Application of the Fokker-Planck equation

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In summary, the conversation discusses the relationship between diffusion MRI, harmonic oscillator hamiltonian, and Fokker Planck equations. The speaker is interested in understanding the common purpose of using Fokker Planck equations and asks for simple applications outside of water molecules in MRI, such as in Mathematical Finance. Two resources are provided for further reading on the use of Fokker Planck equations in Mathematical Finance.
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alecrimi
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Hallo everybody.
Foreword1: I am an engineer not a physicist
Foreword2: I am reading a paper about diffusion MRI who refers to harmonic oscillator hamiltonian.

The paper sometimes mention the Fokker Planck equations. Now, I don't want yet understand the relationshipt between diffusion-hamiltonian and Fokker-Planck, but I will be happy to understand the common purpose to use Fokker-Planck equations.

The definition is : partial differential equations that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. Can you give me simple applications when we need to model the evolution of hte pdf under the influence of forces as in Brownian motion (possibly different from water molecules in MRI)?
 
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1. What is the Fokker-Planck equation?

The Fokker-Planck equation is a partial differential equation that is used to describe the evolution of a probability distribution function over time. It is commonly used in physics and engineering to model stochastic processes, such as diffusion and Brownian motion.

2. What are some applications of the Fokker-Planck equation?

The Fokker-Planck equation has many applications, including modeling the movement of particles in a gas, predicting stock market fluctuations, analyzing the behavior of financial systems, and understanding the dynamics of biological systems.

3. How is the Fokker-Planck equation derived?

The Fokker-Planck equation is derived from the Langevin equation, which describes the stochastic motion of particles in a medium. Using statistical mechanics and the Boltzmann equation, the Fokker-Planck equation can be derived to describe the probability distribution of particle positions over time.

4. What is the relationship between the Fokker-Planck equation and the diffusion equation?

The diffusion equation is a special case of the Fokker-Planck equation, where the drift term is equal to zero. This means that the particles do not have a preferred direction of movement, and they diffuse randomly. The Fokker-Planck equation allows for non-zero drift terms, which can model more complex behaviors.

5. How is the Fokker-Planck equation solved?

There is no general analytical solution for the Fokker-Planck equation, so it is typically solved using numerical methods. These methods involve discretizing the equation and using iterative algorithms to find a numerical solution. The accuracy of the solution depends on the size of the discretization and the convergence of the algorithm used.

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