Bessel function interaction

Therefore the Casimir energy for the two spherical cavities inside a non-relativistic non-interacting fermion background can be approximated in terms of a spherical Bessel function j as... 

On small scales, these sources of pressure determine the evolution of perturbations. Consider once again Eq. 10.10. When the wavenumber k > kj, or conversely when the wavelength is smaller than the Jeans length, pressure dominates over gravity and the fluid oscillates with angular frequency u csk. The detailed solution actually involves Bessel functions when expansion is correctly taken into account, and there are additional complications due to gravitational interactions with any pressureless component such as CDM which can continue to collapse. 

The modified Bessel function, 70, is always positive increasing, the free energy is therefore always negative. That is, the free energy of interacting surfaces is always attractive irrespective of whether the surfaces are held at constant charge or constant potential. 

It should be noted that integration of the exponential term in equation (12) for the BC potential leads to a series of Bessel functions, an expression which is rather inconvenient for numerical calculation that is why in a further development we have restricted ourselves to the Lennard- Jones dispersion and repulsion potential form. Also the repulsion (r ) component in the expression for interaction with the adsorbent bulk was found to be negligibly small and was therefore omitted from the calculations. 

Similar expressions apply to the other types of pairs, and all of them contain the zeroth-order modified Bessel function of the second kind K0(xr), or, simply, the Bessel K0 function. In Figure 1 we show a graph of the smoothly decreasing nonoscillatory Bessel K0 function, which can subsequently be contrasted with the pair potentials w(r) that emerge only after the three free energy terms (polyion self-energy, counterion transfer, and direct pair interaction) are added and minimized, a procedure that involves determination of the functions 0(r) and Q(r). 

In tiiese exfnessions, y is tiie redprocal of tiie unquenched dec time, (Q] is tiie quencher concentration In molecules per angstrom squared, R is the interaction radius, and x, I /D, where f) is tiie mubialdiffusicMi coeffident JoCat) and yo(x) are Zdo-order Bessel functions of tiie first and second kinds, respectively. Even moce complex expressions are needed for the radiation model Because of the difficulties in evaluating [9.16] and [9.17], several approximate analytical expressions have been proposed. ... 

Here, r is the distance between the particles, is the radius of the circular contact line at which the membrane detaches fi om the colloid, a is the angle with respect to the horizontal at which it does so, and the Kj, are modified Bessel functions of the second kind. This solution is analytical, simple, and wrong. Or more accurately, it only holds when r 2 ro, a restriction that excludes the interesting tensionless limit in which 2 oo. The mathematical reason is that superposition in the way celebrated here is not allowed yes, superpositions of solutions to linear equations are still solutions, but superpositions of solutions, each of which only satisfies some part of aU pertinent boundary conditions, generally do not satisfy any boundary condition and are thus not the solutions we are looking for. The physical reason why the superposition ansatz in this case fails is because the presence of one colloid on the membrane, which creates a local dimple, will abet a nearby coUoid to tilt, thereby changing the way in which that second colloid interacts with the membrane and, in turn, the first one. 

In the framework of our formalism, the description of this resonant interaction is essentially analogous to that of inelastic diffraction ( washboard -phonon resonance, see brief discussion in the previous section). The trajectories of translational and rotational motion consist of two elastic branches as well, and the result may be expressed through the Bessel functions of the corresponding classical action increments. 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

Fig. 5 Thermodynamic functions for the hydration of apolar molecules, quasichemical approximation lattice with orientation-dependent interactions after Besseling (1993)...

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