1 Lecture 5 The grand canonical ensemble. Density and energy fluctuations in the grand canonical ensemble: correspondence with other ensembles. Fermi-Dirac statistics. Classical limit. Bose-Einstein statistics.

2 The grand canonical ensemble. dw s (N s ) N s d  s (N s )d  s (N s ) s N s We want the probability dw s (N s ) of a state of the subsystem in which the subsystem contains N s particles and is found in the element d  s (N s ) of its phase space. The notation d  s (N s ) reminds us that the nature of phase space s changes with N s : the number of dimensions will change. subsystem s r total system t microcanonical ensemble with constant energy and constant number of particles. We now consider a subsystem s which can exchange particles and energy with the heat reservoir r, the total system t being represented by a microcanonical ensemble with constant energy and constant number of particles.

3 We do not care about the state of the remainder of the system provided only that (5.1) Then, by analogy with the treatment of the canonical ensemble, (5.2) (5.3) or We expend  r in a power series: (5.4) recalling that (5.5)

4 (5.6) (5.7) (5.8) Dropping the subscript s, we have where A is normalization constant. Writing by convention we have where grand canonical ensemble is the grand canonical ensemble. N   N i  i.  grand potential. If several molecular species are present, N  is replaced by  N i  i. The quantity  is called the grand potential.

5 Grand partition function (5.9) (5.10) The normalization is We define the grand partition function (5.11)

6 Connection with thermodynamic functions Proceeding at the same way that in the case of the canonical ensemble, we get for the entropy (5.12) (5.13) or G, where G is the Gibbs free energy (5.14) (5.15) Now by (5.16) whence (5.17) Let us prove now that  =

7 (5.18) G Np  p  p  G N Now G may be written as N times a function of p and  along. Both p and  are intrinsic variables and do not change value when two identical systems are combined in one. For fixed p and , G is proportional to N and consequently (5.19) g Gibbs free energy per particle where g is the Gibbs free energy per particle. In this case, we have (5.20) whence (5.21) Then from (5.13) (5.22) (5.13)

8 and by comparing with (5.13) we see that (5.23) Other thermodynamic quantities may be calculated from . We can easily get (5.24) (5.25) (5.26) (5.27) (5.28)

9 Fermi-statistics and Bose Statistics The occupation numbers, or number of particles in each one-particle state are strongly restricted by a general principle of quantum mechanics. The wave function of a system of identical particles must be either symmetrical (Bose) or antisymmetrical (Fermi) in permutation of a particle of the particle coordinates (including spin). It means that there can be only the following two cases: for Fermi-Dirac Distribution (Fermi-statistics) n=0 or 1 for Bose-Einstein Distribution (Bose-statistics) n=0,1,2, The differences between the two cases are determined by the nature of particle. Particles which follow Fermi-statistics are called Fermi- particles (Fermions) and those which follow Bose-statistics are called Bose- particles (Bosones). Electrons, positrons, protons and neutrons are Fermi-particles, whereas photons are Bosons. Fermion has a spin 1/2 and boson has integral spin. Let us consider this two types of statistics consequently.

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11 There are two possible outcomes: If the result confirms the hypothesis, then you've made a measurement. If the result is contrary to the hypothesis, then you've made a discovery. Born: 8 Aug 1902 in Bristol, England Died: 20 Oct 1984 in Tallahassee, Florida, USA Enrico Fermi Physicist Fermi-Dirac Distribution

12 Fermi-Dirac Distribution As the particles are assumed to be non-interacting it is convenient to discuss the system in terms of the energy states  i  i of one particle in a volume VV. VV. We specify the system by specifying the number of particles ni ni ni ni, occupying the eigenstate ii ii. We classify i i i i in such way that i denotes a single state, not the set of degenerate states which may have the same energy. On the above model the Pauli principle allows only the values n i =1,0 n i =1,0. This is, of course, just the elementary statement of the Pauli principle: a given state may not be occupied by more than one identical particle. The partition function of the system is We consider a system of identical independent non-interacting particles sharing a common volume and obeying antisymmetrical statistics: that is, the spin 1/2 and therefore, according to the Pauli principle, the total wave function is antisymmetrical on interchange of any two particles.

13 (5.29)  {n i }n n i n i 0 or 1. subject to. We note that the  in the exponent runs over all one-particle states of the system; {n i } represents n allowed set of values of the n i ; and runs over all such sets. Each n i may be 0 or 1. Let us consider as an example a system with two states  1 and  2. The upper sum reads (5.30) (5.31) the other sum reads but we have not included the requirement n 1 +n 2 =N. If we take N=1, we have (5.32)

14 For a system with many states and many particles it is difficult analytically to take care of the condition  n i =N. It is more convenient to work with grand canonical ensemble. We have for the grand partition function (5.33) so that (5.34) A simple consideration shows that we may reverse the order of the  and  in (5.34). We note that the significance of the  changes entirely, from {n i }=0,1. Every term, which occurs, for one order will occur for the other order (5.35) (5.36) where

15 Now from the definition of the grand partition function (5.37) we have (5.38)(5.39) where For ni ni restricted to 0,1, we have (5.40) Now (5.41)

16 with it appears reasonable to set (5.42) The same result can be provided by direct use of averaging in the grand canonical ensemble This may be simplified using the form (5.36): (5.43) (5.44)

17 Fermi-Dirac distribution law. or (5.45) in agreement with (5.42). This is the Fermi-Dirac distribution law. It is often written in terms of f(  ), where f is the probability that a state of energy is occupied: (5.46) It is implicit in the derivation that  is the chemical potential. Often  is called the Fermi level, or, for free electron gas, the Fermi energy EF.EF.

18 Classical limit For sufficiently large  we will have (  -  )/kT>>1, and in this limit (5.47) This is just the Boltzmann distribution. The high-energy tail of the Fermi-Dirac distribution is similar to the Boltzmann distribution. The condition for the approximate validity of the Boltzmann distribution for all energies  0 is that (5.48)

19 Bose-Einstein Distribution

20 Bose-Einstein Distribution Particles of integral spin (bosons) must have symmetrical wave functions. There is no limit on the number of particles in a state, but states of the whole system differing only by the interchange of two particles are of course identical and must not be counted as distinct. For bosons we can use the results (5.38) and (5.39), but with n i =0,1,2,3,...., so that where(5.49) Thus (5.50) or This is the Bose-Einstein distribution (5.51) (5.38)

21 We can confirm (5.50) by a direct calculation on n j. Using the previous result we have or in agreement with (5.50). (5.52) (5.53)