...
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 |
|---|
* 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 |
|---|
!Ensemble_of_systems.PNG! h3. 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 |
|---|
;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 |
at the time of the snapshot. The fraction of systems in state
| Latex |
|---|
in state {latex} \[ v \] |
is
| Latex |
|---|
{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 |
|---|
{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 |
|---|
h2. 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 |
|---|
\[ {latex} ** Probability to find a system in statevat timet ** Fraction of time spent in statev h1. Microcanonical Ensemble The boundary conditions of all systems of the microcanonical ensemble are the same. The variablesN, V,andEcannot fluctuate. Each system can only fluctuate between states with fixed energy,E. !Microcanonical_ensemble.PNG! It is possible to get degeneracy from the Shrodinger equation. \hat H \Psi = E \begin {matrix} \underbrace{ (\Psi_1, \Psi_2, ..., \Psi_{\Omega}) } \\ \Omega(E) \end{matrix} Consider the example of the hydrogen atom. Below is an expression of the energy proportionality and the degeneracy whenn=1andn=2 E \prop \frac \] |
Consider the example of the hydrogen atom. Below is an expression of the energy proportionality and the degeneracy when
| Latex |
|---|
\[ n=1 \] |
and
| Latex |
|---|
\[ n=2 \] |
| 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 |
|---|
\[ N h2. What isP_vin microcanonical ensemble? All states should be equally probable with variablesN, V,andEfixed. The termP_vis the probability\] |
and
| Latex |
|---|
\[ E \] |
[ fixed. The term
| Latex |
|---|
\[ P_v \] |
is the probability of being in any
| Latex |
|---|
\[ E \] |
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 |
|---|
\[ of being in anyEstate for a system, and it should be equal to a constant. An expression is below. Each state can be accessed, and one is not more favored. This is related to the principle of _a priori_ probability. There is no information that states should be accessed with different probability. P_v = \frac{1}{\Omega (E) h2. Example Consider an example of a } \] |
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 |
|---|
\[ box and gas. All the atoms are in one corner in the second box. Add to get complete degeneracy. The value of\Omega_1 (E) \] |
is
...
large;
...
there
...
is
...
enormous degeneracy.
| Latex |
|---|
\[ degeneracy. !Microcanonical_ensemble_II.PNG! \Omega_1(E) \gg \Omega_2(E) \] |
Consider a poker hand. There is a lot of equivalence in bad hands. These are dealt most of the time and correspond to
| Latex |
|---|
\[ Consider a poker hand. There is a lot of equivalence in bad hands. These are dealt most of the time and correspond to\Omega_1(E) \] |
.
...
The
...
royal
...
flush corresponds to
| Latex |
|---|
\[ corresponds to\Omega_2(E). It\] |
. 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 |
|---|
\[ N in whichN, V,andEare \] |
and
| Latex |
|---|
\[ E \] |
are fixed,
...
it
...
is
...
equally
...
probable
...
to
...
be
...
in
...
any of its
| Latex |
|---|
\[ of its\Omega (E) \] |
possible
...
quantum
...
states.
Summary
The variables
| Latex |
|---|
\[ N h2. Summary The variablesN, V,andEare fixed * Each system can only fluctuate between states with fixed energy E (like from Schr��dinger's equation) * \] |
and
| Latex |
|---|
\[ E \] |
are fixed
- Each system can only fluctuate between states with fixed energy
(like from Schrodinger's equation)Latex \[ E \] 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 |
|---|
\[ An isolated system (N, V, E= \] |
fixed)
...
is
...
equally
...
probable
...
to
...
be
...
in
...
any of its
| Latex |
|---|
\[ of its\Omega (E) \] |
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 |
|---|
\[ A \] |
members of the ensemble find themselves in the heat bath formed by the
| Latex |
|---|
\[ A-1 \] |
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 |
|---|
\[ \overline {a} \] |
is equal to the number of systems in state
| Latex |
|---|
\[ v \] |
.
| Latex |
|---|
\[ Each of theAmembers of the ensemble find themselves in the heat bath formed by theA-1members. Each system can fluctuate between different microstates. An energy far from average is unlikely. In the picture below, the energy of the ensemble on the right side of the ensemble is fixed, while the energy of a particular system is not fixed and can fluctuate. !Canonical_ensemble.PNG! Take another snapshot. There is interest in the distribution. The term\overline{a}is equal to the number of systems in statev. {a_v} = \overline{a} Below is a table of microstates, energy, and occurence, and a graph. In the graph, equilibrium has occurred, but all states can be accessed. It is possible to access different states some distance from the average energy. The total energy,\epsilon, is fixed and is equal to the integral of the curve. As the number of systems increases, the curve becomes sharper. \mbox{microstate} 1 2 3 \nu \mbox{energy} E_1 E_2 E_3 E_{\nu} \mbox{occurence} a_1 a_2 a_3 a_{\nu} \] |
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 |
|---|
\[ \epsilon \] |
, 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 |
|---|
\[ !Occurence_versus_energy.PNG! h2. Constraints Below are constraints. The first is the sum of the occupation number. The second constraint is possible due to the system being isolated. \sum_v a_v = A \] |
| Wiki Markup |
|---|
{html}
<P>Â </P>{html} |
| Latex |
|---|
\[ \sum_v a_v E_v = \epsilon \] |
The term
| Latex |
|---|
\[ P_v \] |
is the probability of finding the system in state
| Latex |
|---|
\[ v \] |
. It is possible to use snapshot probability. There are many distributions that satisfy the boundary conditions. There is a better way to find
| Latex |
|---|
\[ P_v \] |
, and a relation is below. It corresponds to the average distribution. This is associated with a crucial insight.
