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Physical Review A

Energy transfer by diffusion and the dipole-dipole interaction: A new perturbation expansion

Paolo Sibani, Jørgen Boiden Pedersen

Published Mar 1, 1984• 9 citations

Abstract

The transfer of electronic energy by diffusion and by the dipole-dipole interaction in solids and liquids is modeled by a diffusion equation with an ${r}^{\ensuremath{-}6}$ sink term plus a boundary condition at the distance of closest approach. The boundary condition is either absorptive, giving rise to diffusive transfer, or reflective, in which case the transfer is direct but modulated by diffusion. We solve the equation by a perturbation expansion in which the expansion parameter is inverse time, i.e., the $t\ensuremath{\rightarrow}\ensuremath{\infty}$ limit (the steady-state value) is reproduced exactly. The perturbation method gives accurate results for physically relevant values of the characteristic parameters and simple closed-form expressions for the time-dependent rate coefficient and the survival probability of an isolated pair are obtained for the first time when the diffusional transfer is larger than the direct transfer.

DOI: 10.1103/physreva.29.1403

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The transfer of electronic energy by diffusion and by the dipole-dipole interaction in solids and liquids is modeled by a diffusion equation with an

{r}^{\ensuremath{-}6}

sink term plus a boundary condition at the distance of closest approach. The boundary condition is either absorptive, giving rise to diffusive transfer, or reflective, in which case the transfer is direct but modulated by diffusion. We solve the equation by a perturbation expansion in which the expansion parameter is inverse time, i.e., the

t\ensuremath{\rightarrow}\ensuremath{\infty}

limit (the steady-state value) is reproduced exactly. The perturbation method gives accurate results for physically relevant values of the characteristic parameters and simple closed-form expressions for the time-dependent rate coefficient and the survival probability of an isolated pair are obtained for the first time when the diffusional transfer is larger than the direct transfer.

The four-state random walk (4RW) model, wherein the particle is endowed with two states of spin and two states of directional motion in each space coordinate, permits a stochastic solution of the Schrödinger equation (or the equivalent parabolic equation) without resorting to the usual analytical continuation in complex space of the particle trajectories. Analytical expressions are derived here for the various transitional probabilities in a 4RW by employing generating functions and eigenfunction expansions when the particle moves on a 1+1 space-time lattice with two-point boundary conditions. The most general case of dissimilar boundaries with partially reflecting boundary conditions is treated in this paper. The transitional probabilities are all expressed in terms of a finite summation involving trigonometric functions and/or Chebyshev polynomials of the second kind that are characteristics of diffusion and Schrödinger equations, respectively, in the 4RW model. Results for the special case of perfectly absorbing boundaries are compared to numerical values obtained by directly counting paths in the random walk simulations.

Using the transfer matrix method for Axial Next-Nearest Neighbor Ising model without an external field on a closed chain of spins of width 2 in the direction of interaction of only nearest neighbors and length L in the direction of interaction of nearest neighbors and next nearest neighbors, we found exact values of the partition function in a finite strip of length L , free energy, internal energy per node, heat capacity in a finite strip of length L and in the thermodynamic limit. The entire spectrum of the 16x16 transfer matrix and the structure of all transfer matrix eigenvectors are found. The problem of finding the spectrum of the transfer matrix is reduced to finding the spectrum of two matrixes of size 8x8. The characteristic polynomial of the first matrix is decomposed into two linear polynomials of multiplicity two and two square trinomials. The characteristic polynomial of the second matrix is decomposed into two polynomials of the fourth degree, the roots of which are found by the Ferrari method. The free energy in the thermodynamic limit has a very simple form and is expressed in terms of the logarithm of the root of the quadratic equation. Two theorems on the physical characteristics of the ANNNI model in a finite closed strip of length L and in the thermodynamic limit are formulated and proved. An example of calculation is given for some values of the model parameters in the thermodynamic limit, heat capacity graphs are plotted. A comparative analysis of the heat capacity for the double-stranded ANNNI model and the ANNNI model on the entire flat lattice is given. The closest connection and similarity of the physical characteristics of the two-chain model and the model on the entire flat lattice is shown.

We introduce and solve from first principles a continuous-time quantum walk with absorption generated by a Lindblad boundary sink of arbitrary strength. Projecting onto the surviving walker sector maps the problem to a non-Hermitian tight-binding Hamiltonian with a rank-one imaginary defect on the semi-infinite line. We obtain closed-form expressions for the exact propagator, from which the first-passage statistics follow. Weak coupling limits absorption through inefficient transfer into the sink, whereas for strong dissipation, occupation at the boundary is suppressed by Zeno-like reflection, accompanied by the emergence of a localized non-Hermitian mode. Despite the different physical origins of these suppression mechanisms, their respective absorption probabilities exhibit an asymptotic duality. The evolution is conveniently visualized in phase space, where the localized mode has an exponentially confined Wigner representation near the edge site.