermodynamic parameters of the system or with a deviation of some of the internal variables from their equilibrium values [1].
Specific feature of light-induced transport phenomena is that they take place at homogeneous thermodynamic parameters of gas and under certain conditions and can generate temperature and pressure gradients [2]. These light-induced gradients stimulate their own fluxes of mass and heat in the gas. Theoretical description of these phenomena at the phenomenological level is impossible since the mechanisms of light-induced transport phenomena are associated only with the molecular-kinetic properties of the gas. Development of Onsager kinetic theory of these processes is the main objective of this study.
Laser radiation under resonant interaction with the gaseous medium behaves as the «two-faced Janus». The radiation can simultaneously violate the gas equilibrium by internal and translational degrees of freedom. This means that radiation stimulates two types of fluxes, «scalar» flux, characterizing change of population of the particles energy levels compared to their equilibrium values and vector fluxes of mass and heat of the gas.
Disequilibrium due to translational degrees of freedom is the result of asymmetry in the distribution functions of the excited and unexcited particles related to zero-component of the particles velocity vector in the direction of radiation propagation ( Figure 1). In accordance with the Doppler effect, radiation interacts only with those particles having Doppler shift kv (k is the wave vector, v is the velocity vector of the particle) compensates detuning  of radiation frequency  relatively to ω mn , the mn frequency transition from the ground state of particle n to excited state m. In linear approximation, when the magnitude of the particle velocity is much less than light speed, Doppler shift is determined by the expression Third, using theorems of the non-equilibrium thermodynamics we consider the possible stationary states of the gas.

Statement of the problem
We consider one-component gas in the capillary. Let capillary length L is much greater than its radius 0 r (L >> 0 r ). The resonant optical radiation is uniform over the cross section of the capillary and propagates along the axis of the capillary as a traveling monochromatic wave of  frequency. In addition, the state of the gas is disturbed by the longitudinal temperature and pressure gradients.
We will use the two-level approximation according to which the particles can be either in the ground state n, or in the excited state m. Particles absorbing radiation transit to an excited state. They change their transport properties, in particular, the collision cross-sections and accommodative characteristics. The process of radiative decay of the excited m level with frequency Г m occurs simultaneously with the induced transitions. Thus, the gas can be considered as a mixture consisting of particles of equal mass m 0 but with different interaction cross-sections and accommodative properties. The components continuously exchange the particles.
The state of gas is described by distribution functions of excited m f and unexcited n f particles that satisfy the Boltzmann kinetic equations including terms responsible for the interaction of particles with radiation [10]: Here Г is homogeneous half-width of the absorption line, E is the electric field amplitude of the light wave, The boundary conditions for the distribution functions of particles have the form [11]: where n is inward normal to the surface of the capillary, ()   (4) reciprocity relation expressing the principle of detailed balance on the gas-surface border where 0 i f is the equilibrium Maxwell-Boltzmann distribution function, i G is the degree of degeneracy.
Integrating (5) over velocity space   0  vn and summing over the states j, taking into account the normalization (4), we obtain We choose a cylindrical coordinate system (r, ϕ, z), so that the polar axis z is directed along the axis of the capillary.
In the weak-field approximation ( ) and at small pressure and temperature gradients, the distribution functions of excited and unexcited particles are insignificantly different from local equilibrium Maxwell-Boltzmann distributions: The local equilibrium Maxwell-Boltzmann distribution looks like Here   Tz,     , i n z n z are the local values of the gas temperature, the total particle number density, and the population of the i-th level, respectively; B k is the Boltzmann constant.
Due to linearization (7) and equation (6), the boundary conditions (3) on the wall of the capillary for the perturbation functions take the following form: To make a separate analysis for vector fluxes directed along the capillary and scalar fluxes associated only with the change of the level population we distinguish even and odd parts of the longitudinal component of the particles IOP Based on the assumptions made the kinetic equations (2) are linearized with respect to perturbation functions i h and for steady gas state are written as: where the following notations are used: Here One can see [12] that the linearized collision integral The linearized kinetic equations (11) are separated into odd ("minus") and even ("plus") relatively velocity z v . The first equations describe vector fluxes that are associated with transfer of mass, heat and entropy of the gas along the capillary, secondwith the scalar fluxes describing the pressure anisotropy and radiation-collisional heating or cooling gas [10].

