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Lecture 22. Ideal Bose and Fermi gas (Ch. 7) the grand partition function of ideal quantum gas: Gibbs factor fermions: n i = 0 or 1bosons: n i = 0, 1, 2,..... Outline 1.Fermi-Dirac statistics (of fermions) 2.Bose-Einstein statistics (of bosons) 3.Maxwell-Boltzmann statistics 4.Comparison of FD, BE and MB.

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

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Fermi-Dirac Distribution Fermi-Dirac distribution The 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)

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Fermi-Dirac Distribution At T = 0, 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 ~ k B T. T =0 ~ k B T = ( with respect to ) 1 0 n=N/V – the average density of particles The 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:

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

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Bose-Einstein Distribution Bose-Einstein distribution The 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( ).

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Comparison of FD and BE Distributions Maxwell-Boltzmann distribution:

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Maxwell-Boltzmann Distribution (ideal gas model) Maxwell-Boltzmann distribution The 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:

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Comparison of FD, BE and MB Distribution

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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.

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Comparison between Distributions Boltzmann Fermi Dirac Bose Einstein indistinguishable Z=(Z 1 ) N /N! n K <<1 spin doesn’t matter localized particles don’t overlap gas molecules at low densities “unlimited” number of particles per state n K <<1 indistinguishable integer spin 0,1,2 … bosons wavefunctions overlap total symmetric photons 4 He atoms unlimited number of particles per state indistinguishable half-integer spin 1/2,3/2,5/2 … fermions wavefunctions overlap total anti-symmetric free electrons in metals electrons in white dwarfs never more than 1 particle per state

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“The Course Summary” EnsembleMacrostateProbabilityThermodynamics micro- canonical U, V, N (T fluctuates) canonical T, V, N (U fluctuates) grand canonical T, V, (N, U fluctuate) The grand potential (the Landau free energy) is a generalization of F=-k B T lnZ systems 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 μ. Thus, we need to use to eliminate μ in terms of T and n=N/V. - the appearance of μ as a variable, while computationally very convenient for the grand canonical ensemble, is not natural. Thermodynamic properties of

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