2-D Brownian motion with correlated noise

In summary, the conversation discusses the 2-D diffusion equation with a correlation term between two noises, η(t) and ζ(t). The analytical solution may not be known, but it can be approximated using numerical or perturbation methods. Further research is needed to fully understand the behavior of this system.
  • #1
Dessert
9
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dx/dt = η(t)
dy/dt = ζ(t)

where
<η(t)>=<ζ(t)>=0
<η(t)η(t')> = 2Dδ(t-t')
<ζ(t)ζ(t')> = 2Dδ(t-t')

If <η(t)ζ(t')> = 0, we have the standard 2-D diffusion equation and the analytical solution is known.

If <η(t)ζ(t')> = 2Dδ(t-t'), or η(t) = ζ(t), we can transform it into a 1-D problem and the analytical solution is also known.

What if <η(t)ζ(t')> = 2Dcδ(t-t') where 0<c<1 which is correlation of the two noises? We can still write down a 2-D diffusion equation, but is the analytical solution known?
 
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  • #2


Hello,

Thank you for sharing this interesting problem. The correlation between the two noises, η(t) and ζ(t), definitely adds a new layer of complexity to the 2-D diffusion equation. In this case, the analytical solution may not be known, but it can be approximated using numerical methods.

To understand the behavior of the system, we can first analyze the correlation term (2Dcδ(t-t')). This term implies that the two noises are not completely independent and affect each other's behavior. This can lead to a more complex diffusion process, where the particles may not move in a purely random manner.

To solve this problem, we can use numerical methods such as Monte Carlo simulations or finite difference methods. These methods involve dividing the system into smaller time and space intervals and solving the equation iteratively. Although this approach may not give an analytical solution, it can provide valuable insights into the behavior of the system.

Additionally, we can also use perturbation methods to approximate the solution. This involves expanding the solution in a series and solving for each term individually. However, this method may not be applicable in all cases and may require certain assumptions to be made.

In conclusion, the analytical solution may not be known for the 2-D diffusion equation with a correlation term, but it can be approximated using numerical or perturbation methods. Further research and analysis are needed to fully understand the behavior of this system.
 

1. What is 2-D Brownian motion with correlated noise?

2-D Brownian motion with correlated noise is a mathematical model that describes the random movement of particles in a two-dimensional space. It takes into account the effects of both Brownian motion, which is the random movement of particles due to collisions with surrounding molecules, and correlated noise, which is a type of noise that is related to a specific physical process.

2. How is correlated noise incorporated into the model?

In 2-D Brownian motion with correlated noise, correlated noise is incorporated by adding a term to the standard Brownian motion equation. This term introduces a correlation parameter that determines how strongly the noise is related to the motion of the particles. The higher the correlation parameter, the more the noise affects the particles' movement.

3. What are the applications of 2-D Brownian motion with correlated noise?

2-D Brownian motion with correlated noise has various applications in different fields of science and engineering. It is commonly used in physics, biology, chemistry, and finance to model the movement of particles in a two-dimensional space. It can also be applied to study the behavior of complex systems, such as the stock market or biological systems.

4. How is 2-D Brownian motion with correlated noise different from other models?

2-D Brownian motion with correlated noise is different from other models because it takes into account the effects of correlated noise, which is often present in real-world systems. This makes it a more realistic model compared to standard Brownian motion models. Additionally, it can be extended to higher dimensions, making it suitable for studying complex systems with multiple variables.

5. What are the limitations of 2-D Brownian motion with correlated noise?

Like any other mathematical model, 2-D Brownian motion with correlated noise has its limitations. One of the main limitations is that it assumes a linear relationship between the noise and the particles' movement. In reality, this may not always be the case, and the correlation may not be accurately represented by a single parameter. Additionally, the model may not be suitable for studying systems with highly nonlinear behavior.

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