Dr Roger Bennett
R.A.Bennett@Reading.ac.uk
Rm. 23 Xtn. 8559
Lecture 19
The Boltzmann Distribution
pr
1 UkT
e
Z
r
Z
¦e
r
Ur
kT
• This is the Boltzmann distribution and gives “the
probability that a system in contact with a heat bath
at temperature T should be in a particular state”.
• r labels all the states of the system. At low
temperature only the lowest states have any chance
of being occupied. As the temperature is raised
higher lying states become more and more likely to
be occupied.
• In this case, in contact with the heat bath, all the
microstates are therefore not equally likely to be
populated.
The Boltzmann Distribution
pr
1 UkT
e
Z
r
Z
¦e
Ur
kT
r
• Usually there are huge numbers of microstates that can
all have the same energy. This is called degeneracy.
• In this case we can do our summations above over each
individual energy level rather than sum over each
individual microstate.
p(U r )
1
g (U r )e
Z
U
r
kT
Z
¦ g (U
Ur
r
)e
Ur
kT
• The summation is now over all the different energies Ur
and g(Ur) is the number of states possessing the energy
Ur. The probability is that of finding the system with
energy Ur.
Entropy in ensembles
• Our system embedded in a heat bath is called a
canonical ensemble (our isolated system on its own
from Lecture 16 is termed a microcanonical
ensemble).
• When isolated the microcanonical ensemble has a
defined internal energy so that the probability of
finding a system in a particular microstate is the
same as any other microstate.
• In a heat bath the energy of the system fluctuates
and the probability of finding any particular
microstate is not equal. Can we now calculate the
entropy for such a system and hence derive
thermodynamic variables from statistical properties?
Entropy in the canonical ensemble
• Embed our system in a heat
bath made up of (M-1)
replica subsystems to the
one were interested in.
• Each subsystem may be in
one of many microstates.
The number of subsystems
in the ith microstate is ni.
• The number of ways of arranging n1 systems of
µstate 1, n2 systems of µstate 2, n3….
M!
W
3 ni !
i
Entropy in the canonical ensemble
S
k ¦ pi ln pi
i
• This is the general definition of entropy and holds even if
the probabilities of each individual microstate are
different.
• If all microstates are equally probable pi= 1/W
(microcanonical ensemble)
S
k ¦ pi ln pi
i
1
1
k ¦ ln
W
i 1W
W
k ln W
• Which brings us nicely back to the Boltzmann relation
Entropy in the canonical ensemble
S
k ¦ pi ln pi
i
pi
1 kUT
e
Z
i
Z
¦e
Ui
kT
i
• The general definition of entropy, in combination
with the Boltzmann distribution allows us to
calculate real properties of the system.
Ui
ln pi ln Z
kT
§ Ui
·
ln Z ¸
S k ¦ pi ln pi k ¦ pi ¨
i
i
© kT
¹
1
U
S
piU i k ln Z ¦ pi
k ln Z
¦
T i
T
i
Helmholtz Free Energy
S
U
k ln Z
T
F
U TS
kT ln Z
• Ū is the average value of the internal energy of the
system.
• (Ū – TS) is the average value of the Helmholtz free
energy, F. This is a function of state that we briefly
mentioned in earlier lectures. It is central to statistical
mechanics.
• The Partition function Z has appeared in our result –it
seems to be much more than a mere normalising factor.
Z acts as a bridge linking the microscopic world of
microstates (quantum states) to the free energy and
hence to all the large scale properties of a system.
Helmholtz Free Energy
F U TS kT ln Z
• F is a state function. Can we now calculate the
thermal properties of our system?
• Lets ignore Ū is an average and assume U = Ū.
• We can now use our thermodynamics knowledge
to define our thermodynamic variables. For a
small reversible change:
F U TS
dF
dU TdS SdT
dF
dQR PdV TdS SdT
dF
TdS PdV TdS SdT
PdV SdT
Helmholtz Free Energy
dF
PdV SdT
F
U TS
kT ln Z
• Employing partial differentials we find:-
§ wF ·
¨
¸
© wV ¹T
§ wF ·
¨
¸
© wT ¹V
P
S
P
§ wF ·
¨
¸
© wV ¹T
§ wT ln Z ·
k¨
¸
© wV ¹T
S
§ wF ·
¨
¸
© wT ¹V
§ wT ln Z ·
k¨
¸
© wT ¹V
• We have intimately related the energies of the
microstates of the system to the pressure and
entropy.
