# How does quantum statistical mechanics explain thermal conductivity?

I am unsure how statistical mechanics can be used to explain the dynamics of so called non-mechanical properties, e.g. Temperature and Entropy (as opposed to say pressure or spatial diffusion).

In particular I was wondering how one can understand thermal conductivity within the framework of statistical mechanics. My confusion is that whilst something like the diffusion coefficient can be related to a velocity-velocity autocorrelation function at the temperature of interest, the thermal conductivity seems to be about changes in a quantity which I understand to be a property of the distribution as a whole.

This is related to other questions that I have found about fluctuations in free-energy, Fluctuations of free energy in quantum statistical mechanics, but this seemed to conclude that it was a meaningless question, which is clearly not true for thermal conductivity since it is a well established phenomenon.

Thermal conductivity controls diffusive transport of energy, and is not fundamentally different from diffusion constants (diffusive transport of impurities), or viscosity (diffusive transport of momentum). Consider the retarded correlation function of the energy current $$G^{ij}_R(\omega,k) = \int d^3x\,dt\, e^{-i(\omega t-kx)}\,\langle[ \jmath^i_\epsilon(0),\jmath^j_\epsilon(x,t)]\rangle \Theta(t)\, .$$ Then the Kubo relation for the thermal conductivity is $$\kappa = \lim_{\omega\to 0}\; \frac{1}{3\omega T}\, {\rm Im}\, G_R^{ii}(\omega,0)$$ The energy current can be determined within a microscopic theory. For example, for weakly interaction fermions (or bosons) $$\jmath^i_\epsilon = -\frac{1}{2m} \left( \partial_0\psi^\dagger\partial^i\psi + \partial^i\psi^\dagger\partial^0\psi \right)$$ All of this is analogous to diffusion and viscosity, where the Kubo formula involves the particle current and the momentum current, respectively.
There is a (semi) classical limit, in which thermal conductivity can be understood in terms of the distribution function $$f(x,p,t)$$. The energy current is $$\jmath^i_\epsilon = \int d\Gamma_p\, \epsilon_p v^i_p f(x,p,t)$$ where $$d\Gamma_p=d^3p/(2\pi)^3$$, $$\epsilon_p=p^2/(2m)$$ and $$v_p=p/m$$. Diffusive transport involves slight perturbations of the distribution function away from thermal equilibrium $$f(x,p) = f^0(x,p) \left( 1 + \frac{\psi(x,p)}{T} \right)$$ where $$f^0(x,p)$$ is the equilibrium distribution (Boltzmann, Fermi-Dirac, or Bose-Einstein), and $$\psi(x,p)$$ is a small deviation driven by a temperature gradient. That means that we look for solustions of the Boltzmann equation of the form $$\psi(x,p) = \chi v^i_p\partial^i T$$ The coefficient $$\chi$$ can be found by the standard Chapman-Enskog procedure. Inserting the result into the formula for the energy current, and comparing with Fourier’s law $$\jmath^i_\epsilon = -\kappa\partial^i T$$ determines $$\kappa$$.
Brief postscript: (related to fluctuations) There is a recent paper in Physical Review Letters (arxiv version here) that points out that one can determine the thermal conductivity from the density-density correlation function. The basic observation is that the density correlator not only couples to sound, but also to diffusive heat waves. The corresponding diffusion constant depends on the specific heat and the thermal conductivity. This means that if the specific heat is known we can determine $$\kappa$$.