Renewal stochastic processes with correlated events: phase transitions along time evolution

Area of Science:

• Statistical Mechanics
• Stochastic Processes
• Probability Theory

Background:

• Renewal stochastic processes are fundamental in modeling event occurrences over time.
• Existing models often assume independent events, limiting their applicability to complex systems.
• The statistical-mechanical structure of interacting systems offers a novel perspective for analyzing these processes.

Purpose of the Study:

• To develop a statistical-mechanical framework for renewal stochastic processes with nonindependent events.
• To investigate the occurrence of phase transitions in time-dependent event distributions.
• To demonstrate the applicability of this framework to specific models, including the Hamiltonian mean-field model.

Main Methods:

• Interpreting the event density distribution as a microcanonical partition function.
• Utilizing generating functions and Laplace transforms within a statistical-mechanical context.
• Applying large deviations theory and deriving an Euler relation for the entropy (Massieu potential).

Main Results:

• A unified framework is established where partition functions relate via standard statistical-mechanical principles.
• Processes exhibiting phase transitions are identified, characterized by a second-order transition.
• The Hamiltonian mean-field model demonstrates clustering at short times and independent behavior at long times.

Conclusions:

• The statistical-mechanical approach provides a powerful tool for analyzing complex renewal processes.
• The framework successfully models phenomena like opinion dynamics and event clustering.
• Similar analytical schemes are applicable to random-walk processes, highlighting the broad utility of this methodology.