Macrostate and Microstate in statistical Mechanics

We need a statistical approach to deal with a system containing large number of particles. Because of the mutual interactions between the particles, it is impossible to know the dynamics of such number of particles together simply by solving their equations of motion. Statistical Mechanics gives the better results for complicated systems. We can get the idea of the nature of a Thermodynamic system (e.g. a gas in a vessel) by knowing its Pressure, Temperature and Volume. But, to know the dynamics of the particles or molecules in that system we need to know the concept of macrostate and microstate of the system. In this post, I am going to explain the idea of Macrostate and Microstate in Statistical Mechanics.

Contents in this article:

  1. What is Macrostate?
  2. What is a Microstate?
  3. Explanation with examples of Macrostate and Microstate in Statistical Physics
  4. How to find number of mircostate?

What is Macrostate in Statistical Mechanics?

The term Macro refers to something large. A Macrostate in statistical Physics is a system that contains large number of particles. A gas system inside a cylinder or any vessel is an example of Macrostate in Physics. Because, this gas system contains large number of molecules in it.

What is a Microstate in Statistical Physics?

Particles in a Macrostate can be arranged in different ways. Each arrangement is a Microstate. The number of Microstate of a system is defined as the number of possible combinations of the particles in that system.

This can be understood in following example of Macrostate and Microstate in Physics.

Why is Microstate important in Statistical Mechanics?

We can know the behavior of a Thermodynamic system by knowing its macroscopic parameters like Pressure, Temperature and Volume. But, some other properties like, Entropy of the system can be known only if we know the dynamics of the microscopic particles inside the system. Therefore, we need to study the Microstates of a Thermodynamic system.

If W be the number of microstates in a system, then the entropy of the system is, S = K ln(W). Where, K is the Boltzmann constant.

Explain the difference between Macrostate and Microstate with example

We consider a bucket that contains three cells. So, the bucket is a Macrostate of these cells. Let, we have three balls of different colors, say Red (R), Blue (B) and Green (G). These balls can be placed in the cells in different combinations.

Let Red, Blue and Green balls are placed in cell-1, cell-2 and cell-3 respectively. Then the combination is (R, B, G). Similarly we can get another combination (B, R, G) by placing Blue ball in cell-1, Red ball in cell-2 and Green ball in cell-3. In this way, we can get six different combinations. Again, all three balls can be placed at the single cell by making other two empty. Let all three balls are placed in cell-1. The the combination is (RBG, 0, 0). Similarly, we can get another two combination for other two cells.

So, inside the same Macrostate (Bucket), the particles (Balls) can be arranged in different combinations. These combinations are microstates. So, there are Nine microstates in this microstate system.

How to find number of Microstate in a system?

We already have explained the concept of Macrostate and Microstate with an example. I hope you have got some idea from there. You just need to find the different arrangement. That’s it. But this arrangement depends on the type of particles – whether they are distinguishable or not. If the particles are distinguishable, then they can be arranged in more ways and hence the number microstates will be higher.

In our example of a bucket with three balls, the balls are distinguishable as they have different colors. If we take three balls of same color, say White (W), then instead of (R, B, G), (B, R, G), (R, G, B), etc. we will get (W, W, W), (W, W, W), (W, W, W), etc. That means the six different combinations are appearing as a single combination (W, W, W) in case of indistinguishable particles.

So, this system gives Nine microstates for distinguishable particles and gives only four microstates for indistinguishable particles.

Problems on Macrostate and Microstate

Question: A system has three states of different energies. If two electrons occupy in this system, what the maximum number of microstates in this system.

[Hints: Electrons are indistinguishable]

This is all from this post on Macrostate and Microstate in statistical mechanics. Hope I have explained it properly. If you still have any doubt on this topic you can ask me in the comment section.

Thank you!

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