Section 15.3: Understanding Probability Distributions
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The best distribution of many particles across regions of phase space is that which maximizes the number of ways the particles can be arranged in these regions (or boxes for the purpose of this animation) for a given energy and given number of particles. This is the most probable state. The best distribution depends on whether these particles are classical particles, bosons or fermions. Why does the type of particle matter? It matters because of the properties of the type. The difference between fermions and the others is perhaps the most obvious: fermions cannot occupy the same state. So, if a given "box" (volume in phase space) has 40 states, only 40 fermions can be in that box while more than 40 bosons or classical particles can be in that box. The counting of number of particle arrangements is a bit different for classical and quantum particles (either bosons or fermions) because classical particles are identical, but distinguishable, while quantum particles are indistinguishable. For boxes with 40 states, each, then, we must maximize the number of possible arrangements of particles in the 40 states among the boxes available. Restart.
This animation shows a systems of 11 boxes, labeled 0 to 10, with energy equivalent to the box label. Each box has 40 states and there are 100 particles to distribute among the boxes. Select an average energy per particle and a distribution (B-E is Bose-Einstein for bosons, F-D is Fermi-Dirac for fermions, and M-B is Maxwell-Boltzmann for a classical system or collection of distinguishable particles). The black bar is a ratio of the probability of the particles in the configuration shown compared with the probability of particles in the most probable configuration for that energy and type of particle. In order for the total number of particles and the total energy to remain constant, decreasing particles from a box puts one particle in a higher energy box and one in a lower energy box. Similarly, increasing particles in a box requires one particle to move from each neighboring box. To begin, decrease the particles in the box they all initially begin in. Adjust the particle distribution to increase the total probability for the distribution of particles in the boxes.
Notice that your probability ratio increases as you spread the particles out more. At an average energy per particle of 5, what looks like it will be the best distribution? How do the bosons compare with the fermions and the classical particles? What about for an energy per particle of 1?
Mathematically, the count of the number of particle arrangements is called the multiplicity of states, W, and is given by the following for the three different cases:1
WB-E = Π (ni + gi − 1) !/ni!(gi − 1) ! (15.1)
WF-D = Π gi !/ni ! (gi − ni) ! (15.2)
WM-B = N! Π (gi)ni/ni ! (15.3)
where gi is the number of states in the ith box, ni is the number of particles in the ith box, N is the total number of particles and Π signifies that we must multiply each ith term together from i = 0 to i = N. So, for this animation, gi = 40 for all boxes and N = 100. Note that the multiplicity, W is related to entropy, S: S = kB lnW. Taking the above expressions for the possible arrangements of particles and maximizing them, we find the following particle distributions:2
nB-E i = gi/(exp(α + βεi) − 1), (15.4)
nF-D i = gi/(exp(α + βεi) + 1), (15.5)
nM-B i = gi/exp(α + βεi), (15.6)
where α and β are set so that Σni = N and Σniεi = E. Solving these for our particular set up (11 boxes, 100 particles, and 40 states in each box), you can click below to see the ideal distribution for the energy/particle of interest (change the energy/particle above):
How do the ideal distributions compare at different values of average energy per particle? Notice that the biggest differences occur for the lowest energies. Why?
1See Kittel and Kroemer, Thermal Physics, 2nd edition (1980), for details of calculation of probabilities of distributions: fermions, pp. 10-15, bosons, p. 25 and classical systems, pp. 75-76.
2Actually, we generally maximize ln W so we can use Stirling's approximation to handle factorials of large numbers. We then set d(lnW) = 0. To ensure that the number of particles stays fixed, as does the energy, we add the terms αΣdni = 0 and βΣεidni = 0 to d(lnW) = 0 and solve for ni that maximizes lnW.
Section by Anne J. Cox and William F. Junkin III.