Question

Difference between micro canonical,canonical,and grand canonical

Answer 1

All three ensembles allow you to use some properties of a system - energy or temperature, number of particles or chemical potential - to calculate other useful properties, such as pressure, volume or surface tension.

If you know the energy, use the microcanonical ensemble
Do you have N particles in a completely insulated box, with total energy E?
Imagine you're inside of a completely insulated box. No heat flows in or out. But you can work out how the energy inside the box will be distributed between different particles - and from the energy distribution you can derive all the thermodynamic properties of the system.

If you know the temperature, use the canonical ensemble
If you put your N particles in a warm, non-insulated bath held at temperature T. Heat energy can flow in and out of the system, and the total energy isn't constant.
But you can still guess the energy distribution using the partition function. For example you know you have a ground state energy 0, then the lowest excited state has energy E1, and there are two more states with energy E2, etc. An example might be an atom, with maybe 10 electrons and a whole set of orbitals to put them in.
Here you don't know the total energy (you get that by adding up the energies of the filled states) but you do know the total temperature, and you work with statistics on the principle that low-energy states are more likely to be filled than high-energy ones, at finite temperatures. You then know where the particles most likely are (in which states) and can derive the thermodynamics.

This ensemble is the most frequently used because it's much easier to measure an object's temperature (and use the canonical ensemble) than to measure it's energy (and use the microcanonical ensemble).

If you don't know how many particles, use the grand canonical ensemble
This is similar to the canonical ensemble, except that the number of particles is not constant. This ensemble is useful if you have a chemical reaction (where two particles react and turn into one, which might decay and turn back into two), or a sticky surface (where a particle can adsorb and leave the system, or detach and re-enter).

Chemical potential works similarly to temperature. You use the temperature to calculate the probability that the system has a certain amount of energy, and (using the same math) you use the chemical potential to calculate the probability that the system has a certain number of particles.
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