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What does one mean by canonical and micro canonical ensemble in statistical mechanics? Can one elaborate on this in a very simple way with examples? Pardon me, if it is a very simple thing; I am a math student.

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Microcanonical ensemble means an isolated system with defined energy. The system may be found only in microscopic state with the adequate energy, with equal probability.

Canonical ensemble means a system attached to the "temperature reservoir", which may supply/take infinite amount of energy. The system may be found in any microscopic state, with probability proportional to the Boltzmann factor.

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Thanks valdo for the very concise and clear answer. –  Ashok Nov 16 '11 at 5:42

The canonical ensembles is the collection of a large number of essentially independent system having the some temperature. Volume and the some number of identical particles. The individual system of this ensembles are seperated by rigid impremeable and conducting walls.

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In the 'microcanonical ensemble', you describe the macroscopic state of a system by giving a probability measure on the space of microscopic states. This probability measure maximizes entropy, subject to the constraint that the energy is constant.

In the 'canonical ensemble', you describe the macroscopic state of a system with a probability measure which maximizes entropy subject to the constraint the expectation value of the energy is constant.

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In a microcanonical ensemble isn't it true that all microstates are equally probable and hence the probability distribution is just flat? –  user6818 Apr 14 at 1:53

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