Download 1 16. The grand canonical ensemble theory for a system in

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16. The grand canonical ensemble theory for a system in equilibrium with a heat/particle
reservoir
(Hiroshi Matsuoka)
In this chapter, we will discuss the third approach to calculate thermal properties of a
microscopic model (the canonical ensemble approach has been discussed in Ch.12 and 13 while
the microcanonical ensemble approach has been briefly discussed in Sec.15.1). Traditionally,
this approach is called “the grand canonical ensemble theory.” Two main motivations for having
yet another approach are: (1) to be able to deal with identical or indistinguishable particles
without resorting to the 1 N! trick used for the monatomic and diatomic ideal gas models as this
trick does not work at very low temperatures nor at very high densities; (2) to be able to treat two
types of identical particles, bosons and fermions, within a single theoretical framework with a
minimum amount of work. The first and the second motivations are actually closely connected
because the difference between bosons and fermions becomes most notable at very low
temperatures or at very high densities. The grand canonical ensemble approach is thus tailored
to meet these demands.
The microscopic models we will discuss in the next two chapters
1. The ideal “phonon” (boson) gas model for lattice waves in a solid
Our main goal in studying this model is to calculate its molar heat capacity at constant
volume:
#% DT3 (T << ! )
cv = $
%&3" R (T >> ! )
where ! is the characteristic temperature for the phonons or the Debye temperature that
separates the low-temperature regime from the high-temperature regime for the phonons.
2. The free “electron” (fermion) gas model for conduction electrons in a simple metal
Our main goal in studying this model is to calculate its molar heat capacity at constant
volume:
"$ !T
cv = #
$%(3 2) R
(T << TFermi )
(T >> TFermi )
2
where TF is the characteristic temperature for the conduction electrons or the Fermi temperature
that separates the low-temperature regime from the high-temperature regime for the electrons.
16.1 A system in equilibrium with a heat/particle reservoir
In the grand canonical ensemble approach, we consider a macroscopic system of volume V
that is attached to a heat/particle reservoir at temperature T and with chemical potential µ˜ .
Chemical potential: an intensive state variable that controls particle number density
Just as heat flows from a high-temperature to a low-temperature system, particles flow from
a high- µ˜ to a low- µ˜ system. When two systems are in thermal and “diffusive” contact with
each other so that they can exchange energy through heat and particles through “diffusion,” they
will eventually reach thermal and diffusive equilibrium, which is characterized by (1) the equal
temperatures and (2) the equal chemical potentials in the systems. Just as temperature is an
intensive quantity, chemical potential is also an intensive quantity. For more detail on chemical
potential, see Appendix #10.
As we will see below, the main role of the chemical potential in the grand canonical
ensemble approach is to control the number density or the molar volume of a system. In other
words, the molar volume can be regarded as a function of T and µ˜ :
v = v (T, µ˜ )
On the macroscopic level
We measure U and N or n as functions of T, V and µ˜
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On the microscopic level
Energy and particles are being exchanged between the system and the reservoir so that the
total energy E and the number Nˆ of particles in the system fluctuate around their average values,
U and N, which we observe on the macroscopic level.
16.2 The grand canonical ensemble theory based on an extensive state variable called
thermodynamic potential
In the grand canonical ensemble approach, we calculate a macroscopic state variable called
“the thermodynamic potential” ! from a microscopic quantity called “grand partition function”
! through the following relation:
!( T, V, µ˜ ) = "k BT ln #( T, V, µ˜ / N Avogadro) ,
which provides a critical link between the macroscopic level and the microscopic level, just as
the following relation does in the canonical ensemble approach.
F( T, V,n ) = !k BT ln Z(T ,V,nN Avogadro) .
In macroscopic thermodynamics, the thermodynamic potential is defined by
! " U # TS # µ˜ n.
As U, S, and n are all extensive while T and µ˜ are intensive, the thermodynamic potential is an
extensive state variable.
16.3 Thermal properties can be calculated directly from the thermodynamic potential
Using the “extended” form of the fundamental equation of thermodynamics,
dU = TdS ! PdV + µ˜ dn,
we find
d! = " SdT " PdV " ndµ˜ .
Derivation
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d! = dU " d (TS) " d( µ˜ n)
= (TdS " PdV + µ˜ dn) + (TdS + SdT ) + (µ˜ dn + ndµ˜ )
= "SdT " PdV " ndµ˜
Comparing this equation, d! = " SdT " PdV " ndµ˜ , with the following calculus equation,
# " !&
# "! &
# " !&
(( dµ˜ ,
d! = %
( dT + %
( dV + %%
$ "T ' V ,µ˜
$ "V ' T ,µ˜
$ "µ˜ ' T ,V
we obtain
$ "# '
S (T, V, µ˜ ) = ! &
) ,
% "T ( V ,µ˜
$ " #'
P(T ,V, µ˜ ) = !&
) .
