Heat Conduction in Solids as Radiative Heat Transfer

What is the physics of heat conduction in a solid like a metal? The trivial story is that "heat flows from warm to cold" or "there is a flux of heat from warm to cold" which scales with the temperature difference or gradient. 

But heat is not a substance like water in a river flowing from high-altitude/warm to low-altitude/cold, which connects to the caloric theory and also to phlogiston theory presenting fire as form of substance, both debunked at the end of the 18th century. 

In any case there is a law of physics named Fourier's Law:

• $q =- \nabla u$          (F)

which combined with a law of conservation 

• $\nabla\cdot q = f$

leads to the following heat equation (here in stationary state for simplicity) in the form of Poisson's equation

• $-\Delta u = f$.                    (H)

where $u(x)$ is temperature and $f(x)$ heat source depending on a space variable $x$, and $q(x)$ is named "heat flux" although it has no physical meaning; heat is not any substance which flows or is in a state of flux. 

Let us now seek the physics of (F) in the case of metallic body as a lattice of atoms, and so seek an explanation of the observation that the temperature distribution $u(x)$ of the body tends to an equilibrium state with $u(x)=U$ with $U$ a constant (assuming no interaction with the surrounding and no internal heating for simplicity). 

We thus ask: 

• What is the physics of the process towards equilibrium with constant temperature?
• How is heat transferred from warm to cold?
• Why is (F) valid?

We then recall our analysis of radiative transfer of energy at distance in a system of bodies/parts separated in space which (without external forcing) leads to an equilibrium state with all bodies having the same temperature, based on the following physical model:

• Each body is a vibrating lattice of atoms described by a wave equation with small radiative damping. 
• The bodies interact by electromagnetic waves through resonance. 
• There is a high-frequency cut-off increasing linearly with temperature with the effect that heat transfer mediated by electromagnetic waves between two bodies, is one-way from high temperature to low temperature. 

The key is here the high-frequency cut-off increasing with temperature, which makes heat transfer one-way. The cut-off can be seen as a form of finite precision threshold allowing coordinated lattice vibration only below cut-off, thus allowing a high temperature lattice to carry higher frequencies. It is like a warmed-up opera soprano being able to reach higher frequencies.

We can view metallic body as a system composed of parts/atoms interacting by 

electromagnetic waves at small distance. 

Heat conduction will then come out as a special case of electromagnetic heat transfer 

between atoms of different temperature with high-frequency cut-off guaranteeing one-way transport as expressed by (F) and exposed above. 

Note that the reference text Conduction of Heat in Solids by Carslaw and Jaeger presents (F) as an ad hoc physical law without physics.

Recall that the standard explanation of radiative heat transfer from warm to cold is based on statistics without physics, which if used to explain heat conduction would again invoke statistics without physics thus not very convincing. 

Also note that the standard explanation of heat transfer in a gas involves collisions of molecules of different kinetic energy, which is not applicable to a metal with atoms in a lattice.

PS1 Fluid flow in a river from higher to lower altitude is driven by pressure. "Heat flow" from warm to cold is not driven by pressure and so the physics is different. 

PS2 Also compare with one-way osmotic transport of material driven by pressure.