Quantum Matter Seminar

Linear response theory (LRT) is widely used, but it has subtle points which are not always appreciated. For example, while Kubo formulas for electric conductivity, thermal conductivity, viscosity, etc., are supposed to describe properties of non-equilibrium steady states (NESS) which can exist only in open systems, most derivations of Kubo formulas assume a closed system. This point has been forcefully expressed by N. van Kampen who argued that a coupling to the environment is needed to ensure macroscopic linearity of response. A related question is what sets the range of applicability of the standard LRT theory. To illuminate these questions, I will discuss a simple but physically reasonable model of electric conduction: a classical particle in a smooth random potential and a dissipative coupling to the environment in the form of Markovian noise and friction. The corresponding NESS can be determined analytically when both the disorder and dissipation are weak. I will show that the range of applicability of LRT shrinks to zero as one switches off dissipation, thereby validating van Kampen's objection. Further, the Principle of Minimum Entropy Production does not determine the NESS beyond linear order in the electric field, and the NESS strongly depends on the dissipative coupling already at quadratic order in the electric field. If time permits, I will also discuss a version of the model where the potential is periodic and argue that a classical particle in a smooth periodic potential behaves radically differently from the well understood Lorentz gas.