Relaxation time approximation

This integral equation can be solved exactly with the stress relaxation time approximated as... 

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... 

Assuming a single relaxation time approximation, the diffusion coefficient takes the form... 

The single relaxation time approximation for all elementary reaction rates in a cycle with two limiting reactions is... 

With the additional assumption of the Poisson distribution for the probability of n jumps in a time t (mean time between jumps r), Equations 8 and 9 can be used in existing lattice diffusion theories to calculate relaxation times (approximately, i.e., within 10%) (16,18). 

The single relaxation time approximation corresponds to a stochastic model in which the fluctuating force on a molecule has a Lorentzian spectrum. Thus if the fluctuating force is a Gaussian-Markov process, it follows that the memory function must have this simple form.64 Of course it would be naive to assume that this exponential memory will accurately account for the dynamical behavior on liquids. It should be regarded as a simple model which has certain qualitative features that we expect real memory functions to have. It decays to zero and, moreover, is of a sufficiently simple mathematical form that the velocity autocorrelation function,... 

Thus the single relaxation time approximation to the memory function is... 

The single relaxation time approximation can be applied to the angular momentum memory function in a Completely analogous way.68 Kj(t) can be interpreted as the time-correlation function of the random torque acting on the molecule. If this random torque has a Lorentzian spectrum or, more restrictively, is a Gaussian-Markov process, Kj(t) is exponential. 

Generally, the BTE with single relaxation time approximation can be written as Equation (15). In order to simulate flow in the porous media, we consider the drag effect of the medium and present the LBM equation by the following form of the statistical average ... 

The Self-Consistent Relaxation Time Approximation (SCRTA)... 

However, upgrading of the theory was not stopped. After the first attempt the MPK2 [247] and MPK3 [126] have arisen whose kernels are different. As can be seen from Table V, the kernel for MPK2 is actually defined as a solution of the transcendent equation. Another one, for MPK3, appears to be identical to that provided by MET. More recently a new theory named the self-consistent relaxation time approximation (SCRTA), was developed by Gopich and Szabo [180]. In the irreversible case the inverse relaxation time of SCRTA, kfc, is defined by the transcendent equation for kf shown in Table V. It is equivalent to the one expressed by the Laplace transformation of k(t) ck(kf) = 1 [177]. In spite of this complication, SCRTA is competitive in the concentration corrections with other theories. 

The results for SNR evaluated with the effective relaxation time approximation in the same linear response framework [i.e., by Eq. (4.281)], are shown in Figure 4.24. One observes a good agreement with respect to the main cusp of the function R(T). However, in the low-temperature range the existing deviations from the exact solution [they do not resolve in the susceptibility graphs of Figure 4.21(a)] become noticeable see the relative positions of the dots and curve 3 for 1/a < 0.1. 

Hall Effect and Magnetoresistance in the Relaxation-Time Approximation... 

The zeros of this integral are the same as for the Bessel function Jo (7). Therefore, from the positions of the resistivity maxima tb can be extracted. The complete equation for the conductivity using the relaxation time approximation is given by [164]... 

Ladd, A., B. Moran, and W.G. Hoover, Lattice Thermal Conductivity A Comparison of Molecular Dynamics andAnharmonic Lattice Dynamics. Physical Review B, 1986. 34 p. 5058-5064. McGaughey, A.J. and M. Kaviany, Quantitative Validation of the Boltzmann Transport Equation Phonon Thermal Conductivity Model Under the Single-Mode Relaxation Time Approximation. Physical Review B, 2004. 69(9) p. 094303(1)-094303(11). 

Additionally for the simulations presented below, a simple relaxation time approximation was used for the spin up and spin down carriers. 

In the Wigner spin dependent simulations we deal with a simple relaxation time approximation and treat transport only by electrons. 

For the design of thermoelectric materials, it is necessary to obtain analytical transport properties, such as Seeheck coefficient, electrical conductivity and thermal conductivity. For this work, the hand theory using the relaxation time approximation of Boltzmaim s equation based upon simple semiconductor band model (Figure 1) has been adopted [2]. 

Using the relaxation time approximation of Boltzmann s equation, the expression for the properties (electrical conductivity, Seebeck coefficient, and Lorenz number) of the holes and electrons are given by... 

Here the suffix e denotes electron and the corresponding expression for holes is obtained by substituting Tj - T] - Xg in above equations(the expression of hole contribution for thermopower is required to change the sign in addition to the substitution). We have used the relaxation time approximation in above expressions and assumed the energy dependence of relaxation time is given by[5] ... 

It is clear that in the most general form as described in Eq. 8.45, the scattering terms pose difficulty for solving. Therefore, the relaxation-time approximation is usually made for convenience, in which case the equation of radiative transfer reduces to... 

Gopich, I.V., Szabo, A. Kinetics of reversible diffusion influenced reactions The self-consistent relaxation time approximation. J. Ghem. Phys. 2002,117, 507. 

Nature of current The nature of the driving force for the electron current is not definitely known. For example, the electron mobility is not expected to be a constant because the relaxation time approximation, which is used to derive Eq. (18), is not necessarily valid. 

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