...
There
...
is
...
real
...
statistical
...
mechanics
...
today.
...
An
...
ensemble
...
is
...
defined
...
and
...
an
...
average
...
calculated.
...
Microcanonical
...
&
...
Canonical
...
Ensembles
...
It
...
has
...
been
...
demonstrated
...
that
...
there
...
is
...
a
...
huge
...
number
...
of
...
microstates.
...
It
...
is
...
possible
...
to
...
connect
...
thermodynamics
...
with
...
the
...
complexity
...
of
...
the
...
microscopic
...
world.
...
Below
...
are
...
definitions.
...
Microstate
A microstate is a particular state of a system specified at the atomic level. This could be described by the many-body wavefunction. A system is something that over time fluctuates between different microstates. An example includes the gas from last time. There is immense complexity. Fix the variables <math>T, V,</math>
...
and
...
<math>N</math>.
...
With
...
only
...
these
...
variables
...
known,
...
there
...
is
...
no
...
idea
...
what
...
microstate
...
the
...
system
...
is
...
in
...
at
...
the
...
many
...
body
...
wavefunction
...
level.
...
Consider
...
a
...
solid.
...
It
...
is
...
possible
...
to
...
access
...
different
...
configurational
...
states,
...
and
...
two
...
are
...
shown
...
below.
...
Diffusion
...
is
...
a
...
process
...
that
...
results
...
in
...
changes
...
in
...
state
...
over
...
time.
...
X-ray
...
images
...
may
...
not
...
be
...
clear
...
due
...
to
...
the
...
diffusion
...
of
...
atoms
...
and
...
systems
...
accessing
...
different
...
microstates.
...
There
...
may
...
be
...
vibrational
...
excitations,
...
and
...
electronic
...
excitations
...
correspond
...
to
...
the
...
excitation
...
of
...
electrons
...
to
...
different
...
levels.
...
These
...
excitations
...
specify
...
what
...
state
...
the
...
solid
...
is
...
in.
...
Any
...
combination
...
of
...
excitations
...
specify
...
the
...
microstate
...
of
...
a
...
system.
Summary
A particular state of a system specified at the atomic level (many function wavebody level
| Latex |
|---|
!Two_configurational_states.PNG! h2. Summary A particular state of a system specified at the atomic level (many function wavebody level {latex} \[ \Psi_{\mbox{manybody}} \] {latex} |
.
...
The
...
system
...
over
...
time
...
fluctuates
...
betwen
...
different microstates
- There is immense complexity in a gas
- Solid
- Configurational states
- Vibrational excitations
- electronic excitations
- There is usually a combination of these excitations, which results in immense complexity
- Excitations specify the microstate of the system
Why Ensembles?
A goal is to find
| Latex |
|---|
microstates * There is immense complexity in a gas * Solid ** Configurational states ** Vibrational excitations ** electronic excitations *** There is usually a combination of these excitations, which results in immense complexity *** Excitations specify the microstate of the system h1. Why Ensembles? A goal is to find {latex} \[ P_v \] {latex} |
.
...
Thermodynamic
...
variables
...
are
...
time
...
averages.
...
Sum
...
over
...
the
...
state
...
using
...
the
...
schrodinger
...
equation
...
to
...
find
...
energy
...
and
...
multiplying
...
by
...
probability.
...
To
...
facilitate
...
averages,
...
ensembles
...
are
...
introduced.
...
Ensembles
...
are
...
collections
...
of
...
systems.
...
Each
...
is
...
very
...
large,
...
and
...
they
...
are
...
macroscopically
...
identical.
...
Look
...
at
...
the
...
whole
...
and
...
see
...
what
...
states
...
the
...
system
...
could
...
be
...
in.
...
Below
...
is
...
a
...
diagram
...
of
...
systems
...
and
...
ensembles.
...
Each
...
box
...
represents
...
a
...
system,
...
and
...
the
...
collection
...
of
...
systems
...
is
...
an
...
ensemble.
...
Each
...
box
...
could
...
represent
...
the
...
class,
...
and
...
v
...
could
...
represent
...
the
...
sleep
...
state.
...
Look
...
at
...
the
...
properties
...
of
...
the
...
ensemble.
...
There
...
are
| Latex |
|---|
} \[ A \] {latex} |
;macroscopically
...
identical
...
systems.
...
Eventually
| Latex |
|---|
} \[ A \] {latex} |
is
...
taken
...
to
...
go
...
to
| Latex |
|---|
} \[ \infty \] {latex} |
.
...
Each
...
system
...
evolves
...
over
...
time.
...
Deriving P_v
...