Response functions –2nd derivatives of
free energy
• Moduli of elasticity are the stress/strain or
force/unit area divided by the fractional
deformation (Lecture 8):-
K
§ wP ·
V ¨
¸
© wV ¹T
§ w2F ·
V¨
¸
2
© wV ¹T
• Most usefully the heat capacity at constant
volume:2
§w F ·
§ wS ·
§ wQ ·
CV ¨
T ¨ 2 ¸
¸ T¨
¸
© wT ¹V
© wT ¹V
© wT ¹V
Mean Energy
• Ū is the average value of the internal energy of
the system. The actual internal energy fluctuates
because we have defined the temperature of the
heat bath.
• How big are the fluctuations? – are they
important?
§ w (T ln Z ) ·
U F TS kT ln Z kT ¨
¸
© wT
¹V
U
Tw ln Z ·
§
kT ln Z kT ¨ ln Z ¸
wT ¹V
©
w ln Z
kT
wT
2
Fluctuations in Internal Energy
• We measure departures from the mean value by
use of standard deviations - as we would with
any distribution.
( 'U ) { U U
2
CV
§ wU ·
¨
¸
© wT ¹V
2
§
¨ w
¨
¨ wT
¨
©
U U
2
¦U i e
i
¦e
r
2
Ui
kT
Ur
kT
·
¸
¸
¸
¸
¹V
Fluctuations in Internal Energy
( 'U ) { U U
2
CV
§
¨ w
¨
¨ wT
¨
©
¦U e
i
i
¦e
Ui
kT
U
r
kT
r
2
·
¸
¸
¸
¸
¹V
( 'U ) { U U
2
2
2
U U
2
>
2
2
1
2
U U
2
kT
kT CV
2
@
Fluctuations in Internal Energy
• We have calculated the variance – the relative fluctuation
'U/ Ū is of most use:-
'U
U
kT 2CV
U
• But Ū and CV are extensive properties proportional to the
size of the system ~ N, the number of particles in the
system. T intensive and is size independent.
'U
1
v
U
N
Fluctuations in Internal Energy
• For typical macroscopic systems with ~1023 particles
fluctuations 'U/ Ū ~ 10-11
'U
1
v
U
N
• Fluctuations are tiny and hence U and Ū can be
considered identical for all practical purposes.
• Macroscopic systems in a heat bath effectively have their
energy determined.
• Similar relationships can be found for other relative
fluctuations of properties of macroscopic systems.
Dr Roger Bennett
R.A.Bennett@Reading.ac.uk
Rm. 23 Xtn. 8559
Lecture 20
Summary – Statistical Mechanics
• Microstate – The state of a system defined microscopically
– a complete description on the atomic scale.
• Macrostate – The state of a stystem of macroscopic size
specified by a few macroscopically observable quantities
only.
• Statistical Weight (W or :) of a macrostate – is the
number of microstates compising the macrostate.
• Postulate of equal a priori probabilities – “for an isolated
system in a definite macrostate, the W microstates
comprising this macrostate occur with equal probability.
• Equilibrium Postulate – “for an isolated macroscopic
system, defined by U,V,N (which are fixed) and variable
parameters D, equilibrium corresponds to those values of D
for which the statistical weight W(U,V,N, D) attains its
maximum.”
Summary – Statistical Mechanics
• Boltzmann definition of the Entropy
S (U ,V , N ,D ) k ln W (U ,V , N ,D )
• Temperature definition
1 § wS (U ,V , N ) ·
¨
¸
T ©
wU
¹V ,N
• Pressure definition
§ wS (U ,V , N ) ·
P T¨
¸
wV
©
¹U , N
• General definition of the Entropy
S
k ¦ pi ln pi
i
Summary – Statistical Mechanics
• Boltzmann Distribution – pi is the probability that a system
at temperature T is in the state i with energy Ui.
pi
1 kUT
e
Z
i
• Partition Function – summed over all microstates
Z
• Mean Energy
U
¦e
U i
kT
i
w ln Z
kT
wT
2
• Helmholtz free energy F = U - TS
Paramagnetic materials
• This is a simple model to develop our
understanding of stat. Mechanics but it proves
to be very significant.
• Paramagnets contain atoms which have
magnetic dipole moments (P). These do not
interact with each other but can respond to an
applied external magnetic (B) field.
• The dipoles can be thought of as independent
(atomic) bar magnets arranged in a crystal
lattice.
Paramagnetic materials
• The dipoles can be thought of as independent (atomic)
bar magnets arranged in a crystal lattice.
