Master Equation For System Research Articles

Electron Energy Distribution Functions of Hydrogen: The Effect of Superelastic Vibrational Collisions and of the Dissociation Process

Abstract Electron energy distribution functions (EDF) of molecular H2 have been calculated by numerically solving the Boltzmann equation including all the inelastic processes with the addition of superelastic vibrational collisions and of the hydrogen atoms coming from the dissociation process. The population densities of the vibrational levels have been obtained both by assuming a Boltz-mann population at a vibrational temperature different from the translational one and by solving a system of vibrational master equations coupled to the Boltzmann equation. The results, which have been compared with those corresponding to a vibrationally cold molecular gas, show that the inclusion of superelastic collisions and of the parent atoms affects the EDF tails without strongly modifying the EDF bulk. As a consequence the quantities affected by the EDF bulk, such as average and characteristic energies, drift velocity, 0-1 vibrational excitation rate are not too much affected by the inclusion of superelastic vibrational collisions and of parent atoms, while a strong influence is observed on the dissociation and ionization rate coefficients which depend on the EDF tail. Calculated dissociation rates, obtained by EDF's which take into account both the presence of vibrationally excited molecules and hydrogen atoms, are in satisfactory agreement with experimental results.

A joint vibro-electronic mechanism in the dissociation of molecular hydrogen in nonequilibrium plasmas

A new mechanism Of H 2 dissociation in electrical discharges (10 11 ⩽ n e ⩽ 10 12 cm −3, 2.10 −16 ⩽ E/ N ⩽ 3.10 −16 V cm 2, 300 ⩽ T g ⩽ 1000 K, 3 ⩽ p ⩽ 30 torr) is presented and discussed. In this mechanism, called joint vibro-electronic mechanism (JVE), the electrons of the discharge create a strong vibrational disequilibrium with respect to the gas temperature ( T g) and promote electronic transitions from all vibrational levels of 1Σ g H 2 state to the repulsive 3Σ u one. Moreover the V-V (vibration-vibration) and V-T (vibration-translation) energy exchanges are considered for building up the vibrational distribution of 1Σ g state. A complete set of e - D cross sections (e + H 2( 1Σ g,ν) → e + H 2 ( 3Σ u) → + 2H, ν = 0,14) is calculated by using an extension of the semiclassical Gryzinski theory in combination with the Franck-Condon principle. Dissociation rates calculated according to JVE are larger either than those obtained by the pure vibrational mechanism (PVM) discussed in our previous work or than those from the direct electronic impact mechanism (DEM) from the ground vibrational level. The behaviour of JVE rates as a function of gas temperature ( T g), of E/ N, of electron density ( n e) and of pressure is then reported. The results show strong differences as compared, with the corresponding values obtained, with PVM. Finally the influence of the atoms as well as their recombination on the dissociation rates is discussed. The results have been obtained by solving a system of vibrational master equations.

Non Equilibrium Dissociation of HF in Electrical Discharges: The Role of Vibrational Excitation

Dissociation rates of HF in electrical discharges have been calculated by solving a system of vibrational master equations, including e-V (electron-vibration), V-V (vibration-vibration) and V-T (vibration-translation) energy exchanges. The results show that large dissociation rates can be obtained by this mechanism only when the e-V pumping rates overcome the V-T deactivating rates. This occurs for electron number densities larger than 1013cm-3. Finally the close connection of the present results with the corresponding ones obtained by using as vibrational excitation source an IR powerfull laser is shown and discussed.

On the structure of exact solutions of discrete masterequations

A concise version of the proof for the graph theoretical representation of the exact solution of the stationary discrete masterequation is given. Further, a new algorithm is developed for the solution of stationary and nonstationary discrete masterequations with next neighbour transition probabilities in the general case without detailed balance. This algorithm reduces the dimension of the system of masterequations to the number of boundary sites and is also appropriate for computer evaluation.

