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Chapter 6
Basic Methods & Results of Statistical Mechanics
Historical Introduction
• Statistical Mechanics developed
by Maxwell, Boltzman, Clausius,
Gibbs.
• Question: If we have individual
molecules – how can there be a
pressure, enthalpy, etc?
2
Maxwell
Key Concept In Statistical Mechanics
Idea: Macroscopic properties are a thermal average of
microscopic properties.
• Replace the system with a set of systems "identical" to
the first and average over all of the systems. We call the
set of systems
“The Statistical Ensemble”.
• Identical Systems means that they are all in the
same thermodynamic state
• To do any calculations we have to first
Choose an Ensemble!
3
Common Statistical Ensembles
• Micro Canonical Ensemble: Isolated Systems.
• Canonical Ensemble: Systems with a fixed number
of molecules in equilibrium with a heat bath.
• Grand Canonical Ensemble: Systems in
equilibrium with a source a heat bath which is also a
source of molecules. Their chemical potential is fixed.
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All Thermodynamic Properties Can Be
Calculated With Any Ensemble
We choose the one most convenient.
For gases: PVT properties – canonical
ensemble
Vapor-liquid equilibrium – grand canonical
ensemble.
5
Properties Of The Canonical and the Grand
Canonical Ensemble
Gibbs showed that the ensemble average was
equivalent to a state average
F   Fn p n
n
(6.10)
Pn=the probability that the system is in a
configuration (state) n.
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Properties of the Canonical Ensemble:
U n
g ne
pn  N
Qcanon
(6.11)
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The Grand Canonical Ensemble:
E n
g ne
pn 
Q grand
(6.12)
with:
E n  Un  N n
(6.13)
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Partition Functions
N
canon
Q =canonical partition function
Qgrand= grand canonical partition function
N
Qcanon
  g ne
U n
n
(6.15)
Qgrand   g ne
n
(6.16)
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E n
Partition Functions
• If you know the volume, temperature, and the
energy levels of the system you can calculate the
partition function.
• If you know T and the partition function you can
calculate all other thermodynamic properties.
Thus, stat mech provides a link between quantum and
thermo. If you know the energy levels you can
calculate partition functions and then calculate
thermodynamic properties.
10
• Partition functions easily calculate from the
properties of the molecules in the system (i.e.
energy levels, atomic masses etc).
• Convenient thermodynamic variables. If you
know the properties of all of the molecules,
you can calculate the partition functions.
• Can then calculate any thermodynamic
property of the system.
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Thermal Averages with Partition
Functions
 pn 
S  k B  pn Ln  
gn 
n

