Multiple scattering, radiative transfer, and parameter regimes

Abstract

We have established the radiative transfer regime in a four-parameter space spanned by amplitude and scale of fluctuations, convergence rate, and spatial range. The derivation of radiative transfer equations from wave equations offers a better understanding of the underlying physics in radiative transfer theory. One approach is based on tools borrowed from quantum field theory (QFT) and employs the Dyson and Bethe–Salpeter equations as the foundation. This has allowed us to investigate the small scale fluctuation regime, $\mathrm{kl}=\mathcal{O}(\epsilon )$. We also investigated this regime using the scale-separated asymptotics approach and found that the conditions on the parameters are the same as in the QFT approach if the convergence rate of the Dyson series is $\mathcal{O}({\epsilon }^2)$. Furthermore, we found that there is a multitude of regimes where radiative transfer manifests. We hence infer that for a fixed spatial range, increase in the scale size of fluctuations leads to decrease in the upper bound of variance of permittivity fluctuations along with diminished convergence rate of the Wigner series. Also, for a fixed scale size and convergence rate, the spatial range where radiative transport manifests decreases with increase in strength of fluctuations.

Keywords:

• Parameter regimes
• radiative transfer
• multiple scattering
• random media

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Substitute (7(7) $P({\mathit{\rho }}_1,{\mathit{\rho }}_2;{\mathit{\rho }}_3,{\mathit{\rho }}_4)\simeq C({\mathit{\rho }}_1-{\mathit{\rho }}_2)\delta ({\mathit{\rho }}_1-{\mathit{\rho }}_3)\delta ({\mathit{\rho }}_2-{\mathit{\rho }}_4)$(7) ) in (6(6) $\begin{array}{rl}& M(\mathbf{R},\mathbf{R}+{\mathbf{r}}^{\prime })\delta (\mathbf{R}-{\mathbf{R}}^{\prime }+{\mathbf{r}}^{\prime }/2)-M^{\ast }(\mathbf{R},\mathbf{R}-{\mathbf{r}}^{\prime })\delta (\mathbf{R}-{\mathbf{R}}^{\prime }-{\mathbf{r}}^{\prime }/2)\\ & =\int G_m(\mathbf{R},\mathit{\rho })P(\mathit{\rho },\mathbf{R};{\mathbf{R}}^{\prime }+{\mathbf{r}}^{\prime }/2,{\mathbf{R}}^{\prime }-{\mathbf{r}}^{\prime }/2)\mathrm{d}\mathit{\rho }\\ & -\int G_m^{\ast }(\mathbf{R},\mathit{\rho })P(\mathbf{R},\mathit{\rho };{\mathbf{R}}^{\prime }+{\mathbf{r}}^{\prime }/2,{\mathbf{R}}^{\prime }-{\mathbf{r}}^{\prime }/2)\mathrm{d}\mathit{\rho }\end{array}$(6) ) to arrive at (8(8) $M({\mathbf{r}}_1,{\mathbf{r}}_2)\simeq C({\mathbf{r}}_1-{\mathbf{r}}_2)G_m({\mathbf{r}}_1,{\mathbf{r}}_2)$(8) )

2 ${\hat{D}}_1$ and ${\hat{D}}_2$ are Dyson operators (cf. (3(3) $\hat{D}{G}_m=-\delta (\mathbf{r}-{\mathbf{r}}^{\prime })$(3) )) corresponding to spatial variables ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, respectively.

3 We employ this procedure to systematically estimate the error involved in making the quasi-homogeneity assumption.

4 Since the optical theorem is an exact identity involving the mass operator and the intensity operator, it follows from (6(6) $\begin{array}{rl}& M(\mathbf{R},\mathbf{R}+{\mathbf{r}}^{\prime })\delta (\mathbf{R}-{\mathbf{R}}^{\prime }+{\mathbf{r}}^{\prime }/2)-M^{\ast }(\mathbf{R},\mathbf{R}-{\mathbf{r}}^{\prime })\delta (\mathbf{R}-{\mathbf{R}}^{\prime }-{\mathbf{r}}^{\prime }/2)\\ & =\int G_m(\mathbf{R},\mathit{\rho })P(\mathit{\rho },\mathbf{R};{\mathbf{R}}^{\prime }+{\mathbf{r}}^{\prime }/2,{\mathbf{R}}^{\prime }-{\mathbf{r}}^{\prime }/2)\mathrm{d}\mathit{\rho }\\ & -\int G_m^{\ast }(\mathbf{R},\mathit{\rho })P(\mathbf{R},\mathit{\rho };{\mathbf{R}}^{\prime }+{\mathbf{r}}^{\prime }/2,{\mathbf{R}}^{\prime }-{\mathbf{r}}^{\prime }/2)\mathrm{d}\mathit{\rho }\end{array}$(6) ), (7(7) $P({\mathit{\rho }}_1,{\mathit{\rho }}_2;{\mathit{\rho }}_3,{\mathit{\rho }}_4)\simeq C({\mathit{\rho }}_1-{\mathit{\rho }}_2)\delta ({\mathit{\rho }}_1-{\mathit{\rho }}_3)\delta ({\mathit{\rho }}_2-{\mathit{\rho }}_4)$(7) ), and (8(8) $M({\mathbf{r}}_1,{\mathbf{r}}_2)\simeq C({\mathbf{r}}_1-{\mathbf{r}}_2)G_m({\mathbf{r}}_1,{\mathbf{r}}_2)$(8) ) that the validity domain of the Kraichnan approximation to the mass operator will establish the validity domain of the ladder approximation of the intensity operator.

5 There is nothing special about the exponential correlation used here. We could have picked any other correction function. The reason for selecting one of them is to carry out explicit calculation for parameter estimation.

6 studied using the QFT approach

Additional information

Funding

This work was supported by Air Force Office of Scientific Research[].