How can Entropy be related to Number of Microstates?
When I was first taught Boltzman's Law, I was very surprised. Entropy is a thermodynamic quantity, something like heat, temperature, and all that, what does it have to do with the position and momentum of all the particles? The number of microstates; position and momentum of all the particles, it seems more of Newtonian Physics than Thermodynamics. The other thing I was thinking how a quantity of big importance like entropy can be related to a number? I explored some of these questions and found the answers very amazing. Some more questions were asked and the answers were extraordinary.
Before I start taking technical terms, I want to say that we will look at the system of inert gas, consisting of N non-interacting particles. They are in a box, the size of the particles is very less than the size of the system; the collisions are elastic so that there is no energy loss in collisions, and the collision time is very less than the time the particles are traveling.
Microstates and Macrostates
We will start our discussion with an introduction to Microstate and Macrostate. A microstate is a complete description of every particle of the system. In our case, the microstate of the system would set of positions and momentum of every particle. A point in 6N dimensional phase space, a point represents a microstate.
A macrostate means a broad description of the system without going into small details such that the position and momentum of every particle. Like specifying the energy, you are saying that I don't know the velocity of every particle but I know the total sum of their squares. This is a broad description that doesn't look into small details.
The thing is there can be more than one microstate corresponding to one macrostate. If I say that some of the velocities of all particles are some constant v. I could mean that one particle is moving with speed v and others are at rest, that is a microstate; or it could mean that a second (different) particle is moving, that is a different microstate. So there are 'many' microstates possible corresponding to one macrostate. Every such microstate is a point in the phase space, so the macrostate is a collection of all such points. We can represent this as a closed volume in the phase space. When we say that the system is a macrostate, the possible microstate of the system lies in this enclosed volume.