Fourier heat theorem

Here Dt is a positive proportionality constant ( diffusion constant for Et), Jfz is z-ward flow induced by the gradient, and superscript e denotes eigenmodt character of the associated force or flow. The proportionality (13.25) corresponds to Fick s first law of diffusion when Et is dominated by mass transport or to Fourier s heat theorem when Et is dominated by heat transport, but it applies here more deeply to the metric eigenvalues that control all transport phenomena. In the near-equilibrium limit (13.25), the local entropy production rate (13.24) is evaluated as... 

Any periodic function (such as the electron density in a crystal which repeats from unit cell to unit cell) can be represented as the sum of cosine (and sine) functions of appropriate amplitudes, phases, and periodicities (frequencies). This theorem was introduced in 1807 by Baron Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist who pioneered, as a result of his interest in a mathematical theory of heat conduction, the representation of periodic functions by trigonometric series. Fourier showed that a continuous periodic function can be described in terms of the simpler component cosine (or sine) functions (a Fourier series). A Fourier analysis is the mathematical process of dissecting a periodic function into its simpler component cosine waves, thus showing how the periodic function might have been been put together. A simple... 

Figure 17.1 A simple thermal gradient maintained by a constant flow of heat. In the stationary state, the entropy current Js,out — diS/dt + The stationary state can be obtained either as a solution of the Fourier equation for heat conduction or by using the theorem of minimum entropy production. Both lead to a temperature T(x) that is a linear function of the position x...

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