The
...
probability,
| Latex |
|---|
} \[ P_v \] {latex} |
is
...
defined
...
for
...
different
...
kinds
...
of
...
boundary
...
conditions.
...
When
...
looking
...
at
...
the
...
probability
...
that
...
students
...
in
...
a
...
class
...
are
...
asleep,
...
it
...
is
...
possible
...
to
...
take
...
an
...
instantaneous
...
snapshot.
...
The
...
term
| Latex |
|---|
} \[ a_v \] {latex} |
is
...
the
...
occupation
...
number
...
and
...
is
...
defined
...
as
...
the
...
number
...
of
...
systems
...
that
...
are
...
in
...
the
...
state
| Latex |
|---|
} \[ v \] {latex} |
at
...
the
...
time
...
of
...
the
...
snapshot.
...
The
...
fraction
...
of
...
systems
...
in
...
state
| Latex |
|---|
} \[ v \] {latex} is {latex} |
is
| Latex |
|---|
\[ P_v \] {latex}
|
.
...
This
...
is
...
one
...
approximation
...
to
...
get
...
the
...
probability,
...
and
...
it
...
could
...
be
...
a
...
bad
...
approximation.
| Latex |
|---|
} \[ P_v \approx \frac {a_v}{A} \] {latex} |
There
...
are
...
an
...
additional
...
definitions
...
of
| Latex |
|---|
} \[ P_v \] {latex} |
.
...
It
...
is
...
equal
...
to
...
the
...
probability
...
of
...
finding
...
a
...
system
...
in
...
state
...
v
...
at
...
time
...
t
...
or
...
identically
...
it
...
is
...
the
...
fraction
...
of
...
time
...
spent
...
by
...
the
...
system
...
in
...
state
...
v.
...
In
...
the
...
case
...
of
...
the
...
sleep
...
example,
...
it
...
is
...
equal
...
to
...
the
...
fraction
...
of
...
time
...
that
...
any
...
one
...
is
...
in
...
the
...
sleep
...
state.
...
The
...
time
...
average
...
corresponds
...
to
...
looking
...
at
...
a
...
class
...
and
...
seeing
...
for
...
what
...
fraction
...
of
...
time
...
does
...
it
...
find
...
someone
...
in
...
the
...
class
...
asleep.
...
The
...
ensemble
...
average
...
corresponds
...
to
...
looking
...
at
...
the
...
set
...
of
...
identical
...
classes
...
and
...
seeing
...
how
...
many
...
classes
...
have
...
at
...
least
...
one
...
student
...
asleep.
...
There
...
is
...
a
...
correlation
...
between
...
the
...
state
...
average
...
and
...
the
...
time
...
average.
...
There
...
is
...
a
...
need
...
of
...
boundary
...
conditions.
...
Take
...
a
...
picture
...
of
...
a
...
large
...
number
...
of
...
systems,
...
look
...
at
...
everyone,
...
and
...
average.
...
Summary
Recap:
| Latex |
|---|
} \[ E = \sum_V E_V P_V \] {latex} |
To
...
facilitate
...
averages,
...
we
...
introduce
...
"ensembles"
...
that
...
we
...
average
...
over
...
- Averaging
...
- over
...
- many
...
- bodies
...
- rather
...
- than
...
- averaging
...
- over
...
- time
...
- Example:
...
- student
...
- =
...
- system,
...
- v
...
- =sleepstate
...
Ensemble
...
of
...
systems:
...
- 'A'
...
- (a
...
- very
...
- large
...
- number)
...
- macroscopically
...
- identical
...
- systems
...
- Each
...
- system
...
- evolves
...
- over
...
- time
...
Probability:
...
- Take
...
- an
...
- instantaneous
...
- snapshot
...
- Define
Latex \[ a_v \]
...
=
...
- #of
...
- systems
...
- that
...
- are
...
- in
...
- state
...
- v
...
- at
...
- the
...
- time
...
- of
...
- snapshot
...
- Fraction
...
- of
...
- systems
...
- in
...
- state
...
- v
...
- is
...
Latex
...
\[ \frac{a_v}{A} \simeq P_v \]- Probability to find a system in statevat timet
- Fraction of time spent in statev
Microcanonical Ensemble
The boundary conditions of all systems of the microcanonical ensemble are the same. The variablesN, V,andEcannot fluctuate. Each system can only fluctuate between states with fixed energy,E.
