Modified Moment Method Research Articles

Modified-moment method for the Fokker-Planck equation and some aspects of the thermodynamics of irreversible processes.

The modified-moment method is applied to solve the Fokker-Planck equation in such a way that the solution is consistent with the requirements of the thermodynamic laws. The formal solution has an attendant mathematical structure for the entropy which is similar to that obtained from the Boltzmann equation. A generalized Gibbs relation is obtained for such a thermodynamic branch of solution. The formalism therefore may be used to study thermodynamic aspects of stochastic processes underlying the Fokker-Planck equation. The method is shown to yield the same analytical results as other methods in the case of linear processes, and in the case of nonlinear processes the moment series converges sufficiently fast to justify a lower-order truncation of the moment series for the distribution function, if the diffusion constant (D) is less than a critical value. In this case the solution of the Fokker-Planck equation tends to the steady-state distribution function. It is shown that the second variation of the nonequilibrium part of the entropy may be used as a Lyapunov function, which provides local criteria of evolution only.

Generalized hydrodynamics, normal-stress effects, and velocity slips in the cylindrical Couette flow of Lennard-Jones fluids.

In this paper we extend the previous study of flow profiles in cylindrical Couette flow of a Lennard-Jones fluid so as to include the effects of normal stresses neglected in the previous work. The flow is subject to a temperature gradient and thermoviscous effects are taken into consideration. We apply the generalized hydrodynamic equations which are provided by the modified moment method for the Boltzmann equation reported previously. The results of calculation are in good agreement with the Monte Carlo direct simulation results of K. Nanbu [Phys. Fluids 27, 2632 (1984)] and experimental data for all Knudsen numbers. The inclusion of normal stresses also brings about a new aspect to the flow properties which has not been seen before, since the flow profiles exhibit a phase-transition-like behavior as the Knudsen number increases beyond a critical value. There also appear slips in velocity at the inner cylinder wall in the supercritical regime of Knudsen number. At the critical Knudsen number the entropy production for the flow process also exhibits a singular behavior which is reminiscent of a second-order phase transition in thermodynamic systems.

Response to Z. Banach's comments on the modified moment method and irreversible thermodynamics

In this paper the author responds to the comments on the modified moment method and irreversible thermodynamics made by Z. Banach [Physica A 145 (1987) 105]. In this paper Banach suggests a variational method in which the Lagrange multipliers are determined from the constraints alone by disregarding the entropy balance equation. It is shown that since this method does not yield an extended Gibbs relation consistent with the entropy balance equation or the H -theorem, there is no irreversible thermodynamics formalism afforded by the method. Consequently, his criticism cannot be supported from the viewpoint of irreversible thermodynamics. It is also pointed out that neither is there a mathematical and physical support for the criticism he makes on the cumulant expansion for the Boltzmann collision integral.

Nonlinear transport processes and fluid dynamics: Cylindrical Couette flow of Lennard-Jones fluids.

In this paper we report on calculations of flow profiles for cylindrical Couette flow of a Lennard-Jones fluid. The flow is subjected to a temperature gradient and thermoviscous effects are taken into consideration. We apply the generalized fluid dynamic equations which are provided by the modified moment method for the Boltzmann equation reported previously. The results of calculations are in good agreement with the Monte Carlo direct simulation method by K. Nanbu (Phys. Fluids 27, 2632 (1984)) for most of Knudsen numbers for which the simulation data are available.

Nonlinear transport coefficients and plane Couette flow of a viscous, heat-conducting gas between two plates at different temperatures

By using the example of plane Couette flow between two plates maintained at different temperatures, we present a method of calculating flow profiles for rarefied gases. In the method, generalized hydrodynamic equations are derived from the Boltzmann equation. They are then solved with boundary conditions calculated by taking into consideration the interfacial interaction between the surface and the gas molecule. The nonlinear transport coefficients employed in the generalized hydrodynamic equations are obtained from the Boltzmann equation by means of the modified-moment method. The profiles calculated are in agreement with the Liu–Lees theory as long as the boundary values are in agreement. It is found that the viscous-heating effect has a significant influence on the temperature and velocity profiles. The nonlinearity of transport coefficients also has significant effects on the profiles as the Knudsen and Mach numbers increase.

Nonlinear transport processes and fluid dynamics: Effects of thermoviscous coupling and nonlinear transport coefficients on plane Couette flow of Lennard-Jones fluids.

To examine the effects of nonlinear transport processes on the flow properties of a real fluid, we study the plane Couette flow under a temperature gradient in a Lennard-Jones fluid over a range of gas pressure. The analysis is based on the nonlinear transport coefficients derived in the modified moment method. The linear transport coefficients appearing in the theory are those constructed by Ashurst and Hoover from the nonequilibrium molecular-dynamics simulations, while for the equation of state for the Lennard-Jones fluid we use the empirical form proposed by Ree. Examples for the flow characteristics at two extreme conditions when either the gas is very dilute, or when it is very dense, are presented. It is shown that the temperature and velocity profiles for nonlinear transport processes are significantly different from the ones obtained with the linear transport coefficients. In the dilute-gas limit, slip boundary conditions are used which are derived by applying the Langmuir theory of gas-surface interaction. Flow profiles show pronounced boundary-layer structures near the wall when the product of the Knudsen number and the Mach number is sufficiently large.

