Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry

Area of Science:

• Statistical Physics
• Non-equilibrium Systems
• Computational Physics

Background:

• Information Geometry offers insights into stochastic differential equations (SDEs) for non-equilibrium systems.
• Traditional Fokker-Planck equation (FPE) methods face challenges in calculating time-dependent probability density functions (PDFs) and information geometric diagnostics.

Purpose of the Study:

• To develop a novel Monte Carlo (MC) simulation method for SDEs as an alternative to FPE solvers.
• To efficiently calculate time-dependent PDFs and information geometric properties using GPU computing.
• To investigate the behavior of PDFs and identify scaling relations in systems with linear and cubic damping.

Main Methods:

• Developed a new MC SDE simulation technique leveraging GPU computing for accelerated calculations.
• Applied the MC SDE method to systems with linear and cubic damping terms.
• Calculated time-dependent and unequal time joint PDFs, and information geometric diagnostics.

Main Results:

• Successfully reproduced Information Geometric scaling relations using MC SDE simulations.
• Demonstrated the advantage of MC SDE simulation over FPE solvers, particularly for unequal time joint PDFs.
• Observed Gaussian joint PDFs for linear processes and bimodal joint PDFs for cubic processes, indicating finite memory effects in nonlinear systems.
• Identified and investigated several power-law scalings in the characteristics of bimodal PDFs.

Conclusions:

• MC SDE simulation provides an efficient and powerful alternative for studying Information Geometry in non-equilibrium systems.
• Nonlinear damping forces can induce complex PDF structures, such as bimodality, even in stationary states, suggesting finite memory times.
• The developed method enables deeper insights into the statistical properties and geometric structures of complex stochastic systems.