# Why do we need different ensembles in statistical mechanics?

Why do we study these different ensembles, microcanonical, canonical, grand canonical ensemble ? Are they used for studying different physical system or scenarios?(e.g. in some system you can only treat it as mirocanonical, and in other cases you can only apply canonical ensemble) Do they have the same result at thermodynamic limit?

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Another way to say this is as follows:

1) Microcanonical Ensemble: Isolated system -- No transfer energy in any form (as heat, as work, with matter, or as radiation)

2) Canonnical Ensemble: Closed system --- Energy transfer allowed as heat, but no work, no matter, no radiation.

3) Grand-canonical Ensemble: Open system -- Energy transfer allowed as heat and with matter (no work, and radiation)

4) NPT Ensemble: Energy transfer as heat and work (no matter, no radiation)

The first three are more prevalent in pedagogical teaching of statistical mechanics because of their history and importance in the development of/from thermodynamics, which itself owes a lot to heat-engine development. But other ensembles can be made based on what out of the four kinds of energy transfers you allow. Furthermore, in matter transfers you can have matter of different types (different molecules, electrons, nuclei and so forth). Then if you consider quantum effects you go further in quantum statistical mechanics and so on.

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Because different ensembles apply to different physical situations, number of particles, volume, temperature are fixed in some situations but not in every situation. For equilibrium statistical mechanics they give the same results as long as fluctuations are small

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If somebody tells you what the entropy is as a function of energy, volume, and number of particles, you have all the information you need (for a standard plain vanilla system). It is not necessary to define any other ensemble, but it is convenient. If your system for instance is in contact with a big other system ("reservoir") with which it can exchange energy, then you can either describe system plus reservoir microcanonically, or you describe only your system canonically. The latter is clearly more convenient, since you need not bother about the internal "workings" of the reservoir. For the purpose of your problem the entire reservoir is perfectly well characterized by a single number: its temperature.

The mathematical machinery of Legendre transforms provides a neat way to change from a thermodynamic potential (such as the entropy) to other potentials in which derivatives of the original thermodynamic potential become the new variables, and this transformation is being done without losing information. So, at the end of the day, this is just mathematical convenience: represent the necessary thermodynamic information in ways that are easier to handle in a given situation characterized by a particular set of constraints.

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