The (NTp) ensemble

To demonstrate the superiority of the modern derivation of the Maxwell-Boltzmann distribution, we derive the distribution for the (NTp) ensemble. Using the modern derivation, this is not harder than for the (NTV) emsemble, while the derivation using the classical derivation it is very hard.

The phase-points are (,,V).



The entropy is maximized:

where , , and are Lagrange multipliers.

By standard arguments


The partition function is


This completes the derivation, but let us again continue with a derivation of expressions for the magnitude of the fluctuations.

By differentiation of


by T:

where C'p is the heat capacity caused by the potential energy and is the thermal expansion

This equation shows that the energy will fluctuate in a finite system.

By differentiation of


by V

where is the isothermal compressibility

This equation shows that the volume will fluctuate in a finite system.

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Author Per Stoltze stoltze@fysik.dtu.dk