The equation of entropy balance
Consider a volume V of gas contained in the capillary tube of length L. We define its entropy in conventional manner [1]: The rate of entropy change in the respective volume of gas is equal to: (14) Taking into account the kinetic equation (2) we obtain the entropy balance equation in the form: R P is the total entropy production in the considered volume due to radiation interaction with the gas: c P is the total entropy production due to the interaction of gas particles with each other: Consider the stationary state of the system when the entropy of the gas volume does not change in time. In this case, the entropy balance equation (15) can be written as: Using linearization of distribution functions (7) we obtain the entropy balance equation for stationary weakly nonequilibrium state of the system.
Taking into account (11) and (12) the equation (19) is divided into equations with vector («-») and scalar («+») fluxes: We consider only vector fluxes. To simplify the equations hereafter we omit the superscript «-» in all the physical quantities, believing that function () iz h v is odd relatively to velocity z v .
Using the Gauss theorem and decomposition   2 ln 1 2 When writing the second integral in the right-hand side of equation (21) the integration is performed only over the lateral surface w S of the capillary, since the integrals over face sections S in volume V in the stationary state are equal in magnitude and are opposite in sign. The entropy production due to the interaction of radiation with the gas can be written as the sum of the entropy production due to the radiative decay of the excited level r P and by induced transitions: Entropy production due to intermolecular collisions after linearization takes the form: To find the meaning of the second term of the right side of expression (21) we calculate the entropy flux from wall to gas due to collisions of the gas particles with the surface of the capillary:  The contribution of the second term in the right side of (25) in the entropy flux w P depends not only on the thermal conductivity of the material of the capillary but also from the ratio of the radius of the capillary 0 r to its length L. In the limit 0 Lr  its contribution becomes negligible under any coefficient of thermal conductivity of the wall. In this case in any cross section of the capillary in stationary conditions the state of local equilibrium is established when the normal component of vector of the heat flux density is zero and the temperature of the wall is equal to the temperature of the gas at any point of section. Therefore further, the second term in the right side of (25) we omit. Hence: Substituting (21) -(24) and (26) into (20) we obtain the equation of entropy balance in the gas for the stationary state: The entropy production inside the allocated volume of gas is caused by presence of the following sources of irreversible processes: intermolecular collisions ( c P ), radiative decay of the excited level ( r P ) and gas particles interaction with the capillary surface ( w P ).
From property of symmetry of the collision integral [13] follows that 0 c P  . According to equation (23), we have 0 r P  . Using reciprocity for the scattering probability density (5) and Jensen's inequality for the continuous, strictly convex down functions [13], we get 0 w P  . Thus, the entropy production in the gas due to vector fluxes is nonnegative 0 P  .
We transform the right-hand side of equation (27). We perform integration over the volume V like integration by capillary length L and cross-section area S. We take into account that in stationary conditions fluxes do not depend on the z coordinate, but the intensity of the radiation varies along the z axis due to absorption by gaseous medium. The dependence of z coordinate radiation intensity can be accounted for by introduction of the linear absorption coefficient  . Then the amplitude of the wave radiation electric field and, as a consequence, a square of the Rabi frequency proportional to the intensity of the radiation, in accordance with Bouguer law, can be written as Here V is molecule own velocity with regard to the gas average macroscopic velocity U,  is the laser wavelength, S is the cross-section area of the capillary; 11 , ( ) 1 T T T   were used in ref. [14] to describe non-isothermal gas motion in the channel. The force S X is associated with the intensity of optical radiation, that is convenient because it is directly recorded in experiments [7].

Kinetic coefficients
Perturbation functions in linear approximation are a linear combination of generalized forces

Onsager reciprocal relations
According to thermodynamics of nonequilibrium processes cross-coefficients in steady state must satisfy Onsager reciprocal relations: To prove this we use the method proposed in ref. [12] and the modified in this study. Integrate the kinetic equation (11) In (39) -(43) we take into account that the collision integrals for the odd (even) perturbation functions are the odd (even) functions of the velocity vector  v .
Noteworthy that the proof of kinetic coefficients reciprocity that describe the transport processes is associated with exposure to radiation. Let us prove reciprocity relation for light-induced drift and baro -entropic flux The first term on the right side of equation (44)  According to the theorem of the equivalence of the infinitesimal functions [15], the second term on the right-hand side of (46) is the infinitesimal magnitude of the higher (fourth) order of smallness and should be omitted. After applying Gauss transform to the first term in right part of eqn. (46) Substituting (48) and (50) Reciprocity for the scattering kernel (5) gives the equation (53) the form: