Problem 15.2: Compare the temperature of distributions

EJS Simulation currently not available.

This animation allows you to change the total number of particles and the total energy of a particular system of 11 boxes, each with an energy of 0 to 10.  Restart.   You can also change the number of states in a box currently set to 40.  To make a change, you must push return (so the input box is no longer yellow) and you must push the "Set Value and Run" button. The animation calculates the values of α and β in the following expressions so that the total energy is constant and the total number of particles is constant (β = 1/kBT for all distributions):

Bose-Einstein: ni = g/(exp(α + βεi) - 1)
Fermi-Dirac: ni = g/(exp(α + βεi) + 1)
Maxwell-Boltzmann (Classical): ni = g/exp(α + βεi)

where ni is the number of particles in region in phase space, εi is the energy of that region (or box) and g is the number of states per box, also called a density of states.

  1. For situations where the average energy per particle is the same, how does the temperature of Fermi-Dirac distribution compare with a classical system?  Specifically, for an average energy per particle of 5, how do the temperatures compare?  What about for an average energy per particle of 1?
  2. The average energy per particle of electrons in a metal (Fermi gas) are close to the Fermi level at very low temperatures (T near zero).  If the electrons in a metal with a Fermi energy of 5 eV behaved classically (like ideal gas particles), what temperature would be necessary for them to have the same average energy per particle as they actually have at T near zero?

Problem by Anne J. Cox
Script by Anne J. Cox and William F. Junkin III
Applet built using EJS (Francisco Esquembre), Open Source Libraries

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