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1. Fluctuations are important because the number of particles
in a system is much less than Avogadro’s number;
2. Importance of the surface properties. Thermodynamic
quantities no longer scale with the number of atoms in a
system becuase the energy associated with the surface may
be a significant fraction of the total.
Nanosystems generally contain too many atoms to be thought
of as simple mechanical systems, but too few to be described by
bulk properties.
Microscopic view of
bulk properties
Equilibrium thermodynamic
properties are well defined
because
fluctuations
are
negligible in large (N=1023)
systems.
The Boltzmann Distribution
P(1,2)=P(1)P(2).
E(1,2)=E(1)+E(2.).
P( 1 )  Const  exp[  E( 1 )]
,
P( 2 )  Const  exp[  E( 2 )]

1
k BT
Normalized Boltzmann distribution
kB = 1.381·10-23 J·K-1 = 8.62·10-5 eV·K-1
Copyright (c) Stuart Lindsay 2008
Values for kT
• kBT at room temperature (300K) = 4.14 10-21J
• = 25 meV (Much smaller than most bonds)
• = 0.6 Kcal·mol-1
• kBT at room temperature in terms of force· distance
= 4.14 pN·nm
- molecular motors produce forces ca. ten times more over
nm distances
Copyright (c) Stuart Lindsay 2008
Normalized Boltzmann distribution
Er
1
P (r )  exp( 
)
Z
k BT
Er
Z   exp 
k BT
r
partition function
In case of degeneracy g(r):
g (r )
Er
P ( Er ) 
exp( 
)
Z
k BT
The partition function enumerates all the states as a function of
energy, so all the equilibrium properties of a system can be derived
from it.
Copyright (c) Stuart Lindsay 2008
The degeneracy of a state leads naturally to a statistical definition
of Entropy:
• Entropy is proportional to the number of ways (statistical
weight, ) a given macrostate r can occur.
S (r )  k B ln (r )
In terms of the probability of the rth state:
S  k B  p(r ) ln  p(r )
r
Copyright (c) Stuart Lindsay 2008
The Equipartition Theorem
• Average thermal energy of e.g., a harmonic oscillator
1 2 1 2
E  mx  x
2
2
• For a classical system (all energies allowed) replace the
Boltzmann sum with an integral and calculate the product of E
and p(E), e.g. for potential energy:


1 2
1
x   x 2 P( E ( x)) dx 
2
2

 x 2 
1 2
 2 x exp   2k BT dx

x 2
 exp(  2k

B
T
)dx
Copyright (c) Stuart Lindsay 2008
• With a change of variables and a standard integral we find
1 2
1
x  k B T
2
2
• Similarly
1 2
1

mx  k B T
2
2
• The thermal average of any quantity that appears in the
classical Hamiltonian as a quadratic term is
1
k BT
2
Equipartition
theorem
The Equipartition theorem assumes that all degrees of
freedom are in equilibrium with the heat bath and are
independent.
However, it takes coupling between the degrees of freedom to
‘spread’ the thermal energy out evenly and this requires a
non linear response.
Ex. It takes an anharmonic potential to couple vibrational
and translational degres of freedom (V-T energy transfer).
Thermodynamics and Statistical mechanics
• Thermodynamic potentials (“Free Energies”) can be
minimized to obtain the equilibrium properties of a
system.
S  0
Isolated system
A  0
A  E  TS
G  E  PV  TS
Closed system (V,T)
G  0
Copyright (c) Stuart Lindsay 2008
Closed system (p,T)
Thermodynamic potentials in terms of
partition functions
S  k B  p( r ) ln p( r )
r
1
Er
p( r )  exp( 
)
Z
k BT
Er
Z   exp 
k BT
r
E
S (T , V , N )  k B ln Z 
T
A  E  TS
A( T ,V , N )  kBT lnZ( T ,V , N )
Grand Canonical ensemble
Each system is enclosed in a container whose walls are both heat
conducting and permeable to the passage of molecules.
→ transport of matter allowed, N variable
V,T,μ
V,T,μ
V,T,μ
V,T,μ
V,T,μ
V,T,μ
V,T,μ
V,T,μ
V,T,μ
aNr = number of systems in the ensemble that contain N molecules
and are in the state r.
The set of occupation numbers {aNr} is a distribution.
Each possible distribution must satisfy the balance equations:
 aNr  A
N
r
 aNr ENr  Ε
N
Total energy of the ensemble
r
 aNr N  N
N
Number of systems in the ensemble
r
Total number of molecules
For any possible distribution, the number of states is given by:
W ({ aNr }) 
A!
N r aNr !
The distribution that maximizes W is:

{ a Nr }*  e e
e


A
N r e
 E Nr ( V ) N
e
 E Nr ( V )
e
1

kT
 N
 

kT
G  E  PV  TS  N
 is the chemical potential
{ a Nr }* exp N  E Nr 
PNr 

A

Z (T , V ,  )   exp  N  E Nr 
Nr
Gibbs Distribution
Grand Partition
function
G  k BT ln (T , V ,  )
Z ( T ,V ,  )   exp N  E Nr 
Nr
Summing over r states, it is possible to write:
Z ( T ,V ,  )   Q( N ,V ,T )e
N
kT
  Q( N ,V ,T )N
N
N
Canonical
partition function

  e kT
  kT ln 
activity
Ideal Gas: Z for one free particle (N=1)
2n y 
2n x 
2n 
kx 
,ky 
, kz  z
L
L
L


