Transcript Fluctuations - University of Florida

C LOSED isothermal system,
thermo  Statistic ther mo
Q N,V,T 
OPEN isothermal system,
thermo  S tatistic thermo
  V ,T,  
Looking at the natural variables
A  N,V ,T   -kT lnQ  N,V ,T 
pV  -kT ln   ,V ,T 
for a fixed value of N, we can sum over all j
  V,T,     e
N
ENj V 

kt

e
kT
N
j
  Q(N,V ,T )  e

kT
N
N
(V ,T ,  )   Q  N,V ,T   N
N

e kT
N
 N
a2
   kt ln
a1
with activities
   kt ln 
where

is an absolute activity and
is the difference in chemical potential for 2 states
a1
and
a2
An example on the equivalency among ensembles
N distinguishable particles, 2 possible states (E=e with E1=0)
a je
 {aj}={a1,a2,…aN} where aj =0, or 1 and therefore Ev 

j
Microcanonical ensemble: degeneracy of the mth level  number of
ways to distribute m objects in a pool of N (i.e. distribute m quanta to
obtain E total energy)
N!
W  (E, N ) 
 N  m ! m !
1   ln   E, N  
we know that  

 and since me  E,
kT 
E
N
when N is large enough ,   E, N  continuous on m
1 N m
1   ln   E, N  


ln 
   


e
m


e
m
N
N
N
e
 e   1  m  e
m
e 1
and since the energy is E  me
E

where we used
Stirling approximation
eN
e e  1

we can also look at the system in the Can onical ensemble,
lnQ  N,V ,    ln  e   Ev
v
and using Ev   a j e
 ln Q(N,V ,  )  ln
j
N
lnQ(N,V ,  )  ln 
j

a 0,1

e
 ea j

 ln Q(N,V ,  )  ln 1  e
j
 ln Q
from here we can obtain the E  

E 


j
j
e

a 0,1
 ln 1  e
a je
- e

N
N
e  e - e
1  e - e
Ne
E  e
e 1
- e

N
Ensemble
Constants
Fundamental
thermodynamics
Microcanonical
N,V,E
S=kbln
Total differentials
dS 
dE
T
Canonical
N,V,T
A=-kbTlnQ(N,V,T)
Grandcanonical
,V,T
pV=kbTln(,V,T)
Isothermalisobaric
N,p,T
G=-kbTln(N,p,T)

pdV
T

 dN
T
dA  SdT  pdV   dN
d ( pV )  SdT  pdV  Nd 
dG  SdT  Vdp   dN
Useful ensembles at least one extensive variable N,V,or E
Generalized ensemble with only intensive properties, (,p,T)
but –kbTlnZ(,p,T)=0  no fundamental function
Fluctuation:spontaneous deviation of a mechanical variable
from its mean… How much it deviates?
The variance measures the spread of a probability
 X-
distribution about a mean value :
X

2
 X
2
 X
2
X   Pi X i
standard deviation  X 
 P  X 
i
i
2

X
Ergodic hypothesis <time>  <ensemble>
rms fluctuation of X=X(t) is equivalent to X
2

 X-
X

2
What are the fluctuations in the canonical ensemble?
  E
2
E
2
 E
2
  Pj E  E
2
j
j
1

Q(N,V ,T )
 E j  e
2

E
j
  E j ( N ,V )
j

 E
 E  Q  N,V ,T 
1

 E
Q(N,V ,T )


 E

 E
 ln Q  N,V ,T 

 E
j
 Ej  e
  E j ( N ,V )
Q  N,V ,T 
 E
2
2
2
 E
 ln Q
2  E
recalling that E  
and
 kT


T
2
 
2
E
 E

 kT
2
 E
 E
T
 ln Q  N,V ,T 

 E
2
 E
2
 E
2

 E2  kT 2CV
The spread of the fluctuations corresponds to the rate at which the
energy changes with T
For an ideal gas,
Order  E
  Order NkT 
 2 E 
Order  E   Order  kT
  Order kT N
T 



 E Order kT N
E
=
Order NkT 

N
 1011.5  2 3  10 11.5
N
Distribution of energies is like a delta function centered at <E>

 N2  ?
  V,T,     e
N
 ln   V ,T,  

ENj V 

kt

e
kT
N
j
1
N

e

  V,T,   N j kT

ENj V 
kt

e
kT
N
 N
  V,T,  

  V,T,  
2
 2 ln   V,T,  

V,T,


  V,T,  


1



2
2
2
2





  V,T,  
  V,T,  
 2 ln   V,T,  

2
 
 N2  N 2  N 2
2
N
2

2
N
2

2
N
2
2
1  ln   V,T,   1  N


2


 
 N
2

in addition,
1
    p   v 
    p    N  
  
 N    p   v   N    p   v    v  

V ,T 
V ,T 
V ,T 
N ,T  N ,T 
N ,T  N ,T  
1
1  G 
1    1/ v   
 p 
where we used
 

