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Microscopic limits of PDEs modeling macroscopic heat conduction

Ronald Mickens, Talitha Washington

Abstract

We show that it is possible to construct microscopic-level discrete equations from macroscopic modeling PDEs for heat conduction in one space dimension. The significance of this result is that, in general, one starts from microscopic theories and then take their continuum limits to obtain the corresponding macroscopic PDEs, whereas here it is demonstrated that the reverse procedure is also possible. While our focus is on heat conduction, we discuss the applicability of our methodology to other physical systems.

Microscopic limits of PDEs modeling macroscopic heat conduction

Abstract

We show that it is possible to construct microscopic-level discrete equations from macroscopic modeling PDEs for heat conduction in one space dimension. The significance of this result is that, in general, one starts from microscopic theories and then take their continuum limits to obtain the corresponding macroscopic PDEs, whereas here it is demonstrated that the reverse procedure is also possible. While our focus is on heat conduction, we discuss the applicability of our methodology to other physical systems.

Paper Structure

This paper contains 37 equations, 1 figure.

Figures (1)

  • Figure 1: Connections between the micro-level (MIL) and macro-level (MAL) mathematical models for a physical system. The corresponding equations are, respectively, discrete (DE) or Partial differential equations (PDEs). The arrow indicate the methodologies needed to transition between MIL and MAL equations.