## 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/*k*_{B}*T* for all distributions):

Bose-Einstein: |
n_{i} = g/(exp(α + βε_{i}) - 1) |

Fermi-Dirac: |
n_{i} = g/(exp(α + βε_{i}) + 1) |

Maxwell-Boltzmann (Classical): |
n_{i} = g/exp(α + βε_{i}) |

where *n*_{i} is the number of particles in region in phase space, ε_{i }is the energy of that region (or box) and *g* is the number of states per box, also called a density of states.

- 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?
- 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