% "V ( T ,µ˜
and
$ "# '
)) ,
n(T ,V, µ˜ ) = ! &&
% " µ˜ ( T ,V
which allow us to calculate S, P, and n as functions of T, V, and µ˜ . We can also calculate the
internal energy U as a function of T, V, and µ˜ by using
U(T ,V, µ˜ ) = ! + TS + µ˜ n.
It would be more convenient to express all the thermal properties as functions of T, v, and n. In
order to achieve that goal, we need to reexpress the thermodynamic potential as –PV.
The thermodynamic potential is nothing but –PV
As the thermodynamic potential is extensive, it must scale with the system size so that it can
be expressed in the following form:
!( T, V, µ˜ ) = V " (T , µ˜ ) ,
from which we obtain
$ " #'
# (T, V, µ˜ )
P(T ,V, µ˜ ) = !&
) = !* (T , µ˜ ) = !
% "V ( T ,µ˜
V
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so that
!( T, V, µ˜ ) = "P (T, V, µ˜ )V .
The above equation for P in terms of ! ( T, µ˜ ) also implies that we can find the pressure as a
function of T, V, and µ˜ directly from ! or ! .
Derivation of !( T, V, µ˜ ) = V " (T , µ˜ )
Being extensive, ! must satisfy !( T, "V, µ˜ ) = "!( T, V, µ˜ ) .
By choosing ! to be
! = 1 V , we get
1
!( T,1, µ˜ ) = !(T ,V, µ˜ )
V
so that
!( T, V, µ˜ ) = V!(T ,1, µ˜ ) " V# (T, µ˜ ) .
As ! = U " TS " µ˜ n, the above result ! = " PV implies the following “Euler equation,”
U ! TS + PV ! µ˜ n = 0 ,
which simply means that U, T, S, P, V, n, and µ˜ depend on each other.
Chemical potential as a function of T and v: µ˜ = µ˜ (T, v )
Using the following equation,
$ "# '
)) ,
n(T ,V, µ˜ ) = ! &&
% " µ˜ ( T ,V
we find the mole number as a function of T, V, and µ˜ :
n = n(T ,V, µ˜ ) .
Solving this equation for V, we obtain
V = nv( T, µ˜ ) ,
where v is the molar volume as a function of T and µ˜ :
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v = v (T, µ˜ ) .
Solving this equation for µ˜ , we then find µ˜ as a function of T and v:
µ˜ = µ˜ (T, v ) .
State variables as functions of T, v, and n
Using µ˜ = µ˜ (T, v ) , we can replace the independent variable µ˜ for P, S, and U by T and v so
that we can express these state variables as functions of T, V, and n.
$ " #'
"(T ,V, µ˜ )
(1) P(T ,V, µ˜ ) = !&
)
or P = !
= !# (T, µ˜ ) and µ˜ = µ˜ (T, v ) leads to P = P (T, v ) .
% "V ( T ,µ˜
V
$ " #'
$ "* '
$ "* '
(2) S = ! &
) = !V &
) = !nv &
) and µ˜ = µ˜ (T, v ) leads to S = ns(T ,v ) .
% " T ( V ,µ˜
% " T ( µ˜
% " T ( µ˜
(3) U = ! + TS + µ˜ n = nv" (T , µ˜ ) + T {ns(T ,v )} + nµ˜ (T ,v ) leads to U = nu(T, v ) .
$ # !'
$ #
$ #! '
# ' $ !'
)) = "T && T
)) & ) and µ˜ = µ˜ (T, v )
Or U = ! + TS + µ˜ n = ! " T &
) " µ˜ &&
+ µ˜
% #T ( V ,µ˜
% #µ˜ ( T ,V
% #T
# µ˜ ( % T (
also leads to U = nu(T, v ) .
Example: The monatomic ideal gas model as a minimal model for the universality class of
low-density monatomic gases. In the monatomic ideal gas model, we model all the atoms
in the gas to be point particles that do not exert forces on each other. We will later
calculate the thermodynamic potential of this model to be
V
˜ ( )
!( T, V, µ˜ ) = "
RTe µ / RT ,
vQ (T )
where
" 2 !h 2 % 3/ 2
'' .
vQ ( T ) = NAvogadro$$
# mkB T &
(0)
$ "# '
V
˜ (
)
)) =
n(T ,V, µ˜ ) = ! &&
e µ / RT
% " µ˜ ( T ,V vQ (T )
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˜ ( RT )
so that v = vQ (T )e! µ /
"$ v &$
'.
and µ˜ (T ,v ) = ! RT ln #
$% vQ ( T ) $(
Also !( T, v, n) = " nRT .
