Definite matrix: Difference between revisions

Revision as of 19:47, 13 February 2025

Revision as of 20:03, 13 February 2025

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== Applications ==

=== Heat conductivity matrix ===

Fourier's law of heat conduction, giving heat flux <math>\mathbf{q}</math> in terms of the temperature gradient <math>\mathbf{g} = \nabla T</math> is written for anisotropic media as <math>\mathbf{q} = -K \mathbf{g},</math> in which <math>K</math> is the symmetric [[thermal conductivity]] matrix. The negative is inserted in Fourier's law to reflect the expectation that heat will always flow from hot to cold. In other words, since the temperature gradient <math>\mathbf{g}</math> always points from cold to hot, the heat flux <math>\mathbf{q}</math> is expected to have a negative inner product with <math>\mathbf{g}</math> so that <math>\mathbf{q}^\top \mathbf{g} < 0 ~.</math> Substituting Fourier's law then gives this expectation as <math>\mathbf{g}^\top K\mathbf{g} > 0,</math> implying that the conductivity matrix should be positive definite. Ordinarily <math>K</math> should be symmetric, however it becomes nonsymmetric in the presence of a magnetic field as in a [[thermal Hall effect]].

More generally in thermodynamics, the flow of heat and particles is a fully coupled system as described by the [[Onsager reciprocal relations]], and the coupling matrix is required to be positive semi-definite (possibly non-symmetric) in order that entropy production be nonnegative.

== See also ==