Diffusion is a ubiquitous phenomenon. It is a widespread belief that, if the area under a current autocorrelation function converges in time, the corresponding spatiotemporal density dynamics should be diffusive. This may be viewed as a result of the combination of linear response theory with the Einstein relation. However, attempts to derive this statement from first principles are notoriously challenging. We first present a counterexample by constructing a correlation function of some density wave, such that the area under the corresponding current autocorrelation function converges, but the dynamics do not obey a diffusion equation. Then we introduce a method based on the recursion method and the Mori memory formalism that may help to identify diffusion. For a decisive answer, one would have to know infinitely many so-called Lanczos coefficients, which is unattainable in most cases. However, in the examples examined in this paper, we find that the practically computable number of Lanczos coefficients suffices for a strong guess.

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• Diffusion
• Quantum statistical mechanics
• Quantum transport
• Transport phenomena
• 1-dimensional spin chains
• Heisenberg model
• Ising model