$ "# '
" nRT RT
1
nRT RT
˜ ( )
=
=
or P = ! &
) =
RTe µ / RT =
=
.
% " V ( T, µ˜ vQ (T )
V
V
v
V
v
(i)
P=!
(ii)
$ µ˜
$ "# '
V
5'
˜ ( )
S (T, V, µ˜ ) = ! &
) =R
e µ / RT & !
+ )
% "T ( V ,µ˜
% RT 2 (
vQ (T )
(HW#16.3.1: show this)
( !# v %# 5 +
& + -.
so that S = nR ** ln "
#
(
)
v
T
) $ Q #' 2 -,
(iii)
U = ! + TS + µ˜ n =
3
nRT
2
(HW#16.3.2: show this)
or
$ # !'
$ #! '
))
U = ! + TS + µ˜ n = ! " T &
) " µ˜ &&
% #T ( V ,µ˜
% #µ˜ ( T ,V
=
3
V
( )
RT
e µ˜ / RT
2
vQ (T )
(HW#16.3.3: show this)
so that
U=
3
3
nRT = PV
2
2
Answers for the homework questions in Sec.16.3
HW#16.3.1
As vQ ( T ) ! T "3 2 ,
d ( Ax ! )
dvQ (T )
Ax !
3 v (T )
, where
= A!x ! "1 = !
is used.
=! Q
dx
x
dT
2 T
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$ "# '
S (T, V, µ˜ ) = ! &
)
% "T ( V ,µ˜
=R
$ ˜ '
V
V
V
˜ ( )
µ˜ /( RT ) dvQ (T )
µ˜ / ( RT ) ! µ
& 2)
e µ / RT ! R
Te
+
R
Te
% RT (
vQ (T )
vQ (T )2
dT
vQ (T )
=R
$ µ˜
V
5'
e µ˜ / ( RT ) & !
+ )
vQ (T )
% RT 2 (
HW#16.3.2
U = ! + TS + µ˜ n
* #% v '% 5 #% v '%
( + / " RT ln $
(n ,
= "nRT + TnR ,, ln $
%& vQ ( T) %)
+ %& vQ (T ) %) 2 /.
3
= nRT
2
HW#16.3.3
$ # !'
$ #! '
))
U = ! + TS + µ˜ n = ! " T &
) " µ˜ &&
% #T ( V ,µ˜
% #µ˜ ( T ,V
$ #
# ') $ ! '
= "T && T
+ µ˜
& )
# µ˜ )( % T (
% #T
'
$ #
# ' $&
V
µ˜ / ( RT ) )
&
˜
)
= "T & T
+µ
"R
e
)
# µ˜ )( &% vQ (T )
% #T
(
* $
$ ˜ ''
V
V
µ˜ / ( RT ) dvQ ( T )
µ˜ / ( RT ) " µ
= "T , T&& R
e
"
R
e
&
))
,+ % v Q (T )2
dT
vQ (T )
% RT 2 ( )(
$
'V
µ˜ / ( RT ) 1 ) /
&
˜
+ µ & "R
e
vQ (T )
RT )( /.
%
3
V
( )
= RT
e µ˜ / RT
2
vQ (T )
16.4 Macro from micro: the thermodynamics potential of a microscopic model from its
Gibbs sum or the grand partition function
The link between a microscopic model and its thermodynamic potential ! is provided by
what we call the “Gibbs sum” or the “grand partition function” ! , which is a function of the
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temperature T, the volume V, and the chemical potential per particle µ of the model and is
directly related to the thermodynamic potential by
!( T, V, µ˜ ) = "k BT ln #( T, V, µ˜ / N Avogadro)
As with F = !k B T ln Z , we can derive this relation based on the microscopic definition of
entropy (for more detail, see Appendix #10). This relation or the “bridge” between the
macroscopic quantity, the thermodynamic potential, and the microscopic quantity, the Gibbs sum
or the grand partition function, as well as F = !k B T ln Z and S = k B lnW , are the most important
equations in statistical mechanics.
Example: The monatomic ideal gas model as the minimal model for the universality class of
low-density monatomic gases. For the monatomic ideal gas model, we will later
calculate its Gibbs sum or grand partition function to be
ln !(T, V, µ ) "
V
VQ ( T )
e
µ /( k B T )
,
where the quantum volume is given by
" 2!h 2 % 3/ 2
'' .
V Q (T ) = $$
# mkB T &
As we will see later, the above Gibbs sum is valid only when the distinction between
bosons and fermions is irrelevant.
