The grand canonical ensemble'

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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.

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The grand canonical ensemble.
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.
We want the probability dws(Ns) of a state of
the subsystem in which the subsystem contains Ns
particles and is found in the element d?s(Ns) of
its phase space. The notation d?s(Ns) reminds us
that the nature of phase space s changes with Ns
the number of dimensions will change.
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We do not care about the state of the remainder
of the system provided only that
Then, by analogy with the treatment of the
canonical ensemble,
or
We expend ?r in a power series
(5.4)
recalling that
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Dropping the subscript s, we have
(5.5)
where A is normalization constant. Writing by
convention
(5.6)
we have
(5.7)
where
(5.8)
is the grand canonical ensemble.
If several molecular species are present, N? is
replaced by ? Ni?i. The quantity ? is called the
grand potential.
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Grand partition function
The normalization is
(5.9)
We define the grand partition function
(5.10)
(5.11)
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Connection with thermodynamic functions
Proceeding at the same way that in the case of
the canonical ensemble, we get for the entropy
or
G, where G is the Gibbs free energy
Let us prove now that ?
Now by
whence
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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
where g is the Gibbs free energy per particle.
In this case, we have
whence
Then from (5.13)
(5.13)
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and by comparing with (5.13) we see that
Other thermodynamic quantities may be calculated
from ?. We can easily get
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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)
n0 or 1
for Bose-Einstein Distribution (Bose-statistics)
n0,1,2,3......
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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Fermi-Dirac Distribution
Enrico FermiPhysicist 1901 - 1954
Born 8 Aug 1902 in Bristol, EnglandDied 20 Oct
1984 in Tallahassee, Florida, USA
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.   
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Fermi-Dirac Distribution
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.
As the particles are assumed to be
non-interacting it is convenient to discuss the
system in terms of the energy states ?i of one
particle in a volume V. We specify the system by
specifying the number of particles ni , occupying
the eigenstate ?i . We classify ?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 ni1,0. 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
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Let us consider as an example a system with two
states ?1 and ?2. The upper sum reads
the other sum reads
but we have not included the requirement n1n2N.
If we take N1, we have
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For a system with many states and many particles
it is difficult analytically to take care of the
condition ?niN. It is more convenient to work
with grand canonical ensemble. We have for the
grand partition function
so that
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
ni0,1. Every term, which occurs, for one order
will occur for the other order
where
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Now from the definition of the grand partition
function
we have
where
For ni restricted to 0,1, we have
Now
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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)
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Fermi-Dirac distribution law.
or
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
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.
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Classical limit
For sufficiently large ? we will have
(?-?)/kTgtgt1, and in this limit
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
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Bose-Einstein Distribution
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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 ni0,1,2,3,...., so that
Thus
or
This is the Bose-Einstein distribution
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We can confirm (5.50) by a direct calculation on
nj. Using the previous result
we have
or
in agreement with (5.50).