Kinetic theory of nonlinear transport processes in dilute ionized gases subject to an electromagnetic field

The modified moment method is applied to study transport processes in a plasma subject to a homogeneous electromagnetic field. Various balance and evolution equations are derived for macroscopic variables chosen, by starting from a kinetic equation. Extended Gibbs relations are also derived as extensions of the extended Gibbs relation for neutral gases. The evolution equations are solved under the adiabatic approximation in the case of the steady state by using an iterative solution method. The first iterative solution yields various linear transport coefficients which reproduce the Chapman–Enskog formulas for the transport coefficients as the magnetic field vanishes, and the second iterative solution gives rise to a set of transport coefficients which depend on the thermodynamic forces as well as the magnetic field. In the sense that they depend on thermodynamic forces they are nonlinear. As the density tends to zero (the rarefied gas limit), the second order transport coefficients vanish as expected for the transport coefficients for rarefied gases.

Nonlinear heat conductivity in plasmas

It is shown that the modified moment method applied to the Vlasov–Landau equation is adequate for accounting for the heat flux inhibition in laser-driven plasmas, if the temperature profile is properly taken into account in calculating the heat flux. The temperature profiles are obtained by solving the steady-state mass, momentum, and energy balance equations in which the heat flux is a nonlinear function of the temperature gradient derived from the kinetic equation by means of the modified moment method. There can be two different temperature profiles, depending on the Mach number of the fluid speed. When the temperature profiles thus calculated are self-consistently used in the heat flux formula, the heat flux limit factors are derived. The factors are of the same order of magnitude as the experimentally inferred value of 0.03. For example, in the case of the hydrogen plasma the factor is less than 0.05 for the supersonic branch.

Temperature profile and heat flux inhibition in laser-driven plasmas

It is shown that the nonlinear heat flux calculated by means of the modified moment method applied to the Vlasov-Landau kinetic equation is adequate to account for the heat flux inhibition in laser-driven plasmas, if the nonlinear heat flux so obtained is used in the steady state energy balance equation to calculate the temperature profile self-consistently. Heat flux limit formulas are obtained for supersonic and subsonic fluid speeds. The flux limit factors have an upper bound of 0.14 in the case of the hydrogen plasma. This bound would be reduced, if the temperature at the sonic surface was less than the electron temperature at the critical density surface.

Kinetic theory and irreversible thermodynamics of nonlinear transport processes in quantum systems

A theory of nonlinear irreversible processes in quantum systems (gases) is developed by using a quantum mechanical Boltzmann equation which is a generalization of the equation originally used by Nordholm, Uehling, and Uhlenbeck. The Boltzmann equation is solved by the modified moment method and quantum mechanical extensions are thereby obtained for various macroscopic equations, including the extended Gibbs relation, derived previously by means of the classical Boltzmann equation. The theory can yield a nonmonotonic entropy production as a function of fluxes in the system and thus provides us with possibility of describing instability in transport processes in the nonlinear regime. In order to demonstrate it with a simplified model, we consider the threshold switching phenomenon in semiconductors at low temperatures and show that, when the impact ionization recombination of donors is postulated as an underlying microscopic process for the phenomenon, the theory can indeed give a qualitatively correct result for the current-field characteristic and thus furnishes a nonequilibrium statistical mechanical description of the phenomenon. It also lays a foundation for studying the electron gas model for various nonlinear transport processes in metals and semiconductors at low temperatures in full conformation with the requirement of the thermodynamic laws.

Nonlinear field dependence of carrier mobilities and irreversible thermodynamics in semiconductors

Irreversible thermodynamics and transport processes in semiconductors are studied by means of the Boltzmann equation for carrier distribution functions. By solving the equation by the modified moment method, we derive the extended Gibbs relation for entropy change and the evolution equations for fluxes. The steady-state solutions of the evolution equations yield diffusion fluxes and drift velocities which are nonlinear with respect to the field strength. Depending on the wave-number dependences of the scattering cross sections and the affinity of electrons in different valleys in the conduction band, the drift velocity and the mobility can exhibit a plateau or a negative differential mobility in the high-field regime. Calculated carrier drift velocities are compared with experimental data on the electron drift velocities in Si and $n$-Ge. The comparisons are in good agreement and show that the theory generally predicts qualitatively correct field dependences of electron drift velocities.

The effect of a magnetic field on the impurity band density of states in the atomic limit

A marked magnetic field dependence is found for the impurity band density of states in an idealized model of a doped semiconductor. The calculations are based on the Hubbard model in the atomic limit. We have obtained exactly the first eight moments of the distribution using a Gaussian model for the hopping integral, and the first seven moments with a hydrogenic model. A simple Zeeman type response of the spin system to the magnetic field is assumed. The predicted density of states is then obtained using a modified moment method with a trial density of states function obtained from the expected asymptotic behaviour of the distribution. Finally, we adjust the parameters of our models to correspond to phosphorous doped silicon below the metal insulator transition.