It is possible to get degeneracy from the Shrodinger equation.
| Latex |
|---|
{latex} ** Probability to find a system in statevat timet ** Fraction of time spent in statev h1. Microcanonical Ensemble The boundary conditions of all systems of the microcanonical ensemble are the same. The variablesN, V,andEcannot fluctuate. Each system can only fluctuate between states with fixed energy,E. !Microcanonical_ensemble.PNG! It is possible to get degeneracy from the Shrodinger equation. {latex} \[ \hat H \Psi = E \begin {matrix} \underbrace{ (\Psi_1, \Psi_2, ..., \Psi_{\Omega}) } \\ \Omega(E) \end{matrix} \] {latex} |
Consider
...
the
...
example
...
of
...
the
...
hydrogen
...
atom.
...
Below
...
is
...
an
...
expression
...
of
...
the
...
energy
...
proportionality
...
and
...
the
...
degeneracy
...
when
| Latex |
|---|
} \[ n=1 \] {latex} and {latex} |
and
| Latex |
|---|
\[ n=2 \] {latex} |
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ E \alpha \frac {1}{n^2} |
...
\] |
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ \Omega (n=1) = 1 \] |
...
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ \Omega (n=2) = 4 \] |
What is Pv in microcanonical ensemble?
All states should be equally probable with variables
| Latex |
|---|
{latex}
h2. What is Pv in microcanonical ensemble?
All states should be equally probable with variables
{latex} \[ N, V, \] |
and
| Latex |
|---|
{latex} and {latex} \[ E \] {latex} \ |
[
...
fixed.
...
The
...
term
| Latex |
|---|
} \[ P_v \] {latex} |
is
...
the
...
probability
...
of
...
being
...
in
...
any
| Latex |
|---|
} \[ E \] {latex} |
state
...
for
...
a
...
system,
...
and
...
it
...
should
...
be
...
equal
...
to
...
a
...
constant.
...
An
...
expression
...
is
...
below.
...
Each
...
state
...
can
...
be
...
accessed,
...
and
...
one
...
is
...
not
...
more
...
favored.
...
This
...
is
...
related
...
to
...
the
...
principle
...
of
...
a
...
priori
...
probability.
...
There
...
is
...
no
...
information
...
that
...
states
...
should
...
be
...
accessed
...
with
...
different
...
probability.
| Latex |
|---|
} \[ P_v = \frac{1}{\Omega (E)} \] {latex} h2. Example Consider an example of a box and gas. All the atoms |
Example
Consider an example of a box and gas. All the atoms are in one corner in the second box. Add to get complete degeneracy. The value of
| Latex |
|---|
are in one corner in the second box. Add to get complete degeneracy. The value of {latex} \[ \Omega_1 (E) \] {latex} |
is
...
large;
...
there
...
is
...
enormous
...
degeneracy.
| Latex |
|---|
!Microcanonical_ensemble_II.PNG! {latex} \[ \Omega_1(E) \gg \Omega_2(E) \] {latex} |
Consider
...
a
...
poker
...
hand.
...
There
...
is
...
a
...
lot
...
of
...
equivalence
...
in
...
bad
...
hands.
...
These
...
are
...
dealt
...
most
...
of
...
the
...
time
...
and
...
correspond
...
to
| Latex |
|---|
} \[ \Omega_1(E) \] {latex} |
.
...
The
...
royal
...
flush
...
corresponds
...
to
| Latex |
|---|
} \[ \Omega_2(E) \] {latex} |
.
...
It
...
is
...
equally
...
probable,
...
but
...
there
...
are
...
many
...
fewer
...
ways
...
to
...
get
...
the
...
royal
...
flush.
...
There
...
are
...
the
...
same
...
boundary
...
conditions.
...
In
...
an
...
isolated
...
system,
...
in
...
which
| Latex |
|---|
} \[ N, V, \] |
and
| Latex |
|---|
{latex} and {latex} \[ E \] {latex} |
are
...
fixed,
...
it
...
is
...
equally
...
probable
...
to
...
be
...
in
...
any
...
of
...
its
| Latex |
|---|
} \[ \Omega (E) \] {latex} |
possible
...
quantum
...
states.
Summary
The variables
| Latex |
|---|
h2. Summary
The variables
{latex} \[ N, V, \] |
and
| Latex |
|---|
{latex} and {latex} \[ E \] {latex} |
are
...
fixed
...
- Each
...
- system
...
- can
...
- only
...
- fluctuate
...
- between
...
- states
...
- with
...
- fixed
...
- energy
...
Latex
...
\[ E \]
...
(like
...
- from
...
- Schrodinger's
...
- equation)
...
Latex \[ \hat H \Psi = E \Psi \rightarrow E[\Psi_2 .... \Psi_\omega \|\Psi_1|\Psi_1] \]
...
Latex
...
\[ \Omega(E) \]
...
- All states are equally probable, and are given equal weight.
Hydrogen atom
Latex \[ E = \frac{1}{n^2}
...
