Section 15.4: Exploring Classical, Bose-Einstein, and Fermi-Dirac Statistics

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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. After making a change, you must push return (so the input box is no longer yellow) and you must push the "Set Value and Play" button. Each box represents a region in phase space with the same energy as set up in Section 15.3 and shown below:

To match the distributions you found in Section 15.3, keep number of states/box equal to 40 and number of particles equal to 100. 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:

Bose-Einstein ni = g/(exp(α + βεi) − 1)(15.3)
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.3 The constants α and β must be set so that Σni = N and Σniεi = E.

  1. Start with E = 200 and N = 100. The average energy per particle is 2.  How does the plot of ni vary from distribution to distribution?  Explain the differences. Increase the number of particles to 200.
  2. This animation does not have any protection against dropping the energy too low for a Fermi-Dirac distribution to exist. For example, with E = 100 and N = 100, drop the number of states, g, to 20.  This means that only 20 particles can go into each state. If the particles sit in the lowest energy states possible, what is the lowest possible total energy?  What is the last box that is full? and what is the corresponding energy of that box (ε = 0, ε = 1, ε = 2)?  The last box that is filled (when all the lower energy boxes are full and none of the higher energy boxes have particles) is called the Fermi energy.
  3. If the total number of particles, N, is much less than the number of states, g, in each box, how do the distributions compare?  Try several values of total energy and explain (be sure to try values of average energy per particle below 1).
  4. Keeping the number of particles, N, fixed, and the number of states fixed, change the total energy, E.  How does β change?

What is β?  If we start with the fundamental definition of temperature, we find that β = 1/kBT for all distributions as follows: For constant volume and particle number, temperature is defined by the relationship dE = T dS where S is the entropy and E is the internal energy. Finding the distribution of ni as a function of εi (as in the animation) requires maximizing S = kB lnW, where W is the count of the total number of states (see Section 15.3).  In other words, we find the value of  ni that gives d(lnW) = Σ(∂lnW/∂ni)dni = 0. There are, however, two other conditions to meet (which determine the values of α and β ): keeping the number of particles and the total energy fixed. In equation form, this means Σni = N and Σniεi = E or that αΣdni = 0 and βΣεidni = 0. Thus, to find α and β (numerically in this animation), we solve Σ(dlnW/dni)dni − αΣdni − βΣεidni = 0. However, built into this equation is the relationship Σ(dlnW/dni)dni = βΣεidni, or

 dS/kB = βdE     and thus     β = 1/kBT. (15.7)

What about α? Section 15.5 shows how it depends on the type of system.

3Here we use a constant value for the number of states per box, g.  This is valid in the one-dimensional case (assuming the degeneracy does not change with energy or location in phase space).  Section 15.5 develops an expression for g in terms of momentum, p, for continuous distributions, g(p)dp, which can then be used to model systems we observe. Finally, since the energy increases linearly with box number, this is a model of a collection of one-dimensional harmonic oscillators.

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

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