Higher-dimensional transport and diffusion equations, encompassing rigorous diffusion limits, multidimensional diffusion theory, advanced numerical and particle-based simulation methods, and fractional/telegraphic generalizations, provide powerful tools for modeling, analyzing, and simulating complex transport phenomena across a wide range of scientific and engineering fields.

Higher-dimensional transport and diffusion equations include rigorous diffusion limits of transport equations, multidimensional diffusion theory with stochastic and PDE tools, advanced numerical and particle-based simulation methods, fractional and telegraphic transport generalizations, and applications to geophysical and engineered systems. These frameworks provide powerful tools to model, analyze, and simulate complex transport phenomena across many scientific and engineering fields.

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Significant manifestoes in the applications of higher-dimensional equations among transport processes and diffusions can be summarized as follows:

1. Diffusion Approximation of Transport Equations

• Transport equations, often describing particle or individual motion with velocity changes, can be approximated by diffusion equations under suitable scaling limits. This diffusion limit emerges when many velocity jumps occur in a small time interval but with small net displacement, leading to a simplified advection-diffusion or pure diffusion equation in higher dimensions.
• The diffusion approximation is rigorously justified using functional analysis, semigroup theory, and variational formulations, providing existence and uniqueness of solutions in appropriate function spaces.
• Different scalings of parameters controlling jump lengths and waiting times lead to different limit equations: drift-dominated transport, diffusion-dominated, or mixed advection-diffusion regimes.

2. Multi-dimensional Diffusion Theory

• Higher-dimensional diffusion processes are studied for their attainability (whether certain states or hyperplanes can be reached), ergodicity (long-term statistical behavior), and rankings (ordering of diffusive particles), with applications in interacting particle systems and financial models such as the Atlas model of equity markets.
• The theory connects elliptic partial differential equations and stochastic calculus, addressing reflection conditions and uniqueness of diffusions in multidimensional domains.

3. Numerical and Modeling Advances for Advection-Diffusion PDEs

• Advection-diffusion partial differential equations (PDEs) model transport of chemicals or heat in fluid media. High-dimensional PDEs pose computational challenges, which are addressed by model order reduction, Bayesian inference, and distributed robotic sensing to estimate uncertain parameters and fields.
• These approaches enable practical applications such as mobile robots monitoring chemical propagation, demonstrating the relevance of higher-dimensional transport-diffusion models in engineering.

4. Mesoscopic Simulation Models

• Transport dissipative particle dynamics (tDPD) models extend classical particle methods by incorporating concentration fields and modeling advection-diffusion-reaction processes at mesoscopic scales. These models handle boundary conditions accurately and relate particle parameters to effective diffusion coefficients.

5. Fractional and Telegraphic Transport Processes

• Extensions of classical diffusion models include telegraphic transport processes and their fractional generalizations, which better capture finite propagation speeds and anomalous diffusion phenomena in multiple dimensions. These models provide more realistic descriptions of transport in complex media by going beyond the classical diffusion equation.

6. Variational and Projection Methods for Transport-Diffusion Coupling

• Variational formulations and Galerkin projection methods allow coupling of full transport equations with diffusion approximations in heterogeneous media, enabling accurate modeling of regions where diffusion is valid and others where transport effects dominate.