# Number of particles in a microcanonical ensemble

Is it always assumed that, in a microcanonical ensemble, the number of particles is $N \gg 1$ ?

If no, are all the theorems related to the microcanonical description true even if the number of particles is small ?

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Why do you emphasize the microcanonical ensemble? Is not it a more general question of statistical description applicability in physics? –  freude Apr 25 '13 at 15:06
I already know that a canonical ensemble can be microscopic (the heat bath, however, should be macroscopic). –  Arnaud Apr 25 '13 at 15:22
Which "theorems" are you referring to? –  joshphysics Apr 25 '13 at 15:35

## 1 Answer

Numerical simulations are a good example to see with your own eyes that statistical mechanics results can be gotten without an infinite number of molecules in general and in particular in the microcanonical ensemble. However, one has to be aware of the finite size effects and see what is the difference with what you would expect from textbooks analytical calculations.

For instance, if you ever wondered how can a scale measure the weight of a bottle filled with gas, then the answer is in the kinetic theory of gases and says that the scale will in fact record individual momenta exchange from the particles that collide elastically with the bottom of the bottle.

In principle, you can put very few particles in a simulation and observe what the scale would actually measure. You would see sharp peaks every now and then but if you average over a very long time you will observe an average number for the weight that is that of the gas particles in your bottle.

Increasing the number of particles basically increases the signal/noise ratio and in the thermodynamic limit you just observe one number even on very short time scales.

Also, bear in mind that one of the goals of statistical mechanics is to give a rational for thermodynamics and this can only be done if one looks at the thermodynamic limit (to give a strong example, theoretical descriptions of phase transitions require non analycities in the free energy at the transition and these can only occur in the thermodynamic limit).

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