Applications of regime-switching in the nonlinear double-diffusivity (D-D) model

Diffusive mixing of multiple phases is a well-studied topic, particularly in the realm of microfluidics.^1,2 While these studies invariably pointed to the dynamical evolution of an interface characterized by properties of the individual and correlated phases,³ the precise dependence of parameters in this interface growth process had to wait until Aifantis produced a sequence of groundbreaking studies to analytically solve the double-diffusion problem.^4,5 Their “trick” lay in unentangling the coupled diffusion of two phases, “diffusivity paths” in the material science parlance, thereby analyzing their relative growth through a study of the emergent interface dynamics.^6,7 The problem has been revisited in recent times by Lux to understand the maximum likelihood probability of such events where the forward Kolmogorov model was analyzed using a combination of Fourier and Laplace transforms.^8,9 While the Lux method employs the full arsenal of a dynamically evolving probability density function by numerically solving the Fokker-Planck model, it is largely limited to understanding linear diffusive mixing much like the original Aifantis models. The Lux method also lacks the simplicity of the Aifantis description that actually allows a detailed parameterization of the multi-diffusive process.

In a recent work,¹⁰ a theoretical framework was established to approximate a closed-form solution of the ensemble-averaged density profiles and correlation statistics of coupled nonlinear double-diffusive processes, starting from the paradigmatic (linear) Aifantis double-diffusivity (D-D) model,^4–7 and extended to the nonlinear regime in analogy to the paradigmatic Walgraef–Aifantis model.^11–14 Later studies focused on the stochastically driven versions of the D-D and W–A models,^15–17 which established the importance of understanding fluctuation effects in such dynamics.

Over the last few decades, the D-D model has become quite popular in both the applied mathematics and the material science communities due to its interesting mathematical properties of the former^18–20 that allow for robust interpretation of experimental observations for the latter.^21–23 While there is no paucity of numerical estimation of nonlinear models, including those for double diffusion, from the perspective of theoretical modeling, analytical clarity had to give way to quantitative precision.

The present article is the first in line to provide a closed-form approximate solution of the nonlinear model resulting from a combination of the D-D and W–A type models. The D-D:W–A motivated model leads to a coupled system of reaction–diffusion (R–D) equations where the diffusion terms resemble those contained in these models; the linear terms as in the D-D model; while the nonlinear terms resemble the W–A model (third order), but contain, in addition, cross-coupled terms (second order).

While examples of numerically analyzed reaction–diffusion models are available aplenty, there is little or no understanding of the parametric dependence of these processes and the nature of regime-switching transitions they may initiate. While D-D models, both linear and nonlinear, have been extensively studied computationally, studies on dynamical phase transitions of these models are few and far between, barring exceptions.⁹ This is crucial for analyzing movements between states, as well as to comprehend how one (diffusive) state may affect others, an approach that we recently studied from a complementary perspective.¹⁰ 

There is another dimension to the present work. While the present article studies two nonlinearly cross-coupled diffusive processes and how the diffusivities can be extracted exactly from a combination of forward Fourier transform (FT), followed by Laplace transform (LT), and then an inverse Fourier transform (IFT), the mathematical foundation is generic enough to accommodate the more realistic case of forced (e.g., stochastic forcing) nonlinear D-D models as in Refs. 15 and 16. Broadly speaking, the framework will allow us to analyze diffusion equations with nonlinearity,²⁴ both forced and equilibrium models. Also, free boundary problems such as the distribution of temperature in a homogeneous material during phase-transition²⁵ can be studied, i.e., the time evolution of the phase boundaries, the so-called Stefan problem.

The other application of this framework could be in traveling wave and pattern formation which can be mapped onto D-D models, such as the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) model for the propagation of an advantageous gene in a population^26,27 and Gray–Scott model for diffusion of chemical species.^27,28 Similar studies can be found in the aforementioned W–A model of dislocation patterning in cyclically deforming metals, which have been extensively studied numerically in the post-bifurcation regime.^12,13 None of these was designed for closed-form analysis of nonlinearly coupled time-evolving diffusivity models, as is the target of this study.

The plan of the paper is as follows: Sec. II summarizes the model equation and provides a physical explanation of the mechanisms involved. Section III represents the non-dimensionalized representation of the model, the relevant non-dimensional governing equations, and their linear stability analysis. Section IV first reinvents the Lux maximal-likelihood definition of regime-switching diffusions,^8,9 then superimposes the D-D methodology⁷ onto the Lux maximum-likelihood approach, and finally uses this hypothesis to solve for the densities of the phases around the stable fixed point as eigenvalue solutions. Section V summarizes the outcomes of the Chattopadhyay–Aifantis–Lux (CAL) model and discusses its wider implications.

