The (NTV) ensemble

The modern derivation of the Maxwell-Boltzmann distribution is as follows:

Energy and the number of particles are additive properties

The entropy


is maximized:

where and are Lagrange multipliers.

By standard arguments


The partition function is


If H is separable


Q will factorize

This completes the derivation.

However, let us continue with a derivation of the magnitude of the fluctuations in the system:

By differentiation of


with respect to T:

where Cv is the heat capacity.

This equation shows that the energy will fluctuate in a finite system. The fluctuations in the total energy scales as N0.5 while the energy per atom scales as N -0.5.

The magnitude of the fluctuations severely complicates the investigation of thermodynamic behaviour through simulations.

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Author Per Stoltze stoltze@fysik.dtu.dk