\]Latex \[ \Omega (n=1) =1 \]
...
Latex \[ \Omega (n=2) =4 \]
...
Probability
...
of
...
being
...
in
...
any
...
E
...
state
...
for
...
a
...
system
...
- employed
...
- the
...
- principle
...
- of
...
- equal
...
- a
...
- priory
...
- probabilities
...
Latex \[ P_v = \mbox{constant} \]
...
Latex
...
\[ P_v = \frac{1}{\Omega(E)} \]
...
Systems
- Equally more probable
- Some accessed more times because of large degeneracy number
- There's just a lot more ways to get configuration left (like a craphand) than the right (like a straight flush)
Latex \[ \Omega_1 (E) >> \Omega_2 (E) \]
...
An
...
isolated
...
system
...
(
| Latex |
|---|
} \[ N, V, E= \] {latex} |
fixed)
...
is
...
equally
...
probable
...
to
...
be
...
in
...
any
...
of
...
its
| Latex |
|---|
} \[ \Omega (E) \] {latex} |
possible
...
quantum
...
states.
...
Canonical
...
Ensemble
...
There
...
is
...
a
...
different
...
set
...
of
...
boundary
...
conditions
...
in
...
the
...
canonical
...
ensemble.
...
There
...
are
...
heat
...
conducting
...
walls
...
or
...
boundaries
...
of
...
each
...
system.
...
Each
...
of
...
the
| Latex |
|---|
} \[ A \] {latex} |
members
...
of
...
the
...
ensemble
...
find
...
themselves
...
in
...
the
...
heat
...
bath
...
formed
...
by
...
the
| Latex |
|---|
} \[ A-1 \] {latex} |
members.
...
Each
...
system
...
can
...
fluctuate
...
between
...
different
...
microstates.
...
An
...
energy
...
far
...
from
...
average
...
is
...
unlikely.
...
In
...
the
...
picture
...
below,
...
the
...
energy
...
of
...
the
...
ensemble
...
on
...
the
...
right
...
side
...
of
...
the
...
ensemble
...
is
...
fixed,
...
while
...
the
...
energy
...
of
...
a
...
particular
...
system
...
is
...
not
...
fixed
...
and
...
can
...
fluctuate.
Take another snapshot. There is interest in the distribution. The term
| Latex |
|---|
!Canonical_ensemble.PNG! Take another snapshot. There is interest in the distribution. The term {latex} \[ \overline {a} \] {latex} |
is
...
equal
...
to
...
the
...
number
...
of
...
systems
...
in
...
state
| Latex |
|---|
} \[ v \] {latex} . {latex} |
.
| Latex |
|---|
\[ {a_v} = \overline{a} \] {latex}
|
Below
...
is
...
a
...
table
...
of
...
microstates,
...
energy,
...
and
...
occurence,
...
and
...
a
...
graph.
...
In
...
the
...
graph,
...
equilibrium
...
has
...
occurred,
...
but
...
all
...
states
...
can
...
be
...
accessed.
...
It
...
is
...
possible
...
to
...
access
...
different
...
states
...
some
...
distance
...
from
...
the
...
average
...
energy.
...
The
...
total
...
energy,
| Latex |
|---|
} \[ \epsilon \] {latex} |
,
...
is
...
fixed
...
and
...
is
...
equal
...
to
...
the
...
integral
...
of
...
the
...
curve.
...
As
...
the
...
number
...
of
...
systems
...
increases,
...
the
...
curve
...
becomes
...
sharper.
...
|
|---|
...
1 | 2 | 3 |
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
Constraints
Below are constraints. The first is the sum of the occupation number. The second constraint is possible due to the system being isolated.
| Latex |
|---|
} \] {latex} | !Occurence_versus_energy.PNG! h2. Constraints Below are constraints. The first is the sum of the occupation number. The second constraint is possible due to the system being isolated. {latex} \[ \sum_v a_v = A \] {latex} |
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ \sum_v a_v E_v = \epsilon \] |
...
The
...
term
| Latex |
|---|
} \[ P_v \] {latex} |
is
...
the
...
probability
...
of
...
finding
...
the
...
system
...
in
...
state
| Latex |
|---|
} \[ v \] {latex} |
.
...
It
...
is
...
possible
...
to
...
use
...
snapshot
...
probability.
...
There
...
are
...
many
...
distributions
...
that
...
satisfy
...
the
...
boundary
...
conditions.
...
There
...
is
...
a
...
better
...
way
...
to
...
find
| Latex |
|---|
} \[ P_v \] {latex} |
,
...
and
...
a
...
relation
...
is
...
below.
...
It
...
corresponds
...
to
...
the
...
average
...
distribution.
