Phase Space Cells - Statistical Thermodynamics

[ClaireA88]

Homework Statement

1.) Explain why it is necessary to divide phase space into quantified cells of a finite size.
2.) Why is it necessary to know the size of these cells to over come the Gibb's paradox?

Homework Equations

The Attempt at a Solution

1.) I think it's something to do with stopping it having infinite entropy but I can't really elaborate more than that..

2.) Is this just because the paradox states that every particle is indistinguishable from every other particle in the volume? Or is there more to it?

Any help would be appreciated! :)

 

[Jhero]

Thank you for your question. I would like to provide a more detailed explanation for why it is necessary to divide phase space into quantified cells of a finite size and how this helps to overcome the Gibb's paradox.

1.) Phase space is a mathematical concept used to describe the state of a physical system. It is a multi-dimensional space where each point represents a specific combination of the system's position and momentum. In classical mechanics, phase space is continuous and infinite, which means that there are an infinite number of possible states for a system. However, in reality, physical systems are made up of discrete particles and have a finite number of possible states.

By dividing phase space into quantified cells of a finite size, we are essentially discretizing the system and limiting the number of possible states it can be in. This is necessary because it allows us to accurately describe and analyze the behavior of the system. If we were to use a continuous and infinite phase space, it would be impossible to make any meaningful predictions or calculations.

Moreover, dividing phase space into cells also helps to avoid the issue of infinite entropy. Entropy is a measure of the disorder or randomness of a system, and in classical mechanics, it is directly related to the number of microstates (possible arrangements of particles) in a given macrostate (overall state of the system). With an infinite phase space, the number of microstates would also be infinite, resulting in infinite entropy. By dividing phase space into finite cells, we are essentially limiting the number of microstates and preventing infinite entropy.

2.) Now, let's discuss how dividing phase space into quantified cells helps to overcome the Gibb's paradox. The Gibb's paradox states that the entropy of a classical ideal gas would increase if the gas was divided into two equal parts, even though the number of particles and total energy remain the same. This paradox arises when we treat particles in the gas as indistinguishable from one another.

However, by dividing phase space into quantified cells, we are essentially distinguishing between the particles in the gas. Each cell represents a specific combination of position and momentum, and therefore, each particle can be uniquely identified by its cell. This overcomes the Gibb's paradox because now we can distinguish between the particles and their microstates, which results in a different calculation for entropy.

In conclusion, dividing phase space into quantified cells of a finite size is necessary for accurately describing physical systems,