Fourier Poisson

In the case of the reciprocal sum, two methods have been implemented, smooth particle mesh Ewald (SPME) [65] and fast Fourier Poisson (FFP) [66], SPME is based on the realization that the complex exponential in the structure factors can be approximated by a well behaved function with continuous derivatives. For example, in the case of Hermite charge distributions, the structure factor can be approximated by... 

York D, Yang W (1994) The fast Fourier Poisson method for calculating Ewal sums. J Chem Phys... 

Kronig and Brink (K5) used the Fourier-Poisson equation to derive equations for the temperature distribution and heat transfer inside a drop with internal circulation described by Eq. (23). Assuming that diffusion is negligible along internal streamlines and that the isotherms at any particular moment coincide with the streamlines, and disregarding external film resistance, they obtained the following equation for the transfer efficiency ... 

Handles and Baron (H5) proposed another model for the more practical range of Reynolds numbers (about 1000). They assumed that the tangential motion caused by circulation is combined with an assumed random radial motion caused by internal vibration, and determined the eddy diffusivity subsequently used in solving the appropriate Fourier-Poisson equations. They postulated radial stream lines, as shown in Fig. 11, rather than those... 

York, D., and Yang, W. (1994], The Fast Fourier Poisson (FFP] method for calculating Ewald sums,/ Chem. Phys. 101, pp, 3298-3300. 

If a gaussian function is chosen for the charge spread function, and the Poisson equation is solved by Fourier transformation (valid for periodic... 

The generation of photons obeys Poisson statistics where the variance is N and the deviation or noise is. The noise spectral density, N/, is obtained by a Fourier transform of the deviation yielding the following at sampling frequency,... 

The prevailing theory of heat, popularized by Sinieon-Denis Poisson, Antoine Lavoisier and others, was a theory of heat as a substance, caloric. Different materials were said to contain different quantities of caloric. Fourier had been interested in the phenomenon of heat from as early as 1802. Fourier s approach was pragmatic he studied only the flow of heat and did not trouble himself with the vexing question of what the heat actually was. 

Fourier was not without rivals, notably Biot and Poisson, but his work and the resulting hook greatly influenced the later generations of mathematicians and physicists. 

From Poisson s equation (265), we get for the Fourier transform of the potential ... 

Poisson s equation (6.9) can be solved directly by writing the potential, K(r), in terms of its Fourier transform, K(q), that is... 

Although the principles of heat flow have lieen understood and treated mathematically since the early 19th century (Fourier, LaPlace. Poisson, Peclel. Lord Kelvin. Riemann. and many otherst. it was not until nearly... 

Let us consider the nonlocal Poisson equation V(sW) = -4irp in the uniform space. The singular boundary condition on the surface of the solute cavity is neglected. Note that this condition furnishes the mechanisms of the excluded volume effect. The solute is charged and spherical, i.e. p(r) = p(R). The solution T (A ) is obtained by using Fourier transform [6,16] it is valid outside the cavity (R > a),... 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Fourier s law for heat flow rate and Ohm s law for charge flowrate (i.e., electrical current). For three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (Qv/e) = (volumetric charge density/permittivity) and (QG//0 = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m 2) and (K m 2). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

The mean field potential for this system, a solution of the linear Poisson-Boltzmann equation, Eq. (32), will appropriately have the same periodic structure as the surface boundary condition. Thus, we expect that if/ will have the Fourier series,... 

THE POISSON-BOLTZMANN EQUATION FOR A SURFACE 49 where C k) is the Fourier coefficient independent of z. Equation (2.4) thus becomes... 

The question raised by the quasicrystal debate is much deeper than whether they exist or not. To see this, we recall that the interpretation of diffraction experiments on all known translationally invariant crystals, however complicated, depends ultimately on the existence of the Poisson summation formula. This relation asserts that the Fourier trtinsform of the periodic delta function is itself a periodic delta function, whence the term reciprocal space. Explicitly, the Poisson summation formula is... 

Here, ft(x — X ) is a charge distribution centered at the point Xa, while s another charge distribution centered on a different point Xa-, and fit = (rii, li, in,). We now introduce the Fourier representation of the Green s function of Poisson s equation ... 

The Flux Expressions. We begin with the relations between the fluxes and gradients, which serve to define the transport properties. For viscosity the earliest definition was that of Newton (I) in 1687 however about a century and a half elapsed before the most general linear expression for the stress tensor of a Newtonian fluid was developed as a result of the researches by Navier (2), Cauchy (3), Poisson (4), de St. Venant (5), and Stokes (6). For the thermal conductivity of a pure, isotropic material, the linear relationship between heat flux and temperature gradient was proposed by Fourier (7) in 1822. For the difiiisivity in a binary mixture at constant temperature and pressure, the linear relationship between mass flux and concentration gradient was suggested by Pick (8) in 1855, by analogy with thermal conduction. Thus by the mid 1800 s the transport properties in simple systems had been defined. 

Fig. 17 Example of application of 4D HNCACO technique, (a) Pulse sequence. Evolution for CO is in the real-time mode, and for N and CA in semi-constant-time mode (a, = (/, -l- A)/2, 6, = t,(l-A/Wi)/2, C = A(l- 6Amaxi)/2) or constant-time mode (a,- = (A -I- /,)/2, 6,- = 0, c,- = (A—/,)/ 2), where A stands for An ca and Aca-co. respectively, t, is the evolution time in ith dimension and imaxi is the maximal length of evolution time delay. Delays were set as follows An h = 5.4 ms An-ca = 22 ms Aca-co = 6.8 ms. (b) Coherence transfer in the peptide chain. Amide nitrogen and proton frequencies (filled colored rectangles) are fixed during Fourier transformation. Each plane contains CO-CA peak for i and i—1 residue, (c) 2D spectral planes for CsPin protein obtained by SMFT procedure performed on the 4D HNCACO randomly sampled signal (Poisson disk sampling) with fixed Hn and N frequencies obtained from 3D HNCO peak list (d) 2D spectral planes for MBP obtained in the same manner. Reprinted with permission from [81]...

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