Fourier’s Law from Schrödinger Dynamics

We consider a class of one-dimensional chains of weakly coupled many level systems. We present a theory which predicts energy diffusion within these chains for almost all initial states, if some concrete conditions on their Hamiltonians are met. By numerically solving the time dependent Schrödinger equation, we verify this prediction. Close to equilibrium we analyze this behavior in terms of heat conduction and compute the respective coefficient directly from the theory.

Figure 1

Heat conduction model: $N$ coupled subunits with ground level and band of $n$ equally distributed levels ($\Delta E=1$). Black dots refer to used initial states.

Figure 2

Probability to find the excitation in the $\mu =1,2,3$ system. Comparison of the HAM prediction (lines) and the exact Schrödinger solution (dots). ($N=3$, $n=500$, $\lambda =5\times {10}^{-5}$, $\delta ϵ=0.05$).

Figure 3

Deviation of HAM from the exact solution ($D_1^2$ least squares): dependence of $D_1^2$ on $n$ for $N=3$. Inset: dependence of $D_1^2$ on $N$ for $n=500$.

Figure 4

Heat conductivity (16) over temperature for a system with $n=500$, $\delta ϵ=0.05$, and $\lambda =5\times {10}^{-5}$.