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Efficiency of quantum and classical transport on graphs
Oliver Mülken1, Alexander Blumen
1Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg i.Br., Germany. oliver.muelken@physik.uni-freiburg.de
Summary
We introduce a new measure for transport efficiency on graphs, based on the density of states (DOS). Quantum transport is generally more efficient than classical transport, except in specific cases like finite tree graphs.
Area of Science:
- Graph theory
- Quantum mechanics
- Statistical physics
Background:
- Classical and quantum transport phenomena are crucial in various scientific domains.
- Quantifying transport efficiency on complex networks remains a challenge.
Purpose of the Study:
- To develop a novel measure for quantifying the efficiency of classical and quantum mechanical transport processes on graphs.
- To analyze the relationship between transport efficiency and graph properties, specifically the density of states (DOS).
Main Methods:
- The proposed measure relies solely on the density of states (DOS) of a graph.
- Analysis involves examining the behavior of the measure under different DOS distributions, including continuous DOS and small-world networks.
Main Results:
- For continuous DOS, the measure exhibits a power-law behavior where the quantum transport exponent is double the classical exponent.
- For small-world networks, a stretched exponential law is observed, with quantum transport still outperforming classical transport.
- Finite tree graphs with highly degenerate eigenvalues present an exception where classical transport can be more efficient.
Conclusions:
- The density of states (DOS) is a sufficient descriptor for transport efficiency on graphs.
- Quantum transport generally demonstrates superior efficiency over classical transport across various network structures.
- Specific graph structures, like finite trees with degenerate eigenvalues, can invert this efficiency dynamic.
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