• Think
npnnnppnpnpnpnpnpppn
• In a B field each dipole can exist in one of two states –
aligned with the field (spin up) or anti aligned (spin
down).
• Spin up dipoles have an energy -PB, spin down +PB.
• We want to find out how the magnetisation of the
material depends on temperature and applied field.
Paramagnetic materials
• As all the dipoles are independent of each other
we really only need to look at the average
properties of one dipole. We can use all the
other dipoles as the heat bath – Canonical
ensemble.
• We have the two possible microstates and
energies already – get Partition Function Z1 for
our single dipole.
Z
¦e
i
U i
kT
e
PB
kT
e
PB
kT
§ PB ·
2 cosh¨
¸ 2 cosh x
© kT ¹
PB
x{
kT
Paramagnetic materials
• We can now calculate the probability of spin up
versus spin down.
1
1
x
x
1 kUT
p
e
p
e
pi
e
2 cosh x
2 cosh x
Z
i
1 .0
0 .9
0 .8
P rob a bility
0 .7
0 .6
S p in alig ne d (U = - PB)
S p in an tia lign ed ( U = + PB)
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
0 .0
0 .5
1 .0
1 .5
X =PB/kT
2 .0
2 .5
3 .0
Paramagnetic materials
• We can now calculate the mean magnetic
moment of our individual dipole.
1
1
x
p
e x
p
e
cosh x
cosh x
P
Pp Pp
P
(e x e x )
2 cosh x
P
P
2 sinh x
2 cosh x
P tanh x
• And hence the mean energy of the individual
U
PB tanh x
Paramagnetic materials
• We have the values necessary for the individual dipole
and because our dipoles do not interact all other
dipoles must behave similarly.
• A solid of N dipoles therefore has and energy:
U
NU
PNB tanh x
• And a mean magnetic moment (in the direction of the
applied field) of:
M
NP
• Note that U = -MB
PN tanh x
Paramagnetic materials
• The magnetisation L is the magnetic moment
per unit volume
L
M
V
PN tanh x
V
• In the limit of a weak field or high temperature
x<<1 and tanh x | x so:
PNx
L|
V
NP 2
B
VkT
Paramagnetic materials
1 .0
1 .0
0 .9
Pro ba b ili ty
0 .7
S pin Up
S pin Do wn
M ag n e tisa tion
0 .6
0 .5
0 .4
0 .6
0 .4
0 .3
0 .2
0 .2
0 .1
0 .0
0 .0
0 .0
0 .5
1 .0
1 .5
X =Pb /kT
2 .0
2 .5
3 .0
M ag n etis a tio n / NP/V
0 .8
0 .8
Reif, Figure 6-3-1
“Saturation
Magnetization”
“Curie’s Law” of
Paramagnetism: M0 { χH
χ { (N0μ2)/(kT) { “Curie Susceptibility”
Paramagnetic materials –Curie’s Law
• The susceptibility F is the magnetisation per applied
field intensity which for small magnetisations is given
by H = B/P0.
F
L
H
NP P 0
VkT
2
1
Fv
T
• Is Curie’s law. It holds very well for paramagnetic
materials with weakly interacting dipoles. So well that it
can be used for temperature calibration for example
Cerium Magnesium nitrate obeys curies law to 0.01K!
Curie Law
M
1
NgP B gμ B + 2k BT 2
2
2
Ng μ B + 4k BT
where C = Ng2PB2/(4kB) is the Curie constant
Since the magnetic susceptibility is defined as F = M/H
the Curie Law results:
F
C
T
CH
T
F vs. T plot
1/F = T/C gives a straight line of gradient C-1 and intercept zero
FT = C gives a straight line parallel to the X-axis at a constant value of FT
showing the temperature independence of the magnetic moment.
Curie “Law”
1. As single electrons are magnets, if you place them in a magnetic
field, they’ll align with the field. However the energy difference
between aligned with field and against field is << thermal energy
at room temperature. Get random orientation – equal populations
of alignment with/against field.
2. As you lower T, energy difference becomes more important
and the population changes – more align anti-parallel to the field.
3. To explain this behavior, Curie invented a parameter – called
“Magnetic Susceptibility”, χ, – which is a measure of how attracted
a sample is to a magnetic field. Normally measured as an apparent
mass increase. As more electrons align anti-parallel to the field at
low temperature, χ increases. In fact χ is inversely proportional to
the field: this is the Curie “Law”
(1/χ) = CT
C = “The Curie Constant”
Dr Roger Bennett
R.A.Bennett@Reading.ac.uk
Rm. 23 Xtn. 8559
Lecture 21
Heat Capacity
U
NU
PNB tanh x
PB
PNB tanh
kT
• Our paramagnetic solid has an energy that depends upon
temperature – it therefore must have a magnetic heat
capacity.
