Diffusion propagator

Mitra, PP Sen, PN Schwartz, LM Le Doussal, P, Diffusion Propagator as a Probe of the Structure of Porous Media, Physical Review Letters 68, 3555, 1992. 

Since it was proposed in the early 1980s [6, 7], spin-relaxation has been extensively used to determine the surface-to-volume ratio of porous materials [8-10]. Pore structure has been probed by the effect on the diffusion coefficient [11, 12] and the diffusion propagator [13,14], Self-diffusion coefficient measurements as a function of diffusion time provide surface-to-volume ratio information for the early times, and tortuosity for the long times. Recent techniques of two-dimensional NMR of relaxation and diffusion [15-21] have proven particularly interesting for several applications. The development of portable NMR sensors (e.g., NMR logging devices [22] and NMR-MOUSE [23]) and novel concepts for ex situ NMR [24, 25] demonstrate the potential to extend the NMR technology to a broad application of field material testing. 

The term propagator is used to describe the motion of a particle A within the solvent (elastic collisions between A and S). No such term as diffusion is used to describe this motion since that rather determines what is being sought, hardly the beet approach A diffusive propagator is Gn(r, z) — l3 — Dv ] 1, which converts the probability of a particle being at r at the initial time, sa,y p(r), to that at the Laplace transform parameter value z, say p(r, z), i.e. p(r, z) — Go(r, ) p(r), as may be seen by Laplace transforming the diffusion equation. 

With diffusive propagators, the survival probability of a pair of radicals was found to be... 

The next step is to consider the diffusion motion of electron through a granular metal Diffusion motion inside a single grain is given by the usual ladder diagram that results in the diffusion propagator... 

Fig. 2. These diagrams represent (a) Dyson equation for diffusion propagator, (b) interaction vertex dressed by impurity and intergranular scattering, (c) Screened Coulomb interaction.

The correlation between the structure of the clay dispersions and the water mobility is investigated by the use of the self-diffusion propagator Ps(fA f, 0), i.e. the probability density of finding at time A and position f a diffusing probe initially located at a position r ... 

The ratio Vo/B determines the transition from coherent diffusive propagation of wavefunctions (delocalized states) to the trapping of wavefunctions in random potential fluctuations (localized states). If I > Vo, then the electronic states are extended with large mean free path. By tuning the ratio Vq/B, it is possible to have a continuous transition from extended to localized states in 3D systems, with a critical value for Vq/B. Above this critical value, wave-functions fall off exponentially from site to site and the delocalized states cannot exist any more in the system. The states in band tails are the first to get localized, since these rapidly lose the ability for resonant tunnel transport as the randomness of the disorder potential increases. If Vq/B is just below the critical value, then delocalized states at the band center and localized states in the band tails could coexist. 

Simulations of diffusion propagators based on a finite element method. J. Magn. Res., 161, 138-147. 

Hagslatt, H., Jonsson, B., Nyden, M., and Soderman, O. Predictions of pulsed field gradient NMR echo-decays for molecules diffusing in various restrictive geometries. Simulations of diffusion propagators based on a finite element method, /. Magn. Reson., 161,138, 2003. 

The calculations presented here have simply served to show that a limiting form of the kinetic theory expression for the rate kernel can yield the results of configuration space approaches. However, the real promise of the kinetic theory method lies in the fact that it is not restricted to a description in terms of diffusive propagators, and the consequent motion on these space and time scales. 

This reformulation in terms of diffusive propagation and microscopic dynamics in the boundary layer is reminiscent of Noyes s encounter formulation that we briefly described earlier. Now each diffusive encounter is interrupted by sequences of collisions and very short excursions into the fluid. The analysis of nonhydrodynamic effects on the rate kernel can, therefore, be discussed naturally in terms of the encounter formalism. 

TURQ - It would be relatively easy to replace your simple diffusion propagator by a slightly more raffinated G(r,t) involving structural information such as g(r) at t=0. 

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