# Statistical Mechanics Mean Field Model

• Chris B
In summary, the conversation discusses the mean field model and its application to finding the proportionality of M to H1/3 at T=Tc and (Tc - T)1/2 at H=0. The discussion includes approximations and the use of Taylor series expansions, as well as the relationship between Tc and other variables.
Chris B
I don't think I've fully grasped the underlying ideas of this class, so at the moment I'm just sort of flailing for equations to plug stuff into...

## Homework Statement

Show that in the mean field model, M is proportional to H1/3 at T=Tc and that at H=0, M is proportional to (Tc - T)1/2

## Homework Equations

It'll take me forever to write it out here, but M equals some coefficients times tanh(x) where x depends on H, M, and T

## The Attempt at a Solution

In class we approximated M at T=0 by taking the first term in the Taylor series expansion of tanh(x) which turns out to just be x. We were given the hint that we need to take the next term in the series (why?) to do this problem. The next term in the series is -x3/3 so the whole thing is
M=N/V gμbS(x-x3/3)
where x = gμbS(H+λM)/kT
I also know that Tc = λN(gμbS)2/VK
I plugged Tc in for T and by rearranging got something that was proportional to H1/3 minus H/λ
For the next part I'm not sure what to do. If I plug in zero for H the whole thing is zero. I don't see how I'm going to get (Tc - T)1/2 if I'm substituting Tc for T.

These are two different scenarios. In part A, you assume T=Tc. In part B, the only assumption is that H=0.

## 1. What is the Mean Field Model in Statistical Mechanics?

The Mean Field Model is a mathematical approach used to describe the behavior of a large collection of interacting particles or systems. It assumes that each particle or system interacts with all other particles or systems in the same way, resulting in a simplified representation of the overall system.

## 2. How does the Mean Field Model differ from other statistical mechanics models?

The Mean Field Model differs from other models in that it does not take into account the specific interactions between individual particles or systems. Instead, it treats all interactions as a collective average, resulting in a simplified and more manageable model for large systems.

## 3. What are the limitations of the Mean Field Model?

The Mean Field Model has several limitations, including its inability to accurately describe systems with strong correlations or interactions between individual particles. It also assumes a homogeneous system, which may not be valid in some cases.

## 4. How is the Mean Field Model used in practical applications?

The Mean Field Model is commonly used in statistical mechanics to study phase transitions and critical phenomena in systems such as magnets, polymers, and fluids. It is also used in areas such as material science, condensed matter physics, and biology to model complex systems.

## 5. What are the benefits of using the Mean Field Model over other statistical mechanics models?

The Mean Field Model allows for simpler and more computationally efficient calculations compared to other models, making it a useful tool for studying large systems. It also provides insights into the behavior of complex systems and can be used as a starting point for more advanced models.

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