A document discusses the emergence of randomness in solutions of coupled, fully deterministic ODE-PDE (ordinary differential equations-partial differential equations) due to failure of the Lipschitz condition as a new phenomenon. It is possible to exploit the special properties of ordinary differential equations (represented by an arbitrarily chosen, dynamical system) coupled with the corresponding Liouville equations (used to describe the evolution of initial uncertainties in terms of joint probability distribution) in order to simulate stochastic processes with the proscribed probability distributions. The important advantage of the proposed approach is that the simulation does not require a random-number generator.
This work was done by Michail Zak of Caltech for NASA's Jet Propulsion Laboratory.
NPO-45241
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Simulation of Stochastic Processes by Coupled ODE-PDE
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Overview
The document titled "Simulation of Stochastic Processes by Coupled ODE-PDE," produced by NASA's Jet Propulsion Laboratory, presents a novel approach to simulating stochastic processes using a combination of ordinary differential equations (ODEs) and partial differential equations (PDEs). This method addresses the need for accurate Monte Carlo simulations that require specific probability distributions.
The core of the research introduces a unique feedback mechanism derived from the Liouville equation, which is integrated into the dynamics of ODEs. This feedback allows for the simulation of randomness in deterministic systems, leading to the emergence of non-Newtonian properties such as self-generated randomness and entanglement among different samples of the same stochastic process. The document outlines how these phenomena can be modeled mathematically, providing a framework for understanding the dynamics of systems that exhibit stochastic behavior.
Key equations are presented, including the formulation of probability density functions and their evolution over time. The document discusses the analytical solutions to these equations, demonstrating how they converge to prescribed stationary distributions under specific initial conditions. The research emphasizes the importance of coupling the Liouville feedback with the equations of motion, which is a departure from traditional Newtonian physics where such coupling is not typically observed.
The document also highlights the qualitative behavior of the solutions, illustrated through figures that depict the dynamics of the system. The findings suggest that the proposed method can effectively simulate complex stochastic processes, making it a valuable tool for various applications in aerospace and other fields requiring advanced modeling techniques.
In summary, this technical support package provides a comprehensive overview of a new simulation technique that leverages the interplay between ODEs and PDEs to model stochastic processes. It offers insights into the mathematical foundations of the approach, the implications of self-generated randomness, and the potential applications in scientific and engineering domains. The research was conducted under the auspices of NASA, reflecting its commitment to advancing technology and understanding in the field of stochastic modeling.
