First, curious how to derive from the definition of entropy S and mathOgenius provided detailed steps premised on definition of equilibrium: Equilibrium is the most probable state having maximum entropy and maximum microstates.

Here’s a simple Python snippet to simulate the Maxwell-Boltzmann distribution. This example is geared towards gas particles and the speeds they possess, given a certain temperature:

this is plotted by applying formula directly, so how is the formula derived from

Then to count degenerate state: