Short-time large deviation of constrained random acceleration process.

Area of Science:

• Statistical Physics
• Nonlinear Dynamics
• Stochastic Processes

Background:

• The Langevin equation describes particle motion under random acceleration, influenced by Gaussian white noise.
• Understanding the statistical properties of particle trajectories under constraints is crucial in various physical systems.

Purpose of the Study:

• To investigate the short-time distribution of integrated particle positions (functionals) under total and partial trajectory constraints.
• To determine the optimal paths (most probable realizations) for these constrained random acceleration processes.
• To analyze the transition from Gaussian to non-Gaussian behavior in these distributions.

Main Methods:

• Employing the optimal fluctuation method to analyze the distribution P(A) of functionals A = ∫xⁿ(t)dt.
• Analytically solving Euler-Lagrange equations for n=1 and n=2 under constrained action functionals.
• Developing a numerical scheme for n≥3 to find optimal paths.
• Validating theoretical findings through simulations of effective Langevin equations.

Main Results:

• For n=1, P(A) follows a Gaussian distribution with variance proportional to Dt_{f}^{5}.
• For n≥2, P(A) exhibits non-Gaussian behavior, including an essential singularity (-lnP(A)∼A⁻³) in the small-A limit.
• Optimal paths show localization near the initial state followed by a sharp escape to the final position.
• Degenerate optimal paths with differing actions exist for large A, with the minimum action path dominating.

Conclusions:

• The study elucidates the complex statistical behavior of constrained particle motion governed by the Langevin equation.
• Non-Gaussian distributions and unique optimal path characteristics emerge for higher-order functionals (n≥2).
• The findings offer insights into the dynamics of stochastic processes with path-dependent constraints.