Maxwell Boltzmann distribution for a classical ideal gas

i. We assume a dilute gas which is enclosed by a thermally insulated container on all sides.

Dilute gas in a thermally insulated container: Dilute means concentration of gas molecules is low. Insulated implies there is no reasonable flow of heat energy across the walls of the container.

ii. Each molecule is assumed to be a hard sphere which moves randomly in all directions such that its velocity vary from – infinity to + infinity.

Hard sphere: Remember a hard sphere is a classical analogy of a rigid sphere whose surfaces do not deform when an external object comes into contact. This essentially means the incoming object is scattered elastically that is without loss of kinetic energy, only momenta magnitude and directions are changed in accordance with the conservation of linear momentum.

iii. When molecules collide they do not lose energy or time. They bounce off each other so that ‘energy’ and ‘momenta’ are conserved.

We would like to obtain an expression for velocity distribution function. That is we would like to know the fraction of molecules having velocity between v to v+dv for all possible values of velocity.

For this we assume an ensemble of molecules in equilibrium. The ratio of number of molecules in a velocity range to the total number of molecules N gives the probability of finding a molecule in that velocity range.

v. The “phase space” of the ensemble of molecules is defined by a 6 N dimensional space, which constitutes of 3 N spatial components and 3 N velocity components of the N molecules in equilibrium. For a more advanced concept of phase space check the following statistical mechanics lecture.

Inherent ability = difficulty * accomplishment.

All of Physics is this “Inherent ability = difficulty * accomplishment”. Thats just intuitive but can easily be seen to correspond mathematically with the Principle of least action.

First the edifice: whats the problem? The problem is given you move in straight line when every direction is same around you, which direction will you chose? While you are waiting for a good answer from astrologers intelligent people already give a good hint. Think you have some inherent ability which is fixed.

fixed: which changes only if estimated wrong.

That inherent ability is actually action. Accomplishments are adjusted for difficulties, you waded through a swamp 5 meters you would have accomplished in sand 8 meters with that given inherent ability called action. Because action is abstract we have been sticking to time and path-length, but they are not as fundamental, they are merely specifics.