Master equation -> diffusion equation

In summary, the conversation discusses the derivation of the diffusion equation from the Master equation for a 1D chain. The Master equation states that the rate of change of particle density at a certain point is equal to the rate at which particles move between neighboring points. By taking the limit as the distance between chain links approaches zero, the expression can be simplified to the diffusion equation, which describes the spread of particles in a continuous medium. The conversation also touches on the use of derivatives in this derivation.
  • #1
Mikhail_MR
17
0

Homework Statement


I am trying to understand the derivation of the diffusion equation from the Master equation for a 1D chain. We have an endless 1D discrete chain. State from ##n## can jump to ##n+1## and ##n-1## with equal probabilities. The distance between chain links is ##a##.

Homework Equations


Master equation: $$\frac{d\rho_n}{dt} = W\left(-2\rho_n + \rho_{n+1} + \rho_{n-1}\right)$$

The Attempt at a Solution


With ##\Delta n = 1## we have $$\frac{d\rho_n}{dt}=Wa^2 \frac{(\rho_{n-1}-\rho_n) + (\rho_{n+1}-\rho_n)}{(\Delta n)^2 a^2},$$ where we could write ##(\Delta n)^2 a^2 = (\Delta x)^2##, since ##na=x##. In limit ##\Delta x \rightarrow 0## we get $$\frac{d\rho_n}{dt} = Wa^2 \frac{\partial^2 \rho_n}{\partial x^2}$$.
Unfortunately, I can not understand the last step. The definition of a derivative is ##f(x_0+\Delta x) - f(x_0) /\Delta x##, but I do not understand how it is used here.

Any help would be appreciated.
 
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  • #2
Using your definition (with brackets inserted correctly), what is the expression for dρ/dx at x0 = n-1? What is the expression at x0 = n? So what is d(dρ/dx)/dx ?
 
  • #3
##d\rho/dx## would be ##\frac{\rho_n - \rho_{n-1}}{\Delta x}## at ##x_0=n-1## and ##\frac{\rho_{n+1} - \rho_{n}}{\Delta x}## at ##x_0=n##.
Then: $$d(d\rho/dx)dx= [(d\rho/dx)_{x_0=n} - (d\rho/dx)_{x_0=n-1}] / \Delta x$$. Oh, now I see it. Thank you!
 
  • #4
A better way of doing this would be in terms of central differences: ##d\rho/dx## would be ##\frac{\rho_n - \rho_{n-1}}{\Delta x}## at ##x_0=n-1/2## and ##\frac{\rho_{n+1} - \rho_{n}}{\Delta x}## at ##x_0=n+1/2##
 

Related to Master equation -> diffusion equation

1. What is the Master Equation?

The Master Equation is a mathematical framework used to describe the time evolution of a system with multiple states. It takes into account the probabilities of transitioning between states and the rates at which these transitions occur.

2. What is the Diffusion Equation?

The Diffusion Equation is a partial differential equation that describes how a quantity, such as heat or concentration, diffuses over time and space.

3. How are the Master Equation and Diffusion Equation related?

The Diffusion Equation can be derived from the Master Equation by assuming a large number of states and small transition rates. It is a more simplified version of the Master Equation that is easier to solve.

4. What are some real-world applications of the Master Equation and Diffusion Equation?

The Master Equation and Diffusion Equation have a wide range of applications in fields such as physics, chemistry, biology, and economics. They are used to model processes such as chemical reactions, population dynamics, and heat transfer.

5. What are the limitations of the Master Equation and Diffusion Equation?

The Master Equation and Diffusion Equation have some limitations, such as assuming a large number of states and small transition rates, which may not always be applicable in real-world systems. They also do not take into account external factors that may affect the system's behavior.

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