...
This
...
is
...
associated
...
with
...
a
...
crucial
...
insight.
| Latex |
|---|
} \[ P_v = \frac{\overline{a_v}}{A} \] {latex} h2. Crucial |
Crucial Insight
An assumption is that the entire canonical ensemble is isolated. No energy can escape, and the energy
| Latex |
|---|
Insight An assumption is that the entire canonical ensemble is isolated. No energy can escape, and the energy {latex} \[ \epsilon \] {latex} |
is
...
constant.
...
Every
...
distribution
...
of
| Latex |
|---|
} \[ \overline{a} \] {latex} |
that
...
satisfies
...
the
...
boundary
...
conditions
...
is
...
equally
...
probable.
...
it
...
is
...
possible
...
to
...
write
...
many
...
body
...
wavefunction
...
because
...
the
...
energy
...
of
...
the
...
entire
...
ensemble
...
is
...
fixed.
...
The
...
principle
...
of
...
equal
...
...
...
probabilities
...
is
...
applied.
...
Look
...
at
...
the
...
whole
...
distribution
...
that
...
satisfies
...
the
...
boundarty
...
condition.
...
Each
...
distribution
...
of
...
occurance
...
numbers
...
must
...
be
...
given
...
equal
...
weights.
Summary
The variables
| Latex |
|---|
h2. Summary The variables {latex} \[ N, V, T \] {latex} |
are
...
fixed
...
- There
...
- are
...
- heat
...
- conducting
...
- bondaries
...
- of
...
- each
...
- system
...
- Each
...
- of
...
- the
...
Latex
...
\[ A \]
...
(=
...
- large
...
- number)
...
- members
...
- finds
...
- itself
...
- in
...
- a
...
- heat
...
- bath,
...
- formed
...
- by
...
- the
...
Latex
...
\[ (A - 1) \]
...
other
...
- members
...
- Take
...
- snapshot;
...
- get
...
- distribution
...
Latex
...
\[ a_v = \overline{a} \]
...
(=
...
- #
...
- of
...
- systems
...
- in
...
- state
...
Latex
...
)\[ v \]
Constraints
The total energy,
| Latex |
|---|
{latex} ) Constraints The total energy, {latex} \[ \epsilon \] {latex} |
,
...
is
...
fixed
| Latex |
|---|
} \[ \sum_v a_v = A \sum_v a_v E_v = \epsilon \] {latex} |
(isolated
...
!)
...
Probability
| Latex |
|---|
} \[ P_v \simeq \frac{a_v}{A} \] {latex} |
is
...
an
...
approximation.
...
- Better
...
- to
...
- use
...
Latex
...
\[ P_v = \frac{\overline{a_v}}{A} \]
...
,
...
- the
...
- averaged
...
- distribution
...
- There
...
- is
...
- an
...
- assumption
...
- that
...
- the
...
- whole
...
- canonical
...
- ensemble
...
- is
...
- isolated
...
- and
...
- that
...
- energy
...
Latex
...
\[ \epsilon \]
...
is
...
- constant.
...
- Every
...
- distribution
...
- of
...
Latex
...
\[ \overline{a} \]
...
that
...
- satisfies
...
- the
...
- boundary
...
- conditions
...
- is
...
- equally
...
- probable.
...
- We
...
- are
...
- applying
...
- the
...
- principle
...
- of
...
- equal
...
...
...
- probabilities,
...
- and
...
- each
...
- distribution
...
- of
...
- occurance
...
- numbers
...
- must
...
- be
...
- given
...
- equal
...
- weights.
...
Some
...
Math
...
Consider
...
every
...
possible
...
distribution
...
consistent
...
with
...
boundary
...
conditions,
...
and
...
for
...
each
...
distribution
...
consider
...
every
...
possible
...
permutation.
...
The
...
term
| Latex |
|---|
} \[ w (\overline{a}) \] {latex} |
is
...
equal
...
to
...
the
...
number
...
of
...
ways
...
to
...
obtain
...
a
...
distribution
| Latex |
|---|
} \[ \overline{a} \] {latex} |
,
...
where
| Latex |
|---|
} \[ a_v \] {latex} |
is
...
the
...
number
...
of
...
systems
...
in
...
state
| Latex |
|---|
} \[ v \] {latex} |
.
...
A
...
bad
...
hand
...
in
...
poker
...
is
...
defined
...
by
...
a
...
large
...
number
...
of
| Latex |
|---|
} \[ a \] {latex} |
.
...
Use
...
the
...
multinomial
...
distribution.
| Latex |
|---|
} \[ a_i = \overline{a} \] {latex} |
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ w (\overline{a}) = \frac{ A! }{ a_1! a_2! a_3! ..... a_v! } \] |
...
