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Quantization and fractional quantization of currents in periodically driven stochastic systems. I. Average currents
Vladimir Y Chernyak1, John R Klein, Nikolai A Sinitsyn
1Department of Chemistry, Wayne State University, 5101 Cass Avenue, Detroit, Michigan 48202, USA.
The Journal of Chemical Physics
|April 24, 2012
Summary
Particle motion on graphs with time-varying rates shows quantized currents under specific conditions. This study reveals topological invariants governing average current generation, offering new insights into stochastic processes.
Area of Science:
- Statistical physics
- Graph theory
- Quantum mechanics
Background:
- Stochastic processes describe particle movement.
- Time-dependent systems exhibit complex dynamics.
- Detailed balance is a fundamental concept in equilibrium statistical mechanics.
Purpose of the Study:
- To investigate quantized currents in Markovian stochastic motion on graphs.
- To explore the role of periodic time-dependent transition rates.
- To connect current quantization to topological invariants.
Main Methods:
- Analysis of Markovian stochastic motion on finite graphs.
- Application of adiabatic driving and low-temperature conditions.
- Utilizing Kirchhoff's theorem for current calculation.
- Developing a quantitative theory for current quantization.
Main Results:
- Demonstrated average current quantization (full or fractional) under adiabatic driving.
- Established a connection between current quantization and topological invariants.
- Derived an explicit formula for average generated current.
Conclusions:
- Periodic transition rates satisfying detailed balance can lead to quantized currents.
- Topological invariants provide a framework for understanding current quantization.
- The derived formula offers a tool for analyzing current quantization effects.
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