N




 
 V 
V=Nv

N   p N ,T
V

v

N ,T
V ,T 

1
1 V 
 p 
 V N
  2 

N
  V N ,T V  N 
2
1
V 2  p 
V 1
 2
 2

N  V N ,T N T
2
2

N


1
N

V
N
 N2 
  2 kbT 

kbT T

 
V
 p  V
Isothermal compressibility
1 V
1 nkT
1
for an ideal gas PV=nkT   


2
V p
V p
p
N
N

1
N
N
kT 
V

kT
Vp

1
N
For a canonical ensemble, even thought there are fluctuations,
The energy is distributed uniformly. Each system is most likely
to be found with energy <E> canonical ensemble equivalent
to microcanonical (where E is constant)
Fluctuations in N show that a grand canonical ensemble is
most likely to be found with <N> particles  grand canonical
canonical ensemble equivalent to canonical (where N is
constant)
What is the probability of finding a particular value of E?
P(E)
P  E   (E )  e

E
kT
as E  , (E) and e

e
E


E
kT

E
kT
E
 P  E  must have a maximum for some value E 
E
we also know that
 0  spread of the
E
P(E)
energy values around E is extremely small
 E  must be very close to E
E*=<E>
Let’s count…
Consider a quantum mechanical system where we can write
H=
H
i
(non-interacting particles)
i
from QM we know that if H =
and thus E   Ei
 H , we can propose   
i
i
i
i
The most useful example is degrees of freedom:
H = H translation  H rotation  H vibration  H electronic
with E  E translation  E rotation  E vibration  E electronic
other examples include H for quasiparticles
H phonons , H excitons , H polarons etc...
to obtain thermodynamic functions we hav e to
1st learn to count all possible states  partition function
i
Canonical ensemble of DISTIGUISHABLE particles/quasi-particles:
a,b,…n.
H = Hn 
n
where the energy of each quasi-particle is given by e
N
n
where the letters label
and the energy of a given state j is E j   e j , individual particles
Q(N,V ,T )   e
Ej
 e
j
 e
n
n
a
b
c
N
   e   e   e   ...e   
e
i
n e j 
j
 j
j
j
j
 distinguishable particles,each particle j value,
is independent of the j values of the others
j
Q(N,V ,T ) 

N
n
b
a 

ei
kT
e

e j
kT
a 
N
 e
n
and since all q
kT
a   b 
q q
 ...  q
i
a
j

ei
have the same e values
j
Q(N,V ,T )  q

n  N

N 
Imagine a system with N=1000 degrees of freedom
(1000 quasi particles)
Each particle can be in one of 5 microstates
 There are 51000 states to be sampled (and counted!!!)
Using the factorization due to equal-but-distinguishable particles,
we only need to enumerate 5 states to evaluate q
 conversion of one N-body problem to N, 1-body problems
Molecular partition function
q
q
q
q
q
molecule
translation
rotation vibration
electronic
...
where each partition function is described
by i ts own energies  q
translation
 e
j
 e translation
j
If the particles are INDISTINGUISHABLE
Q(N,V ,T )   e
j
Ej
 e
a
b
c
N
   e   e   e   ...e   
 j
j
j
j

j
indistinguishable particles ,each particle j
value, depends on the j values of the ot hers
we cannot use
 ...
j
k
l
FERMIONS : All indices j; k,…, l must be different. Hence
summations over indices depend on each other.
BOSONS : Indices j;k;…;l need not all be different. Permutations
like j; k;…;l and k;j;…;l refer to identical states and must occur
only once in the summation.
INDISTINGUISHABLE particles 
e i( a )  e (j b ) 
 e (n) 
 e (N )  e  e 
k
l
i
e 
j
k
el
A particular (and common) case:
T,d number of available energy states >> N
 each an every particle is in a different state
Boltzman Statistics
e e  e  e
i
j
k
l
we have to consider those distribution that are equivalent, that is
e1  e 2  e 3  e 4  e 2  e1  e 3  e 4  e1  e 3  e 2  e 4
There are N! of these combinations which can be subtracted
from the pool of microstates by dividing by N!
q

Q(N,V ,T ) 
N
N!
Boltzman number of 1-particle state >>number of particles
How many 1-particle states?
Remember the sphere used to explain degeneracy?
number of 1-particle states with an energy lower than e = number
of lattice points enclosed by the sphere in the positive octant:
3
2
 e   1  4  R 3   1   8m e  a 3
8 3

6  h
for thermal particles,

3
e  e  kT per particle
2
For the Bolztman condition to hold,

3
2
6
h




  12mk T 
1

 e   1   8m 3kT 
6  h
 e 
N

2 
V
3
2
1  8m 3kT  V

6  h 2  N
which will occur for T  m
and  
3
2
1