The starting point of this work is the nonlinear double diffusion (D-D) model that was recently analyzed to calculate the transition between two diffusive states, the so-called “transmissbility” model. In Ref. 10, the emphasis was on arriving at an approximate formulation of the overlap strength of diffusive state “1” on that of diffusive state “2,” estimated from the correlation strength of the dynamical process that was analyzed from a mapping of the technique used to calculate Reproduction number is epidemic models.^29–31 While being able to capture the cross correlation of one diffusive process with the other, the approach had a key limitation in that it could not provide a measure of the regime-switching processes between the two diffusive systems. The present work attempts to understand this mechanism using a modified Fourier Transform method in the light of the Lux formalism.⁹ 

The approach adopted in this Chattopadhyay–Aifantis–Lux (CAL) method has advantages over the one from Lux and the original W–A model. First, the solution presented in this article applies to the nonlinear D-D model close to the stable fixed points of the linear D-D model that is already beyond the Lux prescription which is tailored for the linear model. Second, the nonlinearity considered here is more general than in the W–A model as it contains all forms of (non-conserved) cross couplings up to the third order. The highest (cubic) nonlinearity considered here generalizes the Ginzburg–Landau (G–L) formulation for phase transition. Thus, our model accommodates not only amplitude instabilities (as in G–L) but also phase instabilities. Moreover, combining Eqs. (20) and (21) with Eq. (23), we have a complete solution of the densities as functions of space and time, together with an ODE-based formulation as shown in Eqs. (22a) and (22b). This needs to be numerically evaluated as shown in Figs. 1 and 2. The plots have been drawn for the steady state using some values from Ref. 21, ⁠, ⁠, ⁠, and ⁠. Values of the cross-coupling parameters have been dimensionally matched against linear parameters and typically considered at two orders of magnitude lower than the linear parameter: ⁠, ⁠, ⁠. The initial conditions too have been value matched against linear solutions: and ⁠. The solutions are highly dependent on initial conditions and especially on the nonlinearity parameters ⁠. The solution in Eq. (23) can also be used to evaluate the spatial evolution of the density functions in the large time (steady state) limit.

The Lux method⁸ offers similar solutions in terms of the Bessel function for a specific choice of initial conditions. Ours is a more generic formulation that applies to cross-coupled diffusivity models.

The model and results presented here have been drawn from a unique amalgam of the original approaches by Aifantis^4,5 and Lux.^8,9 Multiple diffusion of species is not a new topic and is key to understanding the dynamics of interacting active systems, both in continuum systems with excluded-volume effects included³² and in discrete molecular systems.³³ The challenge though is solving such models with nonlinear interactions included. This is where our novel technique can be a major savior. This combination ensures a methodological novelty whereby the Maximum likelihood description can be extended to accommodate nonlinearly coupled diffusive systems, potentially including reaction–diffusion systems. This has key implications for a variety of inhomogeneous fields where mass transport is a governing process. Since the motivation of our model comes from inhomogeneous diffusion in materials¹⁶ and disease spreading^10,31 in epidemiology, the discussion here will be directed toward the financial sector. But before expanding on that topic, we first provide some concluding remarks on the mathematical precept and comparisons between analytical with numerical solutions from the perspective of evolutionary biology and epidemiology.

Clearly, a comparison of the dynamical variable ⁠, motivated by the epidemiological literature, with the autocorrelation function reveals the richness of the dynamics of a reaction–diffusion system, which offers an option of interpolating the results from the epidemic model into the double-diffusion domain, in the process providing a closed-form solution of the latter that has remained elusive thus far. Comparing the time evolution of with the autocorrelation function gives information on the origin of the observed abundance of different species in a reaction–diffusion system; for this, we need to numerically solve coupled equations (20) and (21) and Eq. (23) typically at the point of symmetry (⁠⁠).

Therefore, the introduction of the epidemiologically motivated quantity into the studies of the reaction–diffusion systems can play a crucial role in understanding such systems in more depth. Since this interpolation between two unrelated disciplines only uses the mathematical similarity between two (or multiple) reaction–diffusion species, expressed as double-diffusion in material science, as compared to infection rate growth in epidemiology, the approach is generic enough to be applied to all coupled reaction–diffusion models. At the point of symmetry (⁠ in our model), both quantities (⁠ and autocorrelation) will asymptotically match their values with evolving time allowing for a closed-form mapped (from mathematical biology) solution of the R–D model. As a comparison with the numerical solution confirms close convergence with the approximate mapped solution [based on the formula as a descriptor of the correlation strength of the diffusing variables], the solution provides a handle to studies analyzing higher-order perturbations and relevant bifurcations, also including stochastic terms. Future studies involving spatially forced multi-diffusion and reaction–diffusion models will be considered, particularly with reference to econometric systems.⁹ 

The authors gratefully acknowledge partial financial support from the H2020-MSCA-RISE-2016 program, Grant No. 734485, entitled “Fracture Across Scales and Materials, Processes and Disciplines (FRAMED).” A.K.C. acknowledges insightful comments from A. Konstantinidis.

The authors have no conflicts to disclose.

Amit K. Chattopadhyay: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Elias C. Aifantis: Conceptualization (equal); Funding acquisition (lead); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.