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Bose-Einstein distribution function

Bose-Einstein distribution function. Armed with. We calculate. i n the grandcanonical ensemble. Probability to find the state. With  we can calculate thermal averages:. Again complete thermodynamics via potential. f rom Legendre transformation of Helmholtz free energy.

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Bose-Einstein distribution function

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  1. Bose-Einstein distribution function Armed with We calculate in the grandcanonical ensemble Probability to find the state With  we can calculate thermal averages:

  2. Again complete thermodynamics via potential from Legendre transformation of Helmholtz free energy alternative ways to derive using either  or  for Bose gas identifying average occupation # calculate 1 with Bose-Einstein distribution and using and

  3. For non-interacting bosons with Bose-Einstein distribution function

  4. As a crosscheck we show from that in fact With Simplifying and using

  5. Calculate thermal average using  2 From

  6. Visualizing & discussing the Bose-Einstein distribution function Click here for on-line animation and downloadable live version& source code in Mathematica Inspection of the summations in the partition functions show that convergence requires If lowest single particle energy chemical potential We also see for N=const

  7. Careful analysis (see text page 426) shows that for N=const the fraction N0/N of bosons condensed into the lowest single-particle energy state has the T-dependence: • Bose-Einstein condensation • Phase transition setting at a • Critical temperature Tc • Examples (although with interaction): • Superconductivity • Superfluidity The Bose-Einstein distribution function in the limit such that Boltzmann distribution Let’s find the normalizing factor: normalized Boltzmann factor

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