(6.40)
N
A  k BTLn(Qcanon
)
(6.59)
N
( LnQcanon
)
 U

(6.60)
N
 LnQcanon

 A 
N
S=- 
=k
T
+k
LnQ

B 
B
canon


T

T

V,N

V,N
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(6.61)
N

LnQcanon 
 A 
P  

 =k BT 
 V T,N
 V T,N
(6.62)
 LnQgrand
 PV 
Ν= 
 =k BT 
μ
 μ T,V

N



LnQ

A


canon


  k BT 
 N (6.63)
T,V
 N T,V


T,V
(6.65)
 LnQgrand 
 PV 
S
 +k B Ln(Qgrand )
  k BT 
dT
 T  V,

V,
(6.64)
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Canonical Ensemble Partition Function Z
Starting from the fundamental postulate of equal a priori
probabilities, the following are obtained:
i. the results of classical thermodynamics, plus their
statistical underpinnings;
ii. the means of calculating the thermodynamic parameters
(U, H, F, G, S ) from a single statistical parameter, the
partition function Z (or Q), which may be obtained
from the energy-level scheme for a quantum system.
The partition function for a quantum system in contact with
a heat bath is
Z = i exp(– εi /kT),
where εi is the energy of the i’th state.
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The partition function for a quantum system in
contact with a heat bath is Z = i exp(– εi /kT),
where εi is the energy of the i’th state.
The connection to the macroscopic thermodynamic
function S is through the microscopic parameter Ω
(or ω), which is known as thermodynamic
degeneracy or statistical weight, and gives the
number of microstates in a given macrostate.
The connection between them, known as
Boltzmann’s principle, is S = k lnω.
(S = k lnΩ is carved on Boltzmann’s tombstone).
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Relation of Z to Macroscopic Parameters
Summary of results to be obtained in this section
<U> = – ∂(lnZ)/∂β = – (1/Z)(∂Z/∂β),
CV = <(ΔU)2>/kT2,
where β = 1/kT, with k = Boltzmann’s constant.
S = kβ<U> + k lnZ ,
where <U> = U for a very large system.
F = U – TS = – kT lnZ,
• From dF = S dT – PdV, we obtain
S = – (∂F/∂T)V and P = – (∂F/∂V)T .
Also,
G = F + PV = PV – kT lnZ.
H = U + PV = PV – ∂(lnZ)/∂β.
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Systems of N Particles of the Same Species
• Z = zN for distinguishable particles (e.g. solids);
Z = zN/N for indistinguishable particles (e.g.fluids).
<u> = – ∂(lnz)/∂β = – (1/z)(∂z/∂β), U = N<u>.
cV = <(Δu)2>/kT2, CV = NcV, CP = NcP.
Distinguishable particles: F = Nf = – kT ln zN = – NkT lnz.
Since F = U – TS, so that S = (U – F)/T or S = – (∂F/∂T)V.
Indistinguishable particles: F = – kT ln(zN/N)
= – kT [ln(zN) – ln N] = – NkT [ln(z/N) – 1],
Since for very large N, Stirling’s theorem gives ln N! = N lnN – N.
Also, S = – (∂F/∂T)V and P = (∂F/∂V)T as before.
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Mean Energies and Heat Capacities
• Equations obtained from Z = r exp (– Er),
where  = 1/kT.
•
•
•
•
•
U = rprEr/rpr = – (ln Z)/ = – (1/Z) Z/ .
U2 = rprEr2/rpr = (1/Z) 2Z/2.
Un = rprErn/rpr = (–1)n(1/Z) nZ/n.
(ΔU)2 = U2 – (U)2 = 2lnZ/2 or –  U/ .
CV =  U/T =  U/ . d/dT = – k2.  U/,
or
CV = k2 (ΔU)2 = (ΔU)2/kT2;
i.e. (ΔU)2 = kT2CV .
Notes
Since (ΔU)2 ≥ 0, (i) CV ≥ 0, (ii)  U/T ≥ 0.
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Entropy and Probability
• Consider an ensemble of n replicas of a system.
• If the probability of finding a member in the state r is pr, the
number of systems that would be found in the r’th state is
nr = n pr, if n is large.
• The statistical weight of the ensemble Ωn (n1 systems are in state
1, etc.), is Ωn = n/(n1 n2…nr..),
so that
Sn = k ln n – k r ln nr.
• From Stirling’s theorem,
ln n ≈ n ln n – n,
r ln nr ≈ r nr ln nr – n.
Thus Sn = k {n ln n – r nr ln nr} = k {n ln n – r nr ln n – r nr ln pr},
so that Sn = – k r nr ln pr = – kn r pr ln pr .
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For a single system, S = Sn/n ; i.e. S = – k r pr ln pr .
Ensembles 1
A microcanonical ensemble is a large number of identical
isolated systems.
The thermodynamic degeneracy may be written as ω(U, V, N).
From the fundamental postulate, the probability of finding the
system in the state r is pr = 1/ω.
Thus, S = – k r pr ln pr = k r (1/ω) ln ω
= (k/ω) ln ω r1 = k ln ω.
Statistical parameter: ω(U, V, N).
Thermodynamic parameter: S(U, V, N) [T dS = dU – PdV + μdN].
Connection: S = k ln ω.
Equilibrium condition: S  Smax.
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Ensembles 2
A canonical ensemble consists of a large number of identically
prepared systems, which are in thermal contact with a heat
reservoir at temperature T.
The probability pr of finding the system in the state r is given by
the Boltzmann distribution:
pr = exp(– Er)/Z, where Z = r exp(–Er), and  = 1/kT.
Now S = – k r pr ln pr = – k r [exp(–Er)/Z] ln[exp(–Er)/Z]
= – (k/Z) r exp(–Er) {ln exp(–Er) – ln Z}
= (k/Z) rEr exp(–Er) + (k lnZ)/Z . rexp(–Er),
so that
S = k U + k lnZ = k lnZ + kU.
Thus, S(T, V, N) = k lnZ + U/T and F = U – TS = – kT lnZ.
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Ensembles 3
S(T, V, N) = k lnZ + U/T , F = U – TS = – kT lnZ.
Statistical parameter: Z(T, V, N).
Thermodynamic parameter: F(T, V, N).
Connection: F = – kT ln Z.
Equilibrium condition: F  Fmin.
A grand canonical ensemble is a large number of identical
systems, which interact diffusively with a particle reservoir.
Each system is described by a grand partition function,
G(T, V, μ) = N{r(μN – EN,r)},
where N refers to the number of particles and r to the set of states
associated with a given value of N.
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Statistical Ensembles
• Classical phase space is 6N variables (pi, qi) with a
Hamiltonian function H(q,p,t).
• We may know a few constants of motion such as energy,
number of particles, volume, ...
• The most fundamental way to understand the foundation of
statistical mechanics is by using quantum mechanics:
– In a finite system, there are a countable number of states
with various properties, e.g. energy Ei.
– For each energy interval we can define the density of
states.
g(E)dE = exp(S(E)/kB) dE, where S(E) is the entropy.
– If all we know is the energy, we have to assume that each
state in the interval is equally likely. (Maybe we know the
p or another property)
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Environment
• Perhaps the system is isolated. No contact
with outside world. This is appropriate to
describe a cluster in vacuum.
• Or we have a heat bath: replace surrounding
system with heat bath. All the heat bath does
is occasionally shuffle the system by
exchanging energy,The
particles,
momentum,…..
only distribution
consistent with
a heat bath is a canonical distribution:
Prob(q, p) dqdp  e  H ( q, p ) / Z
See online notes/PDF derivation
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Statistical ensembles
•
•
•
•
•
(E, V, N) microcanonical, constant volume
(T, V, N) canonical, constant volume
(T, P N) canonical, constant pressure
(T, V , μ) grand canonical (variable particle number)
Which is best? It depends on:
– the question you are asking
– the simulation method: MC or MD (MC better for
phase transitions)
– your code.
• Lots of work in recent years on various
ensembles (later).
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Maxwell-Boltzmann Distribution
Prob(q, p) dqdp  e  H ( q , p ) / N !Z
• Z=partition function. Defined so that probability is
Z  exp( E i )
normalized.
• Quantum expression
• Also Z= exp(-β F), F=free energy (more convenient since F
is extensive)
• Classically: H(q,p) = V(q)+ Σi p2i /2mi
• Then the momentum integrals can be performed. One has simply an
uncorrelated Gaussian (Maxwell) distribution of momentum.
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Microcanonical ensemble
E, V and N fixed
S = kB lnW(E,V,N)
Canonical ensemble
T, V and N fixed
F = kBT lnZ(T,V,N)
Grand canonical ensemble
T, V and  fixed
F = kBT ln (T,V,)