Z (T ,V ,1) 
 exp  E (n , n
 2k 2  2 2
E

k x  k y2  k z2
2m
2m
x
y
, nz )
nx , n y , nz
For large systems:
dn 
4Vk dk Vk dk

3
8
2 2
2
2

2 2

V

k 
2
dk
Z (T ,V ,1) 
k exp  
2 
2 0
 2mkBT 
3
2
 mkBT 
Z (T ,V ,1)  
V
2 
 2 
The de Broglie wavelength for a free particle is:
 2 
2


 2 
k x , y ,z
 mkBT 
1
2
So, one particle occupies a quantum volume of about λ3:
2

2


3  
 mkBT
Quantum concentration
(one particle per
wavelength3)



3
2
 mkBT 

2 
 2 
3
2
Ex. Quantum volume for a free electron at 300K
2
3
2
3
 2  2   2 1.113 1068 

  
 
31
 21 
 mkBT   9.1110  4.14 10 
 7.9 1026 m3  79nm3
A sphere of a radius of 2.7nm!
For N non-interacting particles:
N
1 
1

N


Z (T , V , N ) 
exp(


E
)

Z
(
T
,
V
,
1
)

r 

N!  r
N!

Quantum statistics
Z ( T ,V ,  )   Q( N ,V ,T )e
N
N
kT
  Q( N ,V ,T )N
N
Expliciting Q(N,V,T) (i.e. energy distribution):
Q( N ,V ,T )   e
j
 E j
n

i i i
 e

{ nk }
Ej(N,V) = energy states available to a system containing N molecules
εk= molecular quantum states
nk= number of molecules in the kth molecular state when the system
energy is Ej.
E j   nk  k
k
N   nk
k
n
n    ni i


i i i
i i
i
Z ( T ,V ,  )     e
   e


N
N

   e
{ nk }
  i

ni
N { nk }

N { nk } i
  ... e
n1 ,max n2 ,max
n1 0 n2 0

  i ni
i
This last passage originates from the fact that we are summing over
all values of N and that nk ranges over all possible values.
 e
n1 ,max
n1 0
  1
  e 
n1
n2 ,max
n2  0
  2
n2
......  
i
 e
ni ,max
ni 0

  i ni
Fermi-Dirac statistics: ni=0 or 1, ni,max=1

Z FD   1  e
  i

i
N FD
  i
  ln Z 

ln
Z

e


   
 kT 
 
  i
i 1  e
  V ,T
  V ,T
Bose-Einstein statistics: ni=0, 1, 2,… nmax=∞


Z BE    e
i
Where we used:
ni  0
   1  e 
  i 1
  i ni
i
j 
j
1
x

(
1

x
)

j 0
N BE
  i
  ln Z 
e
  ln Z 
   
 kT 
 
  i




1


e
i

V ,T

V ,T

Z FD   1  e
  i


Z BE   1  e
i
i
e  

 
i 1  e
e  

 
1


e
i
i
i
N FD

  i 1
N BE
i
 k e  
E  N   nk  k  
 
1


e
k
k
i
k

pV  kT  ln 1  e  k
k

k
+ = FD
- = BE
Classical limit (λ→0):
At the classical limit (high temperatures or low density) the
number of available molecular quantum states is much greater
than the number of particles.
The average number of molecules in any state is very small
(nk→0, λ→0).
Thermodynamically:
N
 0 (T fixed)
V
N
T  (
fixed)
V
lim N FD or BE  N MB   e
 0
  i
Maxwell-Boltzmann
distribution
i
ni  e
  i
Summing over i:
  i
n


e
i i i
  i
ni e

N
q
N   i e
  i
 q
E  i  i e  i
Quantum gasses
We consider just an ideal gas (non-interacting particles) now subject
to restrictions on how states are counted:
Bosons
Fermions
nr  0,1,2,3.....
n
r
nr  0,1
N
r
Using the Grand partition function:
Z (T , V ,  ) 
 exp  n
1
 n2  ...ni  ..  n11  n2 2  ...ni i  ....
n1 ,n2 ...
With the Gibbs distribution
PNr  P(n1 , n2 ...) 
exp   n1  n2  ...ni  ..  n1 1  n2 2  ...ni  i  ....
Z
Writing numerator and denominator as products:
exp     i ni 
p(n1 , n2 ....)  
Zi
i 1

Single particle
distribution
Fermi-Dirac (FD) statistics:
ni=1 or 0
Z i  1  exp     i 
0  exp    i   1 exp    i 
ni   ni pi ( ni ) 
1  exp    i 
ni
Fermi Dirac thermal
average occupation
1
ni 
exp   i     1
The chemical potential at
T=0 is called the Fermi
energy.
The electronic properties of
most
conductors
are
dominated by quantum
statistics.
For metals  is several eVs!!
Ex. Fermi energy of Na is 3.24 eV.
For εi=μ:
1
ni 
2
kT( 300 K )  0.025eV
3.24
T ( ni  0.5 ) 
 38,800 K
0.025
Bose-Einstein (BE) statistics:
ni=0, 1, 2, 3….
• Summing Zi from n = 0 to :
1
ni 
exp   i     1
<i
As ε approaches μ in the BE distribution, the occupation number
approaches infinity, i.e. bosons condense into one quantum state at
very low temperatures (Bose condensation).
Copyright (c) Stuart Lindsay 2008
Phonons are bosons with no chemical potential (μ=0), so that the
occupation number goes to zero as temperature approaches zero.
μ=0