Diffusion equation from random motion

In summary, the conversation is about a new member introducing themselves and asking for help in finding a derivation of the diffusion equation from the equation of motion with white noise. The member also mentions using a Gaussian form of noise and a delta function in the derivation. They are directed to try a search for "diffusion equation" and are suggested to look for a Langevin equation.
  • #1
sinusoid
1
0
hi all, i am a new member here. nice to meet you all.

i remember seeing a derivation of diffusion equation from the equation of motion \dot{x}=\eta(t) where \eta(t) is white noise. i can't remember where i saw this... could anyone please help me on that?

it should be something like starting with the gaussian form of noise and then inserting a delta function representing the equation of motion... but nothing more i can remember.

thanks very much!
 
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  • #2
Try a google or bing search for "diffusion equation".
 
  • #3
You are looking for a Langevin equation I believe.
 

FAQ: Diffusion equation from random motion

What is the diffusion equation and how is it related to random motion?

The diffusion equation is a mathematical equation that describes the spread of particles or substances through a medium due to random motion. It is based on the concept of Brownian motion, where particles move randomly and continuously due to collisions with other particles. The diffusion equation is a way to mathematically model this type of behavior.

How is the diffusion equation used in scientific research and applications?

The diffusion equation is used in a wide range of scientific fields, including chemistry, physics, biology, and engineering. It is used to study the movement of molecules in a variety of systems, such as gases, liquids, and solids. This equation is also essential in many practical applications, such as predicting the spread of pollutants in the environment, designing drug delivery systems, and understanding heat transfer in materials.

What are the key variables in the diffusion equation?

The diffusion equation has three main variables: time, position, and diffusion coefficient. Time represents the duration of the diffusion process, while position indicates the location of the particles. The diffusion coefficient is a measure of how fast the particles are moving and is influenced by factors such as temperature and concentration gradient.

How is the diffusion coefficient determined for a particular system?

The diffusion coefficient can be experimentally determined by measuring the rate of diffusion in a system and using the diffusion equation to calculate the coefficient. It can also be estimated using theoretical models based on the properties of the particles and the medium they are moving through.

What are some limitations of the diffusion equation?

While the diffusion equation is a powerful tool for studying random motion, it has some limitations. It assumes that the particles are moving in a uniform medium, and it does not account for factors such as particle interactions or external forces. In some cases, more complex equations or simulations may be needed to accurately model diffusion in a system.

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