Variable-order modeling of nonlocal emergence in many-body systems: Application to radiative dispersion

Abstract

We describe a physical framework for analyzing the spectral dynamics of a broad range of media. The framework is built on a variable-order calculus formalism that permits the description of temporally nonlocal behavior. Such emergent behavior is observed in the response of assorted complex media. The analytical features of the formalism are discussed and it is demonstrated how they correspond to the generalization of other well known theories for the description of nonlocal many-body effects. The framework is employed to analyze a set of spectroscopic data for the high-frequency dielectric response of a nanofluidic graphene dispersion and the midinfrared optical response of amorphous quartz silica. A practical application of the analysis is facilitated by a model definition that generalizes the semiclassical Lorentz theory to allow for nonlocal damping effects. The model is derived from a fractional order differential equation of motion. From the analysis, an estimated parametrization for the model structure is obtained. The fidelity of the analysis methodology is validated against optimized parametrizations in a multiobjective (optical and radiative) setting. The results demonstrate the utility of the analysis and indicate a specific well-defined region of nonlocality having a distinct fidelity that encompasses the entire Pareto front. This region is shown to be inaccessible to integer order descriptions of the mean field dynamics.

Temperature-dependent carrier transport: Low-complexity model for the infrared optical and radiative properties of nickel

“…The main result of this work is the seven-degrees-offreedom model obtained by combining (15), (29), and (30) with the temperature-independent parameters from Tbl. I. Coupling this result with the relevant theory from Sec.II A then yields a suite of closed-form expressions that provide a full characterization of the temperaturedependent radiative and optical responses of nickel over the infrared band λ ∈ [1,16] µm and for temperatures over the approximate range T ∈ [0, 1400] K. Examples of closed-form, temperature-dependent spectral emissivity models for nickel are sparsely represented in the literature.…”

“…As the temperature is increased, the primary effect of the temperature-dependent lattice distortion is to apply a smoothing effect about the corner-wavelength that leads, at higher temperatures, to a semiparabolic spectral reflectivity profile. Included in the plot are experimental values for the spectral normal reflectivity computed from the ambient temperature (we assume T = 294 K) measurements of Ordal et al, 44 , (29), and (30) with the optical and radiative relations in Eqs. (19) and (25).…”

“…The energy partitioning property of nonlocal operators in a stationary frequency domain setting has been rigorously investigated in our recent study. 29 Decomposing the proposed model (12) into its real and imaginary components χ φ = χ φ + i χ φ and appealing to the relation ε = 1 + χ = 1 + χ φ + χ β , 30 where the χ β are the contributions from bound states, one has for the temperature-dependent relative permittivity…”

“…In order to provide a basis for the analysis and interpretation of non-integer order differential dynamics in a stationary frequency domain setting, we have recently described a generalized spectroscopic analysis framework. 29 The framework is derived under the formalism of variable order calculus. It allows one to express the linear response of a given system as a variable order transfer function.…”

“…The surface is generated by combining the proposed seven-degrees-of-freedom model in Eqs. (15) - (17), (29), and (30) with the optical and radiative relations in Eqs. (19), (23), and (25).…”