### 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.

## Time Reversal Symmetry

*time-symmetric*they don't change under a transformation of t goes to -t. Even the very microscopic laws of Quantum Mechanics are also time-symmetric.

## Arrow of Time

**The probability of expending a gas is one while contracting to a smaller volume is zero.**That makes sense because if there would have been even a small probability that the gas can contract, the air in our room would contract to a corner and we would die. (Thank God!!)

*All Physical processes happen in such a way that volume in phase space always increases.*(Does that remind you of something?)

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