A quantum particle restricted to a lattice of points has been well studied in many different contexts. In the absence of considering the interaction with its environment, the particle simply undergoes ballistic transport for many suitable Hamiltonian operators. The evolution becomes much more complicated when considering environmental interaction, which leads to the so-called Lindblad master equation. When considering this master equation, the Lindbladian term dominates the dynamics of the particle, leading to diffusive propagation. In this document, we prove diffusion is indeed present in the context of a periodic Hamiltonian. Additionally, we show that the diffusion constant is inversely proportional to the particles' coupling strength with its environment.
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