What is the differences between Fermi Dirac statistic and Maxwell Boltzmann statistic?

Answer 1

the essential difference between M-B statistics and FD statistics (and their quantum cousin, bose-einstein statistics) is that maxwell-boltzmann statistics describe so-called "distinguishable" particles, whereas F-D and B-E statistics are used to describe "indistinguishable" particles.

these statistics are used when trying to describe the statistical behavior of a collection of these particles. sorry if that sounds like a circular definition; i'll try to elaborate.

statistical mechanics is all about the "average" behavior of lots of things - atoms, molecules, or particles. such averages are usually taken by considering every possible state of the system, and the probability of being in each state.

so, for example - you should be familiar with the concept of energy levels if you're asking this question, so you can think about the distribution of a collection of particles among some energy levels.

say you have a collection of 10 particles and there are 2 possible energy states that can be occupied.

if the particles are distinguishable, that means you can put labels on them (particle #1, particle #2, etc). if particles #1,#2,#3,#4,#5 are in state 1, and #6,#7,#8,#9,#10 are in state 2, that is a DIFFERENT distribution than if you were to switch, say, #6 and #1. there are 2^10 = 1024 ways of arranging these 10 particles among 2 states.

however, if the particles are indistinguishable, this means that you cannot label them, and so a collection of 5 particles in state 1 and 5 particles in state 2 is considered a single way of arranging them, even if you switch two particles between levels. there are only 11 different ways of doing this, compared to the 1024 ways for distinguishable particles!

you'll see that there are 2002 different distributions for distinguishable particles, but that there are only 26 possible distributions for indistinguishable particles.

anyway, what you do with this is that you then apply a fundamental theorem of equilibrium statistical mechanics, which basically says that "in a system at equilibrium, all accessible states are equally likely".

if this is true, then for the distinguishable particles case, the system will spend equal amounts of time in all 2002 arrangements. the likelihood of being in any particular arrangement is 1/2002.

HOWEVER, notice that some of these 2002 states have the same NUMBER of particles in a particular energy level. that is, certain distributions of *the number of particles in each state* are more likely than others, and if you keep increasing the number of particles, you will see a single distribution will start to dominate.

in the limit of lots of particles and lots of energy states, you can write that distribution down as an exponential decay. it's called the maxwell-boltzmann distribution function, and it tells you how "populated" a particular energy level is.

for indistinguishable particles, on the other hand, we have to pause and draw a further distinction.

there are 2 types of statistics that apply to indistinguishable particles, one called Bose-Einstein and one called Fermi-Dirac. in the B-E statistics, there are no restrictions on the number of particles (called "bosons" - things like photons, He4 atoms, gluons) in each state, and so we just get our 26 possible arrangements. again we apply our equal probability theorem, and from that we get a bose-einstein distribution function, which is similar in form to the M-B d.f. but behaves drastically different at low temperatures. (look up "bose-einstein condensate" to see how different!)

in the F-D statistics however, the particles are subject to a further restriction known as the "pauli exclusion principle" - that is, a maximum of 2 particles can occupy any state.

if we go back to our 6 particles with 9 quanta example, you'll see that there are only *5* possible arrangements for these particles (called "fermions" - things like electrons, neutrons, protons). again, the requirement of equal probability in all states applies, and we can get a fermi-dirac distribution function, which also looks similar in form to the other 2 but behaves extremely differently. the "max of 2 in each level" requirement shifts the distribution away from the low energy levels.

lots of interesting physics comes about only in the presence of large numbers of "things" - atoms, electrons, whatever. it's (practically) impossible to construct a complicated equation of motion for such a large number of particles, though in principle it could be done. (though anyone who's ever tried solving even a 3-body problem will tell you about how miserable it is.)

so we use these statistical descriptions to describe the "average" behavior of the collection. fermi statistics are very important to the theory of semiconductors, for instance, where lots of interacting electrons cause the formation of a forbidden energy zone called a band gap.

maxwell distributions are a good way to describe the distribution of velocities in a gas of atoms.

also, an experimentalist might use these statistics in reverse. they could measure the distribution of energy levels of some particle, which could tell them about which statistic they obey, and from that one can infer other information about the particle's properties.

generally, you want to use the maxwell-boltzmann statistics/distribution when you are dealing with a "classical" system and the F-D or B-E formulations when you are dealing with a quantum mechanical system. often the hard part is telling the difference between the two.