The thermodynamic potential of this model is then given by
!( T, V, µ˜ ) = "
V
˜ ( )
RTe µ / RT ,
vQ (T )
(HW#16.4.1: show this)
where we have used N AvogadrokB = R and the quantum molar volume vQ ( T ) is related to
the quantum volume V Q (T ) by
" 2 !h2 % 3/ 2
'' .
vQ ( T ) = NAvogadroV Q (T ) = N Avogadro$$
# mk BT &
10
Answer for the homework question in Sec.16.4
HW#16.4.1
!( T, V, µ˜ ) = "k BT ln #( T, V, µ˜ / N Avogadro) = "k BT
= " NAvogadrok BT
="
V
N AvogadroVQ (T )
e
V
µ˜ / ( N
k T)
e Avogadro B
V Q (T )
(
µ˜ / N Avogadrok B T
)
V
RTe µ˜ / ( RT )
vQ (T )
16.5 Energy eigenvalues of a microscopic model
Before we calculate the Gibbs sum or the grand partition function for a microscopic model,
we need to obtain the energy eigenvalues for the model by solving the following Schrödinger
equation for the model:
r r
r
r r
r
H!s ( r1 , r2 ,..., rN ) = Es (V, N )! s ( r1 , r2 ,..., r N )
where the index s is the label or “the quantum number” for the model’s energy eigenstates, each
r r
r
of which is represented by the quantum wavefunction ! s (r1 , r2 ,..., rN ) , which depends on the
positions of the N microscopic particles in the model and is accompanied by a corresponding
energy eigenvalue Es (V, N ) , which depends on the volume V of the model system and the
number N of the microscopic particles, while the Hamiltonian operator H is an operator that
operates on the wavefunction and corresponds to the total energy of the model:
N
r r
r
H = ! Ti + "( r1 , r2 ,..., rN ) ,
i=1
where Ti is the kinetic energy operator for the i-th particle and is given by
Hi = !
h2 # " 2
"2
"2 &
+
+
%
(,
2m $ "x i2 "yi 2 " zi 2 '
r r
r
while !( r1 , r2 ,..., r N ) is the potential energy function for the particles. The potential energy
consists of two parts, one of which is due to the inter-molecular forces among the constituent
molecules while the other part is used to keep the molecules inside the box of volume V.
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16.6 The Gibbs sum or the grand partition function ! the Gibbs factor
The Gibbs sum or the grand partition function is based on a quantity called “Gibbs factor”
and is defined as
# E V, Nˆ " µ Nˆ '
% s
%
(,
!(T, V, µ ) = * * exp $"
%&
k BT
%)
Nˆ = 0 s
(
+
)
where µ is the chemical potential per particle related to the chemical potential µ˜ by
µ=
µ˜
N Avogadro
.
The index s is the label or “the quantum number” for the energy eigenstates of a microscopic
model and runs over all the energy eigenstates each with its corresponding energy eigenvalue
Es , which normally depends on the volume V and the number N of microscopic particles (e.g.,
atoms) making up the system. The exponential factor, exp ! E ! µ Nˆ / k T , is called “the
{(
s
)
B
}
Gibbs factor.” Note that the Gibbs sum or the grand partition function is dimensionless because
the Gibbs factor is dimensionless.
16.7 Macroscopic quantities as statistical averages of microscopic quantities
As in the canonical ensemble approach, the macroscopic thermal quantities such as the
internal energy, the pressure, the number of particles in a macroscopic system are statistical
averages of their corresponding microscopic quantities.
Internal energy as a statistical average of energy eigenvalues
We can show this connection between the internal energy and the microscopic energy
eigenvalues of a microscopic model by substituting the relation between its thermodynamic
potential and grand partition function,
!( T, V, µ˜ ) = "k BT ln #( T, V, µ˜ / N Avogadro)
into the equation for the internal energy in terms of the Helmholtz free energy,
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# "
# "
" &#)&
" & # )&
(( % ( = !T %% T
(( % ( .
U = !T %% T
+ µ˜
+µ
$ "T
" µ˜ ' $ T '
$ "T
"µ ' $ T '
We then obtain
# E V, Nˆ " µ Nˆ '
% s
%
1 +
ˆ
(
U = * * Es V, N exp $"
! Nˆ =0 s
%&
k BT
%)
,
# E V, Nˆ " µ Nˆ '/
+
% s
%1
.
1
(1
= * * E s V, Nˆ . exp $"
!