Remarks on the modified moment method and irreversible thermodynamics

A solution of the Boltzmann equation for irreversible thermodynamic processes is discussed. Some remarks are made on the modified moment method of the solution of Boltzmann equations. (AIP)

Kinetic theory and irreversible thermodynamics for dilute polar gases in an electric field

The modified moment method is applied to study irreversible phenomena in dilute polar gases subject to an external electric field. A suitably generalized Boltzmann equation is employed. Extended Gibbs relations are obtained for such systems and Debye's dielectric relaxation equation is derived therefrom in a linear approximation. The extended Gibbs relation reduces to the equilibrium Gibbs relation obtained by the Groot and Mazur for polarizable systems. The present study thus provides a molecular theory of irreversible thermodynamics of dilute polar gases in an electric field.

Nonlinear transport processes and irreversible thermodynamics of a mixture of ionized and neutral gases subject to an electric field

In this paper, the modified moment method is applied to investigate nonlinear transport processes and irreversible thermodynamics of a dilute mixture of nonpolarizable ionized and neutral gases subject to an external electric field. Theories are developed by means of scalar and tensor temperature distribution functions and the extended Gibbs relations are derived in both theories. The latter relations reduce to the equilibrium Gibbs relation as the fluxes vanish at equilibrium. Various nonlinear transport equations are obtained from which nonlinear transport coefficients can be derived. They all reduce to linear transport coefficients as the magnitudes of the fluxes get smaller. Especially, nonlinear diffusion and mobility coefficients are obtained which depend nonlinearly on the electric field strength, and their mutual relationships are found as generalizations of the Einstein relations.

The modified moment method and theory of nonlinear transport processes in gases: Third order cumulant approximation

By calculating the entropy production to the third order cumulant approximation within the framework of the modified moment method, we examine it for nonlinear transport processes and obtain nonlinear transport coefficients which include ’’Burnett-type’’ coefficients among them. The entropy production obtained has a mathematical form that can admit two minima, one at equilibrium and the other away from equilibrium. It therefore can have a region of decrease as the system is removed farther from equilibrium. This is a phenomenon not existent in linear theory or in the theory based on the first cumulant approximation in the modified moment method.

The modified moment method, irreversible thermodynamics, and the nonlinear viscosity of a dense fluid

The extended Gibbs relation is derived for dense fluids in a nonequilibrium state from the kinetic equation for dense fluids proposed in a previous paper. The method used for derivation is basically the same as that used in the case of dilute (ideal) gases described by the Boltzmann equation. The extended Gibbs relation derived is, as it should be, in the same form as for ideal gases, but the evolution equations for macroscopic variables involved contain contributions due to molecular interactions. This derivation together with that of the corresponding relation for ideal gases previously reported provides us with a mathematical basis to study irreversible thermodynamics for systems at any nonequilibrium state without making the assumption of local equilibrium. The non-Newtonian viscosity is also studied for a dense fluid with the stationary solution of the evolution equation for stress tensor and viscosity formulas are obtained as extensions of the formulas for ideal gases reported previously.

Theory of nonlinear transport processes in a dilute gaseous mixture

The modified moment method is applied for studying nonlinear transport processes in a dilute gaseous mixture described by the Boltzmann equation. The method supplies a variational functional which may be regarded as a nonlinear generalization of the Rayleigh–Onsager variational functional for linear irreversible processes. A variational principle is formulated with the variational functional, which yields a set of nonlinear equations for fluxes as the condition for the functional to be an extremum. By solving the set iteratively, we obtain nonlinear constitutive relations and transport coefficients which reduce to their linear counterpart as thermodynamic gradients decrease in magnitude. The Knudsen gas limits of the transport coefficients obtained are also briefly discussed, which show a slightly different behavior from that predicted by the collisionless Boltzmann equation.

Theory of nonlinear transport processes and irreversible thermodynamics in multicomponent dense fluids

The kinetic equation for a single component dense fluid reported previously is extended to a multicomponent dense fluid and the evolution equations are derived therefrom for macroscopic fluxes. The extended Gibbs relation is derived for dense mixtures from the entropy balance and evolution equations by applying the modified moment method. The evolution equations yield nonlinear transport equations whose solutions in terms of thermodynamic forces give nonlinear constitutive relations as well as nonlinear transport coefficients. The nonlinearity and nonlinear couplings between different irreversible processes generally tend to decrease the transport coefficients according to the results obtained. The non-Newtonian shear viscosity is a typical example.

Influence of resonant charge transfer on ion mobility

The peculiar scattering due to resonant charge transfer causes the ion velocity distribution function to be anisotropic and skewed, especially in strong electrostatic fields. A modified moment method is proposed for dealing with mobility in this case, and shown to be accurate by comparison with model calculations. The method is also applicable to other cases of distorted distribution functions.