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ w (\overline{a}) = \frac{ A! }{ \Pi_v a_v!} \] |
...
| Wiki Markup |
|---|
{html} <P>Â </P>{html} |
...
| Latex |
|---|
...
\[ a_1 \] |
...
| Wiki Markup |
|---|
{html} <P>Â </P>{html} |
...
| Latex |
|---|
...
\[ \mbox{Number of systems in state 1} \] |
...
| Wiki Markup |
|---|
{html} <P>Â </P>{html} |
...
| Latex |
|---|
...
\[ a_{\nu} \] |
...
|
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ \mbox{Number of systems in state v} \] |
...
Below
...
are
...
expressions
...
of
...
the
...
probability
...
to
...
be
...
in
...
a
...
certain
...
state.
...
The
...
term
| Latex |
|---|
} \[ a_v \] {latex} |
is
...
averaged
...
over
...
all
...
possible
...
distributions.
...
Every
...
distribution
...
is
...
given
...
equal
...
weight,
...
and
...
the
...
one
...
with
...
the
...
most
...
permutations
...
is
...
the
...
most
...
favored.
| Latex |
|---|
} \[ P_v = \frac{\overline{a_v}}{A} \] {latex} |
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ P_v = \frac{1}{A} \frac{ \sum_{\overline {a}} \omega (\overline{a}) a_v (\overline{a} ) }{ \sum_{\overline a} \omega (\overline{a}) } \] |
Example
Below is an example of four systems in an ensemble. The term
| Latex |
|---|
} \] {latex} h2. Example Below is an example of four systems in an ensemble. The term {latex} \[ P_1 \] {latex} |
is
...
the
...
probability
...
of
...
any
...
system
...
to
...
be
...
in
...
state
| Latex |
|---|
} \[ 1 \] {latex} . || {latex} |
.
|
|---|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
...
|
| Latex |
|---|
{latex} | {latex} \[ \mbox{Distribution A} \] {latex} |
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ w(A) = \frac{4!}{0!4!} \] |
...
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ w(A) = 1 \] |
...
| Wiki Markup |
|---|
{html} <P>Â </P>{html} |
...
| Latex |
|---|
...
\[ \mbox{Distribution B} \] |
...
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ w(B) = \frac{4!}{1!3!} \] |
...
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ w(B) = 4P_1 = \frac{1}{4} \left ( \frac{1 \cdot 0 + 4 \cdot 1 + 6 \cdot 2 + 4 \cdot 3 + 1 \cdot 4}{1 + 4 + 6 + 4 + 1} \right ) \ |
...
] |
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
...
| Latex |
|---|
...
\[ P_1 = \frac{1}{2} \] |
...
| Wiki Markup |
|---|
{html} <P>Â </P>{html} |
...
Distribution
...
of
...
| Latex |
|---|
...
\[ w(a) \] |
...
The
...
term
| Latex |
|---|
} \[ w(a) \] {latex} |
is
...
the
...
number
...
of
...
permutations
...
for
...
a
...
particular
...
distribution.
...
As
...
the
...
number
...
of
...
systems
...
increases,
...
or
...
asAincreases,
...
the
...
distribution
...
becomes
...
more
...
peaked.
...
Consider
...
the
...
probability.
| Latex |
|---|
} \[ P_{\nu} = \frac{1}{A} \frac{ \sum_{\overline {a}} \omega (\overline{a}) a_{\nu} (\overline{a} ) }{ \sum_{\overline {a}} \omega (\overline{a}) } \] {latex} |
| Latex |
|---|
{latex} \[ P_{\nu} \approx \frac{ \frac{1}{A} w ( \overline{a}^*) a_{\nu}^*}{w ( \overline{a}^*) } \] |
| Latex |
|---|
{latex} {latex} \[ P_{\nu} \approx \frac{a_{\nu}^*}{A} \] {latex} |
(Equation
...
1)
...
Look
...
at
...
the
...
distribution
...
that
...
maximizes
| Latex |
|---|
} \[ w ( \overline{a}) \] {latex} |
,
...
the
...
permutation
...
number.
...
To
...
get
| Latex |
|---|
} \[ a_{\nu} \] {latex} |
,
...
maximize
| Latex |
|---|
} \[ w ( \overline{a} ) \] {latex} |
subject
...
to
...
the
...
constraints
...
below.
| Latex |
|---|
} \[ \sum_{\nu} a_{\nu} - A = 0 \] {latex} {latex} |
| Latex |
|---|
\[ \sum_{\nu} a_{\nu}E_{\nu} - \epsilon = 0 \] {latex}
|
Use
...