• Experimentally one measures the heat capacity at
constant magnetic field intensity H.
CH
§ dQ ·
¨
¸
© wT ¹ H
§ wU ·
¨
¸
© wT ¹ H
w
PB
PNB
tanh
wT
kT
• as our paramagnetic compound is weakly magnetic H =
B/P0 so B is also constant.
Heat Capacity
0 .4
CH / Nk
CH
PB
w
PNB
tanh
wT
kT
2
§ PB ·
2 § PB ·
Nk ¨
¸ sech ¨
¸
© kT ¹
© kT ¹
Ma gnet i c Heat Capa ci ty
0 .2
0 .0
0
1
2
3
1/ X = kT / P B
4
5
Schottky Heat Capacity
2
§ PB ·
2 § PB ·
Nk ¨
¸ sech ¨
¸
© kT ¹
© kT ¹
CH
• This is in fact a general result for the heat capacity in
any two level system. One example we have already
encountered is the Schottky defect.
CH / Nk
0 .4
Ma gnet i c Heat Capa ci ty
0 .2
0 .0
0
1
2
3
1/ X = kT / P B
4
5
Isolated Paramagnetic Solid
• Very similar problem to the one previously treated in a
heat bath. Here we constrain (fix) the total energy U of
the isolated system.
• N total dipoles, n spin-up aligned with the applied B field
• npnnnppnpnpnpnpnpppn
• U is clearly a function of n.
U ( n) ( N n) PB nPB
PB( N 2n)
• A given energy U(n) corresponds to a given number of n
spin up atoms with statistical weight:-
W ( n)
N!
( N n)! n!
Isolated Paramagnetic Solid
• Hence Entropy is given by:-
S ( n)
k ln W ( n)
N!
§
·
k ln¨
¸
© ( N n)! n! ¹
• For large N (~1023) can use Stirling’s Approximation
S ( n)
k >N ln N n ln n ( N n) ln( N n)@
• Should look familiar – it’s the same problem as the
Schottky Vacancy formation.
1
T
wS
wU
wS ( n)
wU ( n)
wS ( n) dn
wn dU
1 wS ( n)
H wn
Isolated Paramagnetic Solid
1
T
wS
wU
1
T
wS ( n)
wU ( n)
wS ( n) dn
wn dU
1 wS ( n)
2 PB wn
U ( n)
PB( N 2n)
k § N n·
ln¨
¸
2 PB © n ¹
• Solve for n, to find the density of spin up atoms:-
n
N
1 x
e
Z1
• This is identical to the probability of finding a spin up
atom in a heat bath we started with last lecture!
Negative Temperature
• From our previous derivation we had
1
T
k § N n·
ln¨
¸
2 PB © n ¹
k
§ n ·
ln¨
¸
2 PB © N n ¹
• If n < N/2 then more than half the dipoles are antiparallel and T becomes negative!
• What is a negative temperature?
• We know that as the temperature Tof the populations
of spin-up and spin-down only become equal!
• A negative temperature state must therefore be hotter
than T=f as its is a more energetic state of the system.
Negative Temperature
• For a negative temperature the entropy and statistical
weight must be decreasing functions of E.
• This can happen if the system possess a state of finite
maximum energy – such as our paramagnet with
U=NPB.
• No systems exist where this happens for all particular
aspects (I.e. vibrational energies, electronic energies
and magnetic energies). However, if one such aspect or
subsystem is effectively decoupled from the others, so
they do not interact, that subsystem may be considered
to reach internal equilibrium without being in equilibrium
with the others.
• This is the case for magnetic systems where the
relaxation times between atomic spins is much quicker
than the relaxation between spins and the vibrational
modes of the lattice.
Negative Temperature
• In the paramagnet the lowest possible energy is
U=-NPB and the highest U=+NPB. These are
both unique microstates so S=0.
• In between we can only reach states with
positive energy with a negative temperature.
1.0
1.0
0.5
En erg y / NPB
En erg y / NPB
S ystem E n erg y
S ystem E n erg y
0.5
0.0
- 0.5
- 1.0
0.0
- 0.5
- 1.0
-5
-4
-3
-2
-1
0
1
T em pe ratu re
2
3
4
5
-20
-10
0
1 / T em pe ratu re
10
20