Acoustic power spectra due to multiple rough surface scattering

The previously derived master equations [J. Acoust. Soc. Am. 62, A181 (1977)] describing the second‐order statistics of normal mode amplitudes due to energy transfer between modes caused by multiple rough surface scattering have been generalized to include quasistatic time‐dependent effects. We obtain a new system of master equations for the quantities 〈an(r,t)an*(r, t + τ)〉, where the an(r,t) are the normal mode amplitudes. We then construct 〈p(r,z,t)p*(r, z, t + τ)〉 by incoherently summing the modes. The Fourier transform (with respect to τ) of this quantity gives the sideband spectra of the pressure signal as a function of range and depth. We find that the spectra are symmetric (asymmetric) according to whether the surface wave spectrum is even (odd) with respect to the horizontal wave number in the direction of propagation. Numerical examples of spectra will be presented. [Work supported by ONR.]

Nonequilibrium vibrational populations and dissociation rates of oxygen in electrical discharges

Nonequilibrium vibrational distributions and dissociation rates of molecular oxygen in both electrical and thermal conditions have been calculated by solving a system of master equations including V-V (vibration-vibration), V-T (vibration-translation) and e-V (electron-vibration) energy exchanges. The dissociation constant under thermal conditions (i.e. without electrons) follows an Arrhenius law with an activation energy of 120 kcal/mole, while the corresponding rates under electrical conditions (5000 ⩽ T e ⩽ 15000 K, 300 ⩽ T g ⩽ 1000 K, 10 11 ⩽ n e ⩽ 10 12 cm −3,5 ⩽ p ⩽ 20 torr) increase with decreasing gas ( T g) and electron ( T e) temperatures and pressure ( p) and with increasing electron density ( n e). These results are explained on the basis of the different interplay of V-V and V-T energy exchanges and are rationalized by means of simplified models proposed in the literature. The accuracy of the present results is discussed paying particular attention to the dependence of V-V and V-T rate coefficients on the vibrational quantum number. A comparison of the calculated dissociation rates with the corresponding ones obtained by the direct electron impact mechanism shows that the present mechanism prevails at low electron and gas temperatures. Finally a comparison is shown between theoretical and experimental dissociation rates under electrical and thermal conditions.

Non-equilibrium dissociation of nitrogen

Dissociation rates of molecular nitrogen in electrical discharges have been calculated by solving a system of master equations, including V-V (vibration-vibration), V-T (vibrationtranslation) and e-V (electron-vibration) energy transfers. The results show that the dissociation rates slightly increase with decreasing gas temperature in the temperature range 1 000-2 000 K at the electron temperature Te = 20 000 K and electron density ne = 1012 cm-3. In these calculations, dissociation rates are two orders of magnitude greater than the corresponding rates obtained by direct electron impact. REVUE DE PHYSIQUE APPLIQUEE TOME 13, MARS 1978, PAGE Classification Physics Abstracts 52.20H-82,20M-82.20R

Nonequilibrium vibrational populations of diatomic species in electrical discharges: Effects on the dissociation rates

Dissociation rates of N 2 in N 2-H 2 mixtures and of H 2 in Ar-H 2 mixtures have been calculated for conditions typical of gas discharges. The results have been obtained by solving a system of master equations including V-V (vibration-vibration), V-T (vibration-translation) and e-V (electron-vibration) energy exchanges. The results show that these rates are much smaller than the corresponding ones for pure gases, as a consequence of the increased importance of V-T processes. A simplified model, which takes into account the main features of the distribution of vibrational levels in an electrical discharge, has been built up to rationalize the present results as well as those derived in our previous work.