%
k
T
%)1
Nˆ =0 s
B
.&
0
(
(
)
)
(
)
(
)
(HW#16.7.1: show this)
which can be expressed as
"
U = Es =
!! p E
s
ˆ =0 s
N
s
,
where ps is defined by
$ E V, Nˆ # µ Nˆ (
& s
&
1
),
ps ! exp %#
"
&'
k BT
&*
(
)
which implies that the internal energy is a “statistical average” Es of the energy eigenvalue Es
over all the energy eigenstates, each of which contributes to the average according to statistical
weight ps , which is proportional to the Gibbs factor.
Macroscopic pressure as a statistical averages of microscopic pressure
We can also show a similar connection between the macroscopic pressure and its
microscopic counterpart:
"
P = Ps =
!! p P ,
Nˆ = 0 s
s
s
(HW#16.7.2: show this)
where the “microscopic” pressure Ps of the model system that is in energy eigenstate s is defined by
$ #E '
Ps ! " & s ) .
% #V ( Nˆ
13
Macroscopic number of particles as a statistical averages of microscopic number of
particles
Since the chemical potential µ˜ and the chemical potential per particle µ are related by
µ˜ n = ( NAvogadroµ )n = µ (nN Avogadro) = µ N , the thermodynamic potential is also related to the
chemical potential per particle by
! = U " TS " µ˜ n = U " TS " µN ,
from which we obtain
d! = " SdT " PdV " Ndµ
so that
$ " #'
)) .
N (T, V, µ ) = ! &&
% "µ ( T ,V
Using this equation, wee can then show a similar connection between the macroscopic
number of particles and its microscopic counterpart:
"
N = Nˆ = ! ! ps Nˆ .
(HW#16.7.3: show this)
Nˆ =0 s
Answer for the homework question in Sec.16.7
HW#16.7.1
# "
# "
" &( # ) &
" &( # ) &
U = !T %% T
+ µ˜
+µ
% ( = !T %% T
% (
(
" µ˜ ' $ T '
"µ (' $ T '
$ "T
$ "T
# "
" &(
k T# "
" &(
= kB T %% T
+µ
ln * = B %% T
+µ
*
(
"µ '
* $ "T
"µ ('
$ "T
7
+ E V, Nˆ ! µNˆ / 4 + E V, Nˆ ! µ Nˆ /
- s
-6 - s
kB T 3
Nˆ 9
=
0 6T ,
0+µ
2 exp ,-!
* N2
kB T
kB T 2
k B T 99
ˆ =0 s
-1 65 -.
-1
.
8
+ E V, Nˆ ! µ Nˆ /
- s
1 3
ˆ
0
= 2 2 Es V, N exp ,!
* Nˆ =0 s
-.
k BT
-1
(
(
)
)
(
(
)
)
14
HW#16.7.2
$ "# '
$ "
'
k T $ "*'
P = !&
ln *) = B &
) = k BT &
)
% " V ( T, µ˜
% "V
( T,µ˜
* % "V ( T ,µ˜
+ E V, Nˆ ! µNˆ / 4
- s
- 6 1 -+ $ "E s ' -/79
kB T 3
,
0
,! &
=
) 02 2 exp -!
6
%
( Nˆ 198
* Nˆ = 0 s
k
T
k
T
.
"
V
5
B
.
1 B
(
)
+ E V , Nˆ ! µNˆ /
+- $ "E ' /- s
1 3
s
= 2 2 ,! &
0
) 0- exp, !
* Nˆ =0 s . % " V ( Nˆ 1
-.
k BT
-1
(
)
HW#16.7.3
$ " #'
$ "
'
k T $ "* '
)) = kB T &&
)
N = ! &&
ln *)) = B &&
* % "µ )( T ,V
% "µ ( T ,V
% "µ
( T ,V
+ E V, Nˆ ! µNˆ /$ ˆ '
- s
- N
kB T 3
)
0&&
=
exp ,!
2
2
* Nˆ = 0 s
-.
kB T
-1% k BT )(
(
)
+ E V, Nˆ ! µ Nˆ /
- s
1 3
ˆ
= 2 2 N exp ,!
0
* Nˆ =0 s
-.
k BT
-1
(
)
16.8 Overall scheme for the grand canonical ensemble approach
The overall scheme for the grand canonical ensemble approach is very similar to that for the
canonical ensemble approach.
N
r r
r
1. Construct a microscopic model with a Hamiltonian: H = ! Ti + "( r1 , r2 ,..., rN ) .
i=1
2. By solving the Schrödinger equation for the model,
r r
r
r r
r
H!s ( r1 , r2 ,..., rN ) = Es (V, N )! s ( r1 , r2 ,..., r N ) ,
calculate its energy eigenvalues: {Es (V, N )}s .
3. Calculate
the
Gibbs
sum
#
ˆ
*
E s V, N " µNˆ &(
%
!(T, V, µ ) = ) ) exp % "
(( .
