Boltzmann equation Fourier transform

All measured SAXS data were analyzed by the generalized indirect Fourier transformation (GIFT) technique with the Boltzmann simplex simulated annealing (BSSA) algorithm [34, 35]. The GIFT calculation is based on the analytical or numerical solution of the Ornstein-Zernike (OZ) equation that describes the interplay between the total (h(r)) and direct (c(r)) correlation functions ... 

Owing to the use of Fourier Transform in the implementation of the exponential of the Laplacian operator, the technique is mainly useful for systems with periodic boundary conditions. Also, note that the technique carmot be used to solve Poisson-Boltzmann equation due to time-independent nature of the equation. However, the Poisson-Boltzmann equation can be solved using finite difference techniques or by a combination of fast Fourier transforms and the finite difference techniques [68, 81]. 

T is measurable in principle by any method which detects the effects on the nuclear magnetization of any perturbation of the Boltzmann populations. It may be measured either statically or dynamically. The original static method was progressive saturation, which exploits equation (1) by increasing until its effects on the resonance are substantial. The method requires an independent measure of the rf field of IBi, which is more conveniently available in the method s Fourier-transform approximation, the (90° — On Observe sequence. Here one obtains from the dependence upon t of the observed reduction in intensity between n = 1 and n = large using the equation... 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

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