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Complexity and nonseparability of classical Liouvillian dynamics
1Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia.
Summary
We introduce separability entropy, a new measure for classical Liouvillian dynamics complexity. Linear growth of this entropy indicates exponentially unstable, nonlinear, and non-Markovian dynamics, a stricter complexity criterion than Kolmogorov-Sinai entropy.
Area of Science:
- Classical dynamics
- Quantum information theory
- Statistical mechanics
Background:
- Characterizing complexity in classical Liouvillian dynamics is crucial for understanding system behavior.
- Existing entropy measures may not fully capture the intricate nature of complex dynamics.
Purpose of the Study:
- To propose a novel and simple complexity indicator for classical Liouvillian dynamics.
- To establish the relationship between this new indicator and established complexity measures.
Main Methods:
- Definition of separability entropy based on the Schmidt decomposition of phase space density.
- Analysis of the conditions under which separability entropy exhibits linear growth.
Main Results:
- Separability entropy quantifies the effective number of terms in a Schmidt decomposition.
- Linear growth of separability entropy is shown to be a stricter criterion for complexity than Kolmogorov-Sinai entropy.
Conclusions:
- Separability entropy offers a new perspective on quantifying complexity in classical dynamics.
- The linear growth of separability entropy necessitates exponentially unstable, nonlinear, and non-Markovian dynamics.
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