Lagrange
...
multipliers,
...
and
...
maximize
| Latex |
|---|
} \[ \ln w ( \overline{a} ) \] {latex} |
in
...
order
...
to
...
be
...
able
...
to
...
use
...
Stirling's
...
approximation.
| Latex |
|---|
} \[ \frac{\partial}{\partial a_{\nu}} \left ( \ln w ( \overline{a}) - \alpha \sum_k a_k - \beta \sum_k a_k E_k \right ) = 0 \] |
| Latex |
|---|
{latex} {latex} \[ w ( \overline{a} ) = \frac{A!}{\pi_k a_k!} \] {latex} {latex} |
| Latex |
|---|
\[ \ln w ( \overline{a}) = \ln A! + \left ( - \ln \pi_k a_k! \right ) = \ln A! - \sum_k \ln a_k! \] {latex}
|
Use
...
Stirling's
...
approximation
...
as
| Latex |
|---|
} \[ A \] {latex} |
and
...
the
...
occupation
...
number,
| Latex |
|---|
} \[ a_k \] {latex} |
,
...
go
...
to
...
infinity.
| Latex |
|---|
} \[ \sum_k \ln a_k! = \sum_k \left( a_k \ln a_k - a_k \right ) \] |
| Latex |
|---|
{latex} {latex} \[ = \sum_K a_K \ln a_K - \sum_K a_K \] {latex} {latex} |
| Latex |
|---|
\[ = \sum_K a_K \ln a_K - A\frac{\partial}{\partial a_{\nu}} \left ( \ln A! - \sum_k a_k \ln a_k + A - \alpha \sum_k a_k - \beta \sum_k a_k E_k \right ) = 0 \] {latex}
{latex} |
| Latex |
|---|
\[ \left ( a_v \to x \mbox{ , } \ln A! - \sum_k a_k \ln a_k + A \to -a_v \ln a_v \mbox{ , }\alpha \sum_k a_k \to \alpha a_v \mbox{ , }\beta \sum_k a_k E_k \to \beta a_v E_k \right )\ln a_v - 1 - \alpha - \beta E_{\nu} = 0 \] |
[
The term
| Latex |
|---|
{latex} \[ The term {latex} \[ a_{\nu} \] {latex} |
is
...
the
...
occupation
...
number
...
that
...
maximizes
...
the
...
expressione
| Latex |
|---|
} \[ \alpha ' \cdot e^{\beta E_{\nu}} \] {latex} |
,
...
where
| Latex |
|---|
} \[ \alpha ' = \alpha + 1 \] {latex} |
.
...
Use
...
constraints
...
to
...
determine
...
the
...
Lagrange
...
multipliers
...
and
...
determine
...
the
...
probability.
| Latex |
|---|
} \[ a_{\nu}=e^{\alpha '} \cdot e^{\beta E_{\nu} \] |
| Latex |
|---|
{latex} {latex} \[ \sum_{\nu} a_{\nu} = A \] {latex} {latex} |
| Latex |
|---|
\[ \sum_{\nu} e^{\alpha '} \cdot e^{\beta E_{\nu} = A \] {latex}
{ |
| Latex |
|---|
latex} \[ e^{\alpha '} = \frac{1}{A} \sum_{\nu} e^{\beta E_{\nu}} \] |
The probability of being in a certain state
| Latex |
|---|
{latex} The probability of being in a certain state {latex} \[ \nu \] {latex} |
can
...
be
...
calculated,
...
and
...
it
...
is
...
still
...
in
...
terms
...
of
...
the
...
second
...
Lagrange
...
multiplier.
...
Plugging
...
back
...
into
...
Equation
...
1:
| Latex |
|---|
} \[ P_v = \frac{a_{\nu}}{A} = \frac{A}{\sum_{\nu} e^{-\beta E_{\nu}}} \cdot \frac{e{\beta E_{\nu}}}{A} = \frac{e^{-\beta E_{\nu}}}{\sum_{\nu} e^{-\beta E_{\nu}} \] |
Partition Function
The denominator is the partition function,
| Latex |
|---|
{latex} h2. Partition Function The denominator is the partition function, {latex} \[ Q = \sum_{\nu}e^{-\beta E_{\nu} \] {latex} |
.
...
It
...
tells
...
us
...
how
...
many
...
states
...
are
...
accessible
...
by
...
the
...
system.
...
