KCM: Normal scattering mechanism and random walk

The microscopic description of diffusion is the brownian motion. In the case of thermal transport the moving particles are the phonons. In the difusive heat transport a phonon moves freely through the material until it collides with a scattering center or another particle (electrons, phonons...). After the collision, a new phonon gets out in a new randomly chosen direction. The distance that the phonon travels without experiencing any collision is called mean free path, and the time that it takes, mean free time.

In the kinetic regime the mean free path depends on the frequency of the phonon. This means that heat transport can be understood as a combination of independent brownian particles each one having a different mean free path. In the left image of Figure 1 it can be seen the trajectories of two different phonons in the kinetic regime. Notice that each one has a different step length.

Figure 1: Random walk of two phonons of different frequency in the kinetic regime (left) where each one has a different
mean free path and in the collective regime (right) where both phonons has the same mean free paths

In the collective regime a change in the phonon behaviour arises because of the presence of the normal scattering (N-scattering). N-scattering is not resistive, and when a two phonons collide, the outcoming phonon has a momentum equal to the sum of the incoming phonons. This prevent the distribution function of the phonon to be relaxed to that of equilibrium. The consequence is that the mean free paths of the phonons tend to homogenize, that is, long mean free paths are reduced and short mean free path are enlarged. In the limit when N-scattering is dominant all the mean free paths are made equal. In figure 1 right it can be observed that both phonons has the same scale length.  

To calculate the kinetic and collective mean free path we only need the times for the different resistive scattering mechanism $\tau_{Ri}$ but not that of the normal scattering. In the kinetic regime, the total mean free time can be obtained using the Mathiessen rule with these resistive times

In the collective regime the relaxation time is obtained by


where Cw is the specific heat of mode w.

The calculation of the collective mean free time is the one expected when every single phonon notices the effect of the collisions from all the rest of the phonons. Notice that the nature of collective mean free path is also only resistive. Normal scattering does not enter in the calculation. Normal scattering is just the mechanism to distribute the energy and its effect is just to create a drag that reduces the mean free path of the long phonons and enlarge the mean free path of the short phonons.

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