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)|
|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.
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