Determine
| Latex |
|---|
} \[ \beta \] {latex} , a measure of thermally accessible states (?). Look |
, a measure of thermally accessible states 
. Look at how the partition function connects to macroscopic thermodynamic variables. Find
| Latex |
|---|
at how the partition function connects to macroscopic thermodynamic variables. Find {latex} \[ \beta \] {latex} |
and
...
find
| Latex |
|---|
} \[ \overline{E} \] {latex} {latex} |
| Latex |
|---|
\[ \overline{E} = \sum_{\nu} P_{\nu} E_{\nu}\overline{E}= \frac{\sum_{\nu} E_{\nu} e^{-\beta E_{\nu}}}{Q} \] {latex}
|
Consider
...
the
...
average
...
pressure.
...
The
...
pressure
...
for
...
one
...
microstate
...
is
| Latex |
|---|
} \[ p_{\nu} \] {latex} . {latex} |
.
| Latex |
|---|
\[ p_{\nu} = \frac{-\partial E_{\nu}}{\partial V} \] {latex}
{latex} |
| Latex |
|---|
\[ p_{\nu} = \sum_{\nu} P_{\nu} p_{\nu} \] {latex}
{latex} |
| Latex |
|---|
\[ p_{\nu} = \frac{ -\sum_{\nu} \left ( \frac{\partial E_{\nu}}{\partial V} \right ) e^{-\beta E_{\nu}}}{Q} \] {latex}
Summary
|
Summary
In the case of a canonical ensemble, the energy of the entire ensemble is fixed. Each state is equally probable, and there is degeneracy. The probability is a function of how many ways to get the distribution. The distribution with the most permutation is the most probable. The graph can become very peaked. Once the distribution is known, do a maximization of
| Latex |
|---|
In the case of a canonical ensemble, the energy of the entire ensemble is fixed. Each state is equally probable, and there is degeneracy. The probability is a function of how many ways to get the distribution. The distribution with the most permutation is the most probable. The graph can become very peaked. Once the distribution is known, do a maximization of {latex} \[ w(\overline{a}) \] {latex} |
.
...
Use
...
Lagrange
...
multipliers
...
and
...
two
...
constraints.
...
The
...
term
| Latex |
|---|
} \[ a_{\nu}^* \] {latex} |
is
...
the
...
distribution
...
that
...
maximizes
...
an
...
expression.
...
This
...
is
...
what
...
is
...
most
...
often
...
found.
...
Go
...
back
...
to
...
the
...
probability,
...
get
...
an
...
expression,
...
and
...
give
...
part
...
of
...
it
...
a
...
name.
...
The
...
term
| Latex |
|---|
} \[ \beta \] {latex} |
is
...
a
...
measure
...
of
...
how
...
many
...
states
...
are
...
thermally
...
accessible.
| Latex |
|---|
} \[ \overline{a} \] {latex} |
Consider
...
every
...
possible
...
distribution
| Latex |
|---|
} \[ a_i = \overline{a} \] {latex} |
(consistent
...
with
...
boundary
...
conditions)
...
- For
...
- each
...
- distribution,
...
- consider
...
- every
...
- possible
...
- permutation
...
- The
...
- number
...
- of
...
- ways
...
- to
...
- obtain
...
Latex
...
\[ \overline{a} \]
...
isLatex \[ \omega (\overline{a}) = \frac{ A! }{ a_1! a_2! a_3! ..... a_v! } = \frac{ A! }{ \Pi_v a_v!} \]
...
,
...
- where
...
Latex
...
\[ a_v \]
...
is
...
- the
...
- number
...
- of
...
- systems
...
- in
...
- state
...
Latex
...
.\[ v \]
Probability
| Latex |
|---|
{latex}
.
Probability
{latex} \[ P_v = \frac{\overline{a_v}}{A} = \frac{1}{A}\frac{ \sum_{\overline a} \omega (\overline{a}) a_v (\overline{a} ) }{ \sum_{overline a}}\omega (\overline{a}) } \] |
Averaging
| Latex |
|---|
{latex} Averaging {latex} \[ a_v \] {latex} |
over
...
all
...
possible
...
distributions.
| Latex |
|---|
} \[ w(\overline a) \] {latex} {latex} |
| Latex |
|---|
\[ w(\overline a) \] {latex}
|
is
...
very
...
peaked
...
around
...
a
...
specific
...
distribution
...
Increase
| Latex |
|---|
} \[ A \] {latex |
and
| Latex |
|---|
} and {latex} \[ \omega{\overline a} \] {latex} |
becomes
...
more
...
peaked
| Latex |
|---|
} \[ a_v \] {latex} |
To
...
get
| Latex |
|---|
} \[a_v \] {latex} |
,
...
maximize
| Latex |
|---|
} \[ w (\overline a) \] {latex} |
subject
...
to
...
constraints.
...
Find
...
the
...
partition
...
function,
| Latex |
|---|
} \[ Q \] {latex} |