Lecture 22. Ideal Bose and Fermi gas (Ch. 7)the grand partition function of ideal quantum gas: Gibbs factorfermions: ni = 0 or 1bosons: ni = 0, 1, 2, .....OutlineFermi-Dirac statistics (of fermions)Bose-Einstein statistics (of bosons)Maxwell-Boltzmann statisticsComparison of FD, BE and MB.

The Partition Function of an Ideal Fermi GasIf the particles are fermions, n can only be 0 or 1:The grand partition function for all particles in the ith single-particle state (the sum is taken over all possible values of ni) :Putting all the levels together, the full partition function is given by:

Fermi-Dirac DistributionFermi-Dirac distributionThe mean number of fermions in a particular state:The probability of a state to be occupied by a fermion:( is determined by T and the particle density)

Fermi-Dirac DistributionAt T = 0, all the states with < have the occupancy = 1, all the states with > have the occupancy = 0 (i.e., they are unoccupied). With increasing T, the step-like function is smeared over the energy range ~ kBT.T =0~ kBT = (with respect to ) 10n=N/V the average density of particlesThe macrostate of such system is completely defined if we know the mean occupancy for all energy levels, which is often called the distribution function:While f(E) is often less than unity, it is not a probability:

The Partition Function of an Ideal Bose GasIf the particles are Bosons, n can be any #, i.e. 0, 1, 2, The grand partition function for all particles in the ith single-particle state (the sum is taken over all possible values of ni) :Putting all the levels together, the full partition function is given by:

Bose-Einstein DistributionBose-Einstein distributionThe mean number of Bosons in a particular state:The probability of a state to be occupied by a Boson:The mean number of particles in a given state for the BEG can exceed unity, it diverges as min().

Comparison of FD and BE DistributionsMaxwell-Boltzmann distribution:

Maxwell-Boltzmann Distribution (ideal gas model)Maxwell-Boltzmann distributionThe mean number of particles in a particular state of N particles in volume V:MB is the low density limit where the difference between FD and BE disappears. Recall the Boltzmann distribution (ch.6) derived from canonical ensemble:

Comparison of FD, BE and MB Distribution

Comparison of FD, BE and MB Distribution (at low density limit)MB is the low density limit where the difference between FD and BE disappears. The difference between FD, BE and MB gets smaller when gets more negative.

Comparison between DistributionsBoltzmannFermiDiracBose EinsteinindistinguishableZ=(Z1)N/N!nK

The Course SummaryThe grand potential(the Landau free energy) is a generalization of F=-kBT lnZsystems are eventually measured with a given density of particles. However, in the grand canonical ensemble, quantities like pressure or N are given as functions of the

natural variables T,V and . Thus, we need to use to eliminate in terms of T and n=N/V. the appearance of as a variable, while computationally very convenient for the grand canonical ensemble, is not natural. Thermodynamic properties of

EnsembleMacrostateProbabilityThermodynamicsmicro-canonicalU, V, N(T fluctuates)canonicalT, V, N(U fluctuates)grand canonicalT, V, (N, U fluctuate)

*The partition functions of different levels are multiplied because they are independent of one another (each level is an independent thermal system, it is filled by the reservoir independently of all other levels).

*The partition functions of different levels are multiplied because they are independent of one another (each level is an independent thermal system, it is filled by the reservoir independently of all other levels).

**