Microscopic description of thermal conduction in steady-state

When transport coefficients are first introduced in most thermal physics texts, the physical system that motivates the discussion is a gas of particles. An imaginary plane is placed normal to the gradient of whatever physical property is under discussion (temperature, particle density, etc). Then a kinetic analysis, based on collisions, is performed of the particles crossing from right to left and vice versa. This is done because the particles "carry" the property of interest. The difference in the number of particles crossing between these two directions means there is a net transport and leads to the derivation of the transport coefficient based upon microscopic considerations.

Sorry, I need to write one equation! The equation for thermal conductivity obtained this way is: $$kappa = frac{1}{3}nvcl$$ Where n is the number of molecules per unit volume, v is the average speed of these molecules, c is the specific heat per molecule and l is the mean free path of the molecules between collisions.

This equation is only approximate. As Reif writes in Fundamentals of Statistical and Thermal Physics: "Our calculation has been very simplified and careless about the exact way various quantities ought to be averaged. Hence the factor of $frac{1}{3}$ is not to be trusted too much. On the other hand, the dependence on the [other] parameters... ought to be correct."

In a solid, as opposed to a gas, we can do something similar because after quantizing the lattice vibrations we obtain quasiparticles called phonons. These can be treated like a gas, and as Kittel argues in Introduction to Solid State Physics, we can take this same equation with now the definition of l as the mean free path of phonons, v their average speed, and n and c are combined to give C, the heat capacity per unit volume.

But this requires some further thought. A phonon is really representative of a normal vibrational mode of the solid. It turns out that if the vibrational modes are perfectly harmonic, then they do not interfere with each other - this means the phonons do not collide with each other and the mean free path, l, is only limited by the crystal dimensions. In a real crystal, this is not true. The modes have anharmonic components and this indeed leads to interactions (collisions) between the different modes.

It even gets a little more complicated than this as a distinction between those types of collisions that can actually reverse the direction a phonon is traveling (Umklapp), versus those that can not do this (Normal) must be made. The former are the important collisions to consider as only they can bring about the equilibrium you noted above.

But in essence this is how one looks at simple thermal transport on a microscopic level in a crystal: collisions of quantized lattice vibrations thought of as a gas of quasiparticles.

An actual visualization of this on a microscopic level would be difficult. Lattice vibrations extend over many planes of atoms. Visualization of two different modes interacting would mean visualizing how each vibration mode is distorted by the other vibration mode. But these interactions bring about what is called "local equilibrium". This means the local phonon distribution (what modes are active and how big amplitudes of the modes are) is in a steady state. This local equilibrium is representative of the temperature at that particular point in your rod. And thus the local equilibrium distributions gradually change as you move from the hot to the cold points in your sample.

I guess you are referring to a rod with two extremes having fixed constant temperatures.

Let's try to answer with different kind of microscopic descriptions...

A particle description

Imagine the rod is a "gas" of particles (atoms) along one dimension.

What happens microscopically, if we make this (wrong: see later) assumption, is that atoms of the rod are moving and the energy they have stored in this "motion" is of the order of $sim k_B T$ (where $k_B$ is a constant) i.e. their energy is connected to the temperature $T$: if they move more this implies the rod is hot and vice versa if they move less.

So, now you start heating up your rod at one side: what happens? Atoms become locally more "hot" i.e. they move more. Because they move more, they are also more likely to leave the hot region towards closer colder region. As they move to colder regions, they give up some of their energy by colliding with other atoms, thus locally "warming up" the atoms they hit. Because you are keeping the side of the rod "hot" (by keeping the bondary at a constant temperature) you have a source of "hot" particles which move away and heat up neighboring regions. This goes on along the rod until you reach the other end of the rod, which will be the colder one.

However, you might object that in this view, hot particles should get depleted from the hot side as they travel towards the cold side, until there isn't any left... one answer might be that particles can also randomly go back so that eventually one reaches an equilibrium with hot particles leaving one side being replaced by colder particles coming from the other side which then heat up again and the cycle starts. This would correspond to a temperature gradient. But is this "cycle" really what is going on?

A bit more precise: vibrations

The above would be true if solids consisted of point particles.

However, solids are not composed of free particles but of fixed atoms in a lattice and temperature does not correspond to atoms "moving around" but rather to atoms "vibrating". What actually travels along the rod are not particles, as in a gas, but vibrations! Atoms at the hot side vibrate a lot and transmit their vibration to neighboring atoms: the vibration propagates towards the cold side of the rod getting weaker and weaker. You can imagine your rod as being constituted by a set of masses along a line connected by springs. If the first mass starts vibrating a lot, the vibration will propagate and dampen along the rod. The flow of the vibration is what we call heat flow.

Fourier equation simply describes this process of mathematically and tells you that the solution for the problem "what happens to a set of masses and springs [=atoms in a lattice] if I fix the initial (max) [hot] and the final (min) [cold] vibration [=temperature]?" is that at equilibrium each spring will vibrate with a frequency ("temperature" in our case) lineraly decreasing from the maximum to minimum one at the extremes.

Quasi - particles

Because Fourier equation is very similar to the diffusion equation for particles, the two points of view I described (particles vs vibration) above can be merged into a single one, that of the "quasi-particle", i.e. objects that mathematically behave as particles diffusing, making computing results easier and sometimes also leading to more intuitive explanations. Read CGS's answer for more on this: quasi-particles can indeed move, flip direction, appear, disappear... they are very weird!

However the "real" behavior of solids is that of "vibrations" propagating (neglecting quantum effects and a lot of other things, of course).