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Variational principle for the Pareto power law
Anirban Chakraborti1, Marco Patriarca
1Laboratoire de Mathématiques Appliquées aux Systèmes, Ecole Centrale Paris, 92290 Châtenay-Malabry, France. anirban.chakraborti@ecp.fr
Physical Review Letters
|April 7, 2010
Summary
Complex systems can exhibit power-law energy distributions through a novel mechanism. This statistical mechanics approach explains energy distribution in systems with varying degrees of freedom, observed in diffusion and wealth exchange models.
Area of Science:
- Statistical Mechanics
- Complex Systems Theory
- Theoretical Physics
Background:
- Power-law distributions are prevalent in many complex systems, but their origins are not always clear.
- Understanding these distributions is crucial for modeling phenomena from finance to physics.
Purpose of the Study:
- To propose and demonstrate a general mechanism for the emergence of power-law energy distributions in complex systems.
- To show this mechanism arises from the superposition of equilibrium energy densities within a conservative mechanical system.
Main Methods:
- Utilized a variational principle based on Boltzmann entropy.
- Applied concepts from canonical equilibrium statistical mechanics.
- Developed an exactly solvable mechanical model of a dimensionally heterogeneous system.
Main Results:
- Demonstrated that a robust power-law tail in energy distribution can appear in equilibrium.
- Showed this occurs in conservative mechanical systems with subsystems of differing degrees of freedom.
- Illustrated the mechanism through examples of free diffusion on complex networks and a kinetic wealth exchange model.
Conclusions:
- The proposed mechanism provides a unified explanation for power-law tails in diverse complex systems.
- The findings are derived strictly within the framework of equilibrium statistical mechanics.
- This work offers a theoretical foundation for understanding energy distribution in heterogeneous systems.
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