%
kB T
Nˆ = 0 s
$
'
(
)
or
grand
partition
function
15
4. Calculate the thermodynamic potential: !( T, V, µ˜ ) = "k BT ln #( T, V, µ˜ / N Avogadro) .
5
Calculate the state variables from the thermodynamic potential:
$ "# '
)) ,
n = ! &&
% "µ˜ ( T ,V
$ "# '
P = !&
) ,
% " V ( T, µ˜
$ " #'
S = !&
) ,
% " T ( V ,µ˜
and
U = ! + TS + µ˜ n
16.9 State variables of a quantum ideal gas model
One-particle states
Consider a single particle (e.g., an atom, an electron, a photon, etc.) in a box of volume V.
Find the energy eigenstates and eigenvalues for the particle. These eigenstates are called “oner
particle states.” Each one-particle state is usually labeled by an integer vector, n = n x , ny ,nz
(
)
and the corresponding energy eigenvalue is ! nr .
Example: the single-atom states
! nr =
h 2" 2
2
2
2
2/ 3 ( nx + ny + nz ) n! = 1, 2, 3, ......, " (! = x, y, z )
2mV
Energy eigenstates for a quantum ideal gas of Nˆ particles
Each energy eigenstate for a quantum ideal gas of Nˆ particles is represented by the numbers
of particles in all the one-particle states.
{!nr }nr:all the one-particle states
r
where !nr is the number of particles in one-particle state n and is called the “occupation
number.” This way, we are not distinguishing or labeling the particles in the gas. All the
particles in the gas are identical so that we can only specify how many of them are in each oneparticle state.
Bosons and fermions
Quantum particles are classified into two groups: bosons and fermions.
Bosons: particles with integral spin or ! = (integer )h .
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Examples: photons, W ± , Z, gluons, pions, phonons, 4 He (2e !, 2p + , 2n )
Any number of bosons can occupy each one-particle state so that
!nr = 0, 1, 2, 3, ... , " .
Fermions: particles with half-integral spin or ! =
(odd integer )
2
h.
Examples: electrons, protons, neutrons, neutrinos, quarks, 3 He ( 2e ! , 2p+ , n)
According to the “Pauli exclusion principle,” a fermion cannot share a oneparticle state with another so that
!nr = 0 or 1,
where
r
0 means that one-particle state n is empty;
r
1 means that one-particle state n is occupied.
The total energy eigenvalues
E({! rn }) =
"
# nr! nr
"
! nr
r
n : all the
one- particle states
The total number of particles
Nˆ ({! nr }) =
r
n : all the
one-particle states
The Gibbs sum or the grand partition function
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%
!(T, V, µ ) = ''
$
#
& nr: all the one-particle states "rn
% E ({" r }) + µNˆ ({" r })(
(
n'
n'
*
* exp ' +
*
'
*
k BT
)
&
)
%
(
%
(, r + µ )" nr' (*
= '' $ # ** exp '' + # n '
*
kB T
& rn'
)
& rn "nr )
% ( ,r + µ ) "r ( 0
n*2
= $ / # exp'' + n
*2
r / r
k
T
&
)1
n . "n
B
( r ) ( )
= $ / # e + , n + µ / k BT
r /
n . " nr
{
}
" rn
0
2
21
For non-interacting particles, the one-particle states are independent of each other so that the
Gibbs sum becomes
! = !nr1 !nr2 !rn1 """ = # !nr
r
n
where !nr = % e
"( # nr " µ ) $ nr / ( k BT )
r
is the Gibbs sum for one-particle state n . This result is analogous
$ nr
to the “decomposition formula” for the partition function for N non-interacting “distinguishable”
particles,
Z = Z1 Z2 Z3 !!! ZN = Z1 N ,
where Z1 is the partition function for a single particle.
Note that for N non-interacting “indistinguishable” particles, we cannot apply the
decomposition formula to their partition function because we must impose a constraint that the
total number of the particles is N while we sum over all possible values for the occupation
r
number !nr of each one-particle state n in the partition function:
$ E ({! r }) '
n'
),
Z = " exp&& #
)
k
T
{ ! nr' }
%
B
(
r
"r ! n = N
n
where because of the constraint
"
r
n
! nr = N , the sums over the occupation numbers are not
independent of each other. In contrast, the Gibbs sum satisfies the decomposition formula
! = "nr ! rn and this is exactly why the grand canonical ensemble approach is more suitable for
the quantum ideal gas model than the canonical ensemble approach.
