Statistical properties of avalanches via the c-record process

Area of Science:

• Statistical Physics
• Complex Systems
• Dynamical Processes

Background:

• Avalanche statistics are crucial for understanding driven systems with disorder.
• Previous models often assumed specific pinning force distributions.
• The connection between avalanche dynamics and record processes in random variables is not fully explored.

Purpose of the Study:

• To investigate the statistical properties of avalanches in a 1D lattice model with i.i.d. pinning forces.
• To establish the relationship between avalanche statistics and a modified record process.
• To determine the influence of the tail behavior of the pinning force distribution on avalanche statistics and process stationarity.

Main Methods:

• Analysis of a particle hopping model on a 1D lattice with continuous, i.i.d. pinning forces.
• Mapping avalanche statistics to inter-record intervals in a modified record process defined by parameter c.
• Derivation of avalanche size distribution and analysis of process stationarity based on the tail decay of f(x).

Main Results:

• For pinning force distributions f(x) with tails decaying slower than exponential, the record process is nonstationary.
• For f(x) with tails decaying faster than exponential, the record process becomes stationary, exhibiting avalanche size distribution π(n) decaying faster than 1/n^2.
• A phase transition from nonstationary to stationary behavior occurs at a critical value c_crit for exponentially decaying f(x)=e^{-x}, with c_crit=1.

Conclusions:

• The tail behavior of pinning forces critically determines the stationarity and statistical properties of avalanches.
• For c>1 (in the exponential tail case), stationary avalanche statistics with a power-law distribution π(n) ~ n^{-1-λ(c)} are observed.
• Correlations between avalanches in the stationary phase resemble earthquake sequences, indicating complex emergent behavior.