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Intrinsic irreversibility of Markovian chains
P Di Porto1, B Crosignani2, E DelRe3
1Dipartimento di Fisica, Università di Roma "La Sapienza," I-00185 Rome, Italy.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 30, 2014
Summary
For many stationary Markov processes, the difference between the final equilibrium state and the current state decreases predictably over time. This vanishing distance reveals a statistical arrow of time beyond energy reduction.
Area of Science:
- Statistical physics
- Probability theory
- Dynamical systems
Background:
- Understanding the long-term behavior of stochastic processes is crucial in various scientific fields.
- Characterizing the convergence to equilibrium in Markov processes is a fundamental problem.
Purpose of the Study:
- To demonstrate that the total variation distance in stationary Markov processes is a strongly monotonic vanishing function.
- To establish a statistical arrow of time for systems with a canonical description.
Main Methods:
- Analysis of stationary Markov processes.
- Calculation of total variation distance between distributions.
- Illustration using paradigmatic processes.
Main Results:
- The total variation distance between the equilibrium distribution and the distribution at time t is a strongly monotonic vanishing function.
- This vanishing function provides a measure of convergence to equilibrium.
Conclusions:
- A statistical arrow of time exists for systems described canonically, independent of the decrease in free energy.
- The findings offer new insights into the temporal evolution and convergence properties of Markov processes.
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