18
( %
Bosons: !" (T ,V, µ ) = # * &
r
*) $nr =0
n
{e
"( ' nr "µ ) /( k B T )
}
$ rn
+
1
- =#
.
r
(
"
'
n "µ ) /( k B T )
-,
r
n 1" e
where we have assumed
e
or
!( " nr !µ ) /( k B T )
<1
r
! nr > µ for all n
so that we can use
"
#x
!
=
!= 0
1
.
1$ x
( x < 1)
Since usually ! nr > 0 and the lowest one-particle energy is close to zero or
! (1, 0,0 ) ~ 0 , then ! nr > µ implies µ ! 0 for bosons..
&
" ( # r " µ ) /( k B T )
Fermions: !+ ( T, V, µ ) = , ( % e n
r (
n ' $ nr =0,1
{
}
$ nr
)
+ = , 1 + e "( # nr "µ ) / ( k BT ) .
+*
r
n
[
]
The thermodynamic potential
[
Bosons: ! " (T ,V, µ ) = " kB T ln #" = kB T % ln 1 " e
r
n
"( $ nr "µ ) / ( k BT )
[
Fermions: ! + (T ,V, µ ) = " kB T ln #+ = "kB T % ln 1+ e
r
n
]
" ( $ rn " µ ) /( k B T )
]
The “classical” regime
The distinction between bosons and fermions becomes irrelevant if
e
!( " nr !µ ) /( k B T )
<<1
so that
! ± " #k BT % e
#( $ nr #µ ) /( k B T )
r
n
= #k BTe
µ / ( kB T )
where we have used
ln (1 + x ) ! x .
( x << 1)
%e
r
n
#$ nr / ( kB T )
,
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Example: The thermodynamic potential for the monatomic ideal gas model in the classical
regime is then given by
!( T, V, µ ) " #k BTe
µ / ( kB T )
Z1 = #
V
µ / (k T )
k BTe B ,
V Q (T )
where Z1 is the partition function for a single atom calculated in Sec.13.2:
Z1 = # e
!" nr / ( kB T )
r
n
=
V
VQ ( T)
,
and
" 2!h 2 % 3/ 2
'' .
V Q (T ) = $$
# mkB T &
This thermodynamic potential is exactly the one we have used in Sec.16.3 to derive the
state variables for the monatomic ideal gas model, including the chemical potential:
"$ v &$
'
µ˜ = ! RT ln #
$% vQ (T ) $(
so that
µ=
µ˜
N Avogadro
=!
"$ v &$
"$ v &$
' = !k BT ln #
',
T ln #
$% vQ (T ) $(
$% vQ ( T ) $(
N Avogadro
R
where we have used R = N Avogadrok B .
! ( " r !µ ) /( k T )
B
The condition for the classical regime e n
<<1 implies ! nr " µ >> k BT . As
the lowest energy eigenvalue is very close to zero, ! ( 1,0,0) ~ 0 , this in turn implies that
µ << !kBT so that
"$ v &$
' << ! kB T
! kB T ln #
%$ vQ (T ) ($
or
! v $
& >>1,
ln ##
&
" vQ (T ) %
which is equivalent to
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!# T %#3 2
v
& >> 1,
="
vQ ( T ) $# TQ (v ) '#
which is satisfied for
(a) High temperatures: T >> TQ (v ) !
2 /3
2 "h2 # NAvogadro&
%
( .
mkB $
v '
(b) Low densities: v >> vQ (T ) .
The Bose-Einstein distribution function as the average occupation number for bosons
The average number for a quantum ideal gas of bosons is given by
$ " #! '
1
)) = + ( * r ! µ ) / ( kB T )
N (T, V, µ ) = ! &&
.
n
r e
% "µ ( T,V
!1
n
If we define the Bose-Einstein distribution function by
f ! ( " ,T, µ ) #
1
( " !µ ) / ( k BT )
e
!1
,
then we find
r
N = Nˆ = "r !nr = "r !nr = "
r f # ($ n ,T, µ )
n
n
n
so that
1
.
!nr = f " (# rn ,T, µ ) = ( # r " µ ) / (k T )
B
e n
"1
The Bose-Einstein distribution function is therefore the average occupation number !nr of oner
particle state n and it tells us how many bosons, on the average, are found in one-particle state
r
n.
f ! decreases as ! is increased so that one-particle states with lower energy are occupied by
more bosons. As the temperature approaches T = 0 , more and more bosons are found in the
lowest-energy one-particle state until all the bosons are accommodated in the lowest-energy one-
21
particle state at T = 0 . This accumulation of bosons in the lowest-energy one-particle state is
called the “Bose- Einstein condensation.”
