Numerical Integration of Langevin Equation

In summary, the Langevin equation is a stochastic differential equation used to describe the motion of a particle in a fluid. Numerical integration is necessary for studying this equation in order to simulate the particle's behavior over time. However, there are challenges in numerically integrating the Langevin equation, such as accurately modeling random forces and choosing an appropriate time step. Common methods for numerical integration include the Euler-Maruyama method, Runge-Kutta method, and stochastic Runge-Kutta method. To validate results, they can be compared to experimental data or sensitivity analyses can be performed. Applications of numerical integration of the Langevin equation include studying particles in fluids, diffusion processes, chemical reactions, and computer simulations in various fields such as physics, chemistry,
  • #1
dsdsuster
30
0
Can anyone explain to me why the gaussian white noise term is multiplied by the square root of the time differential when we turn the Langevin differential equation into a finite difference equation for the purposes of integration?

http://pre.aps.org/pdf/PRE/v50/i6/p4404_1

The step I don't understand is the change in the last term in going from equation 3 to 4.

Any help would be greatly appreciated, thanks!
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Can't access - requires membership.
 
  • #3
equation 3:
Ocr12TW.png


equation 4:
oeahOsB.png


Here are the two equations. Thanks for your help!
 

What is the Langevin equation and why is numerical integration necessary for studying it?

The Langevin equation is a stochastic differential equation that describes the dynamics of a particle in a fluid. It takes into account both the random forces acting on the particle and the deterministic forces that govern its motion. Numerical integration is necessary for studying this equation because it allows us to simulate the behavior of the particle over time and observe how it responds to different conditions.

What are the challenges of numerically integrating the Langevin equation?

One of the main challenges of numerically integrating the Langevin equation is dealing with the random forces, which are usually modeled as white noise. This means that the forces are uncorrelated and have infinite frequency, making it difficult to accurately simulate their effects. Additionally, numerical integration requires choosing an appropriate time step and ensuring that the simulation is stable and accurate.

What are some common numerical integration methods used for the Langevin equation?

Some common methods used for numerically integrating the Langevin equation include the Euler-Maruyama method, the Runge-Kutta method, and the stochastic Runge-Kutta method. These methods vary in their complexity and accuracy, but all aim to approximate the solution of the Langevin equation over a certain time interval.

How do you validate the results of a numerical integration of the Langevin equation?

One way to validate the results of a numerical integration is to compare them to experimental data or analytical solutions, if available. Another approach is to perform sensitivity analyses, where different parameters and initial conditions are varied to see how they affect the results. Additionally, convergence tests can be performed to check the accuracy and stability of the numerical integration method.

What are some applications of numerical integration of the Langevin equation?

Numerical integration of the Langevin equation has a wide range of applications in fields such as physics, chemistry, biology, and engineering. It can be used to study the behavior of particles in fluids, diffusion processes, chemical reactions, and many other systems. It is also commonly used in computer simulations and modeling to understand complex systems and predict their behavior under different conditions.

Similar threads

A Numerical method to Lippman-Schwinger equation
    • Quantum Physics
Replies
1
Views
583
I Smoothing Numerical Differentiation Noise
    • General Math
Replies
28
Views
4K
A Langevin equation - derivative of random force?
    • Classical Physics
Replies
8
Views
1K
A Numerical Hartree Fock with Finite Difference Matrices for Helium
    • Atomic and Condensed Matter
Replies
4
Views
1K
Mathematica Numerical solution of integral equation with parameters
    • MATLAB, Maple, Mathematica, LaTeX
Replies
1
Views
1K
Selecting a numerical method
    • Differential Equations
Replies
6
Views
1K
I Real ODE yields real solution through complex numbers
    • Differential Equations
Replies
3
Views
2K
A Orthogonality of variations in Faddev-Popov method for path integral
    • Quantum Physics
Replies
1
Views
639
I Integration of Poisson's Equation
    • Classical Physics
Replies
9
Views
2K
Engineering Simulation of Circuits Without Kirchhoff's Laws?
    • Engineering and Comp Sci Homework Help
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top