3.21 Lecture 1 Spring 2006: Fields and Gradients; Fluxes; Continuity Equation
Thermodynamics is precise about equilibrium states, but real materials are raely at equilibium
The concept of local equilibrium is applicable to real materials on a micro-scale.
The rates of approach to equilibrium in real materials are found experimentally to depend on gradients of thermodynamic potentials
A scalar field associates a physical quantity with position--e.g. a composition field
A gradient of a scalar field is a vector that quantifies how rapidly the field changes with position
The flux of a substance quantifies the rate at which that substance flows through a unit area. The flux is a vector that is parallel to the local direction of the flow.
The rate of accumulation of an extensive quantity is minus the divergence of the flux of that quantity plus the rate of production of the substance.
For conserved quantities like the number of moles of a complonent in a solution there are no sources or sinks and hence no production of the substance.
For non-conserved quantities like entropy there can be production of the quantity during the course of a spontaneous process.
Numerous kinetic processes are described by linear equations relating fluxes and driving forces.
Methods from linear algebra are often used to simplify the description by using coordinates parallel to crystal axes, or by finding the prinicpal axes.
Matrix eigenvalues, eigenvectors, and similarity transformations are useful tools for describing coupled kinetic processes.