The figure below shows f ! as a function of ! µ and k BT µ (the top curve is for
k BT µ = 0.1, the middle one is for k BT µ = 0.05, and the bottom one is for k BT µ = 0.01).
f ! ( " µ ,k BT µ ) =
1
("
e
µ !1 ) / ( k BT µ )
!1
The Fermi-Dirac distribution functions as the average occupation number of fermions
The average number for a quantum ideal gas of bosons is given by
$ " #+ '
1
)) = + ( * r ! µ ) / ( kB T )
.
N (T, V, µ ) = ! &&
n
r
% "µ ( T,V
+1
n e
If we define the Fermi-Dirac distribution function by
f + ( ! ,T, µ ) "
1
( ! #µ ) / ( k BT )
e
+1
.
then we find
r
N = Nˆ = "r !nr = "r !nr = "
r f + ( # n ,T, µ )
n
so that
n
n
1
!nr = f + ( " rn ,T, µ ) = ( " r # µ ) / (k T )
.
n
B
e
+1
22
The Fermi-Dirac distribution function is therefore the average occupation number !nr of oner
particle state n and it tells us how many fermions, on the average, are found in one-particle state
r
n.
Basic properties of the Fermi-Dirac distribution function
(a) 0 ! f + (" nr ,T, µ ) ! 1. Because of the Pauli exclusion principle, each one-particle state can be
occupied by one particle at most. That is why f + (! ,T , µ ) is at most one.
(b) f + ( ! ,T, µ ) decreases as ! is increased so that one-particle states with lower energy are
occupied by more fermions than those with higher energy.
(c) f + (µ ,T , µ ) = 1 / 2
(d) At T = 0 : f + ( ! ,T, µ ) is a step function:
#%1
f + ( ! ,0, µ ) " $
&%0
(! < µ)
(! > µ)
Physically, this means that at T = 0 , all the one-particle states with energy less than µ are all
occupied while those with energy above µ are all empty.
(e) For ! << µ " k B T : f + ( ! ,T, µ ) " 1
(f) For ! >> µ " k B T : f + ( ! ,T, µ ) " 0
The figure below shows f + as a function of ! µ and k BT µ (for ! µ just below 1, the top curve
is for k BT µ = 0 , the middle one is for k BT µ = 0.01, and the bottom one is for k BT µ = 0.1).
f + ( ! µ ,k BT µ ) =
1
(!
e
µ "1) / ( k BT µ )
+1
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In the classical regime: e
!( " nr !µ ) /( k B T )
f± =
<<1 or e
1
( ! nr "µ ) /( k B T )
e
±1
#e
( ! rn " µ ) / ( kB T )
"( ! nr "µ ) /( k B T )
>>1 so that
$ f classical <<1.
Therefore, both the Bose-Einstein distribution function and the Fermi-Dirac distribution function
practically coincide with exp {! (" nr ! µ ) / (k BT )} when ! nr " µ >> kBT .
f ± & exp{! (" nr ! µ ) / ( kB T )} vs. (" nr ! µ ) / (k BT )
The top curve is the Bose-Einstein distribution function.
The middle curve is the classical distribution function.
The bottom curve is Fermi-Dirac distribution function.
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The internal energy
U = E({! rn }) = # " nr ! rn = # " nr !nr = # " nr f ± (" nr ,T, µ )
r
n
Bosons: U = #
r
n
r
n
! rn
( ! nr "µ ) / ( k BT )
e
Fermions: U = #
r
n
"1
.
! nr
( ! nr "µ ) / ( k BT )
e
r
n
+1
.
We will use these equations to calculate the internal energy for photons, phonons, and electrons
in a simple metal.
The pressure is controlled by the internal energy
Bosons (–) and Fermions (+):
$ " #± '
$ "* r '
$ "* r '
1
P(T ,V, µ ) = !&
) = ! + & n ) ( * rn ! µ ) / ( kB T )
= ! + & n ) , rn
r
r
% "V ( T ,µ
±1
n % "V ( e
n % "V (
For a material particle (e.g., an atom or an electron) in a box of volume V, we find
! nr "
1
1
2 =
L
V 2 /3
so that
# !" rn &
2 " rn
%
(
=)
.
$ !V ' T, µ
3 V
We then find
P=
2
3V
"!
r
n
r
n
# rn =
2
U,
3V
which implies that the equation of state follows from the internal energy U = U(T ,V,n) . We
also find
U=
3
PV .
2
25
For phonons and photons, we will find later
! nr "
1
V 1/ 3
so that
# !" rn &
1 "r
%
( =) n .
$ ! V ' T, µ
3V
We then find
P=
1
3V
"!
r
n
r
n
# rn =
1
U,
3V
which implies that the equation of state follows from the internal energy U = U(T ,V,n) . We
also find
U = 3PV .