Equation, Boltzmann, generalized transport

In the classical Boltzmann equation for electron transport, scattering is included via a dissipation term using x, the average collision time. A master equation approach is basically a generalization of the Boltzmann equation to a fully quantum mechanical system. The master equation is based on the... 

The goal of the presentation in this section is to pose a generalized model comprising all the types of nonequilibrium detectors presented in the literature until now, and applicable to potential novel devices. When deriving the model we start from the semiconductor equations in their general form (Maxwell s equations and Boltzmann s transport equation.). 

J. G. Kirkwood and J. Boss, The Statistical Mechanical Basis of the Boltzmann Equation, in I. Frigogine, ed., Transport Processes in Statistical Mechanics, pp. 1-7, Interscience Publishers, Inc., New York, 1958. Also, J. G. Kirkwood, The Statistical Mechanical Theory of Transport Processes I. General Theory, J, Chem, Phys, 14, 180 (1946) II. Transport in Gases, J, Chem. Phys, 15, 72 (1947). 

About Rep, it decreases as temperature decreases, due to the fact that the number of phonons decreases. A full treatment of the problem, however, can only be obtained by solving the Boltzmann transport equation, which has only been solved for the case of quasi-free electrons. Further information and approximate solutions can be found in ref. [7,106,107], The general result of these calculations shows that at low temperature T < 0D/1O), the thermal resistance Rep is of the form b- T2. 

Eq. (437) may be transformed into a true transport equation for f this transport equation is the generalized linearized Boltzmann equation for f, as it also appears in the theory of thermal transport coefficients. More precisely, we get ... 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

Finally, we study the structure of the generalized Boltzmann operator. It can be expressed in terms of the transport operator, which allows one to obtain the virial expansion of the generalized Boltzmann equation. The remarkable point here is that the generalized Boltzmann operator can be expressed in terms of non-connected contributions to the transport operator. This happens for the correction proportional to c3 (c = concentration) and for the following terms in the virial expansion of the generalized Boltzmann operator. 

If, by internal means, the system is near c2> only a small amount of energy is necessary to drive the system into the highly excited state. Furthermore, we have been able to show that oscillations on the hysteresis are possible for A detailed inspection of the transport equations (generalizeS nonlinear Peierls-Boltzmann equations for phonons) shows that nonlinear kinetics, dissipation and energy supply via transport are indispensable for such a behaviour to occur. 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

Electron transport simulation is performed using earlier developed ensemble Monte Carlo algorithms and procedures, which include self-consistent solution of Poisson and Boltzmann equations [3,4]. In general, in both types of MOSFETs the normal component of electric field at Si/Si02 interface in a certain jc-point of the channel may be calculated using the following expression... 

Local thermodynamic equilibrium in space and time is inherently assumed in the kinetic theory formulation. The length scale that is characteristic of this volume is i whereas the timescale is xr. When either L i, ir or t x, xr or both, the kinetic theory breaks down because local thermodynamic equilibrium cannot be defined within the system. A more fundamental theory is required. The Boltzmann transport equation is a result of such a theory. Its generality is impressive since macroscopic transport behavior such as the Fourier law, Ohm s law, Fick s law, and the hyperbolic heat equation can be derived from this in the macroscale limit. In addition, transport equations such as equation of radiative transfer as well as the set of conservation equations of mass, momentum, and energy can all be derived from the Boltzmann transport equation (BTE). Some of the derivations are shown here. 

The kinetic theory of dense gases began with the work of Enskog, who in 1922, generalized Boltzmann s derivation of the transport equation to apply it to a dense gas of hard spheres. Enskog showed that for dense gases there is a mechanism for the transport of momentum and energy by means of the intermolecular potential, which is not taken into account by the Boltzmann equation at low densities, and he derived expressions for the transport coeffi-... 

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