A theoretical calculation of dissociation rates of molecular hydrogen in electrical discharges

Dissociation rates of molecular hydrogen in electrical discharges have been calculated at different electron ( T e) and gas ( T g) temperatures (10000 ⩽ T e ⩽ 23000 K, 500 ⩽ T g ⩽ 4000 K), at different pressures p (5 ⩽ p ⩽ 50 torr) and electron number densities n e (0 ⩽ n e ⩽ 10 12 cm −3). The results have been obtained by solving a system of master equations, including VT (vibration-translation), VV (vibrationvibration) and eV (electronvibration) microscopic processes. The results obtained at n e ≠ O show a “laser-type mechanism” in the dissociation of molecular hydrogen in electrical discharges. In particular one notices a strong increase of dissociation rates with decreasing gas temperature and pressure. The results show that this mechanism is as important as the mechanism of direct dissociation by electron impact.

Network theory of microscopic and macroscopic behavior of master equation systems

A general microscopic and macroscopic theory is developed for systems which are governed by a (linear) master equation. The theory is based on a network representation of the master equation, and the results are obtained mostly by application of some basic theorems of mathematical graph theory. In the microscopic part of the theory, the construction of a steady state solution of the master equation in terms of graph theoretical elements is described (Kirchhoff's theorem), and it is shown that the master equation satisfies a global asymptotic Liapunov stability criterion with respect to this state. The Glansdorff-Prigogine criterion turns out to be the differential version and thus a special case of the global criterion. In the macroscopic part of the theory, a general prescription is given describing macrostates of the systems arbitrarily far from equilibrium in the language of generalized forces and fluxes of nonlinear irreversible thermodynamics. As a particular result, Onsager's reciprocity relations for the phenomenological coefficients are obtained as coinciding with the reciprocity relations of a near-to-equilibrium network.

Nonlinear fluctuations in master equation systems. I. Velocity correlation function for the Rayleigh model

The influence of nonlinear velocity fluctuations on the velocity correlation function Π (t) is studied for the Rayleigh model of a massive particle in an ideal gas as an example of a master equation system. It is shown that the Mori kernel K (t), which determines the decay of Π (t), has a slow mass-dependent decay on the time scale of the decay of Π (t) and has no well-behaved expansion in the mass ratio. Both features are contrary to standard assumption. The origins of the slow decay are traced to nonlinear fluctuations and the relationship to previous work on requisite conditions for exact exponential decay is discussed. The slow decay of Π (t) is shown to lead to divergent ’’Burnett’’ coefficients in macroscopic friction laws and the resolution of this difficulty is discussed. The relationship of the microscopic ’’bare’’ friction constant to the macroscopic friction constant is considered. Explicit expressions for Π (t) and K (t) for small mass ratio are obtained by mode–mode coupling analysis and perturbation methods. The influence of nonlinear fluctuation effects is found to be numerically negligible despite their long lifetime. The remaining deviation from standard Brownian motion results is examined numerically. The validity of some standard assumptions in mode–mode coupling theory is also examined.

Pair approximation for interface kinetics

A pair approximation method is described for the interface kinetics of a simple cubic (100) surface with no surface migration. Condensation and evaporation are considered explicitly in a system of master equations applied to a pair of sites; and the effect of the neighboring sites is treated with a set of parameters which are determined self-consistently. Runge Kutta numerical integration of the equations yields the time evolution of the surface. The results for a high entropy of transformation indicate a region of metastable states (zero growth rate) for small driving force, and the instantaneous growth rate is a periodic function of time for larger driving force. The period of the variations is the time required to grow a monolayer, and the amplitude is less than that obtained by Temkin using the mean field approximation. The variations appear as artifacts of most approcimation methods which treat a limited number of sites. The average growth rate is less than the limiting Wilson-Frenkel law for all growth conditions studied. This results since the surface becomes quite rough in the presence of a large condensation rate, and therefore its evaporation rate is greater than the evaporation rate of an equilibrium surface assumed in the Frenkel model. The metastable region is absent at low entropies of transformation and here the growth rate is nearly linear in the driving force. Agreement with Monte Carlo data is excellent for a low entropy of transformation and for all materials at high growth rates.