| Latex |
|---|
\[ The termP_vis the probability of finding the system in statev. It is possible to use snapshot probability. There are many distributions that satisfy the boundary conditions. There is a better way to findP_v, and a relation is below. It corresponds to the average distribution. This is associated with a crucial insight. P_v = \frac{\overline{a_v}}{A} </math> <br> </center> h2. Crucial Insight An assumption is that the entire canonical ensemble is isolated. No energy can escape, and the energy <math>\epsilon</math> is constant. Every distribution of <math>\overline {a} that satisfies the boundary conditions is equally probable. it is possible to write many body wavefunction because the energy of the entire ensemble is fixed. The principle of equal [a priori|http:/ \] |
Crucial Insight
An assumption is that the entire canonical ensemble is isolated. No energy can escape, and the energy
| Latex |
|---|
\[ \epsilon \] |
is constant. Every distribution of
| Latex |
|---|
\[ \overline{a} \] |
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 a priori 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 |
|---|
\[/en.wikipedia.org/wiki/A_priori_%28statistics%29] probabilities is applied. Look at the whole distribution that satisfies the boundarty condition. Each distribution of occurance numbers must be given equal weights. h2. Summary The variables N, V, T \] |
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
otherLatex \[ (A - 1) \]
...
- members
...
- Take
...
- snapshot;
...
- get
...
- distribution
Latex \
...
[ a_v
...
= \overline{a}
...
(=\]
...
- #
...
- of
...
- systems
...
- in
...
- state
)Latex \[ v \]
Constraints
The total energy,
| Latex |
|---|
\[ \epsilon \] |
, is fixed
| Latex |
|---|
\[ )<p> </p>ConstraintsThe total energy, <math>\epsilon</math>, is fixed<math>\sum_v a_v = A A</math><math>\sum_v a_v E_v = \epsilon</math> epsilon \] |
(isolated!)
Probability
| Latex |
|---|
\[ PProbability<math>P_v \simeq \frac{a_v}{A} \] |
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
is constant. Every distribution ofLatex \[ \epsilon \]
thatLatex \[ \overline{a} \]
...
- satisfies
...
- the
...
- boundary
...
- conditions
...
- is
...
- equally
...
- probable.
...
- We
...
- are
...
- applying
...
- the
...
- principle
...
- of
...
- equal
...
...
...
- probabilities,
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- and
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- each
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- distribution
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- of
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- occurance
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- numbers
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- must
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- be
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- given
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- equal
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- weights.
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Some
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Math
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Consider
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every
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possible
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distribution
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consistent
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with
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boundary
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conditions,
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and
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for
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each
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distribution
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consider
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every
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possible
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permutation. The term
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\[ w The termw (\overline{a}) </math>\] |
is
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equal
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to
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the
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number
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of
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ways
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to
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obtain
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a distribution
| Latex |
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\[ distribution <math>\overline{a} \] |
, where
| Latex |
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\[ a_v \] |
is the number of systems in state
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\[ v \] |
. A bad hand in poker is defined by a large number of
| Latex |
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\[ a \] |
. Use the multinomial distribution.
| Latex |
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\[ wherea_vis the number of systems in statev. A bad hand in poker is defined by a large number ofa. Use the multinonmial distribution. \{ a_i \} = \overline{a}</math><br><math>w (\overline{a})\] |
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<P>Â </P>{html} |
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\[ w (\overline{a}) = \frac{ A! }{ a_1! a_2! a_3! ..... a_v! } \] |
...
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\[ w (\overline{a}) = \frac{ A! }{ \Pi_v a_v!} |
...
\] |
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<P>Â </P>{html} |
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\[ a_1 \] |
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\[ \mbox{Number of systems in state 1} \] |
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\[ a_{\nu} \] |
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\[ \mbox{Number of systems in state v} \] |
Below are expressions of the probability to be in a certain state. The term
| Latex |
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\[ a_v \] |
is averaged over all possible distributions. Every distribution is given equal weight, and the one with the most permutations is the most favored.
| Latex |
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\[ v} Below are expressions of the probability to be in a certain state. The terma_vis averaged over all possible distributions. Every distribution is given equal weight, and the one with the most permutations is the most favored. P_v = \frac{\overline{a_v}}{A} P_v = \frac{1}{A} \\] |
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\[ P_v = \frac{1}{A} \frac{ \sum_{\overline {a}} \omega (\overline{a}) a_v (\overline{a} ) }{ \sum_{\overline a} \omega (\overline{a}) } |
...
\] |
Example
Below is an example of four systems in an ensemble. The term
| Latex |
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\[ P_1 \] |
is the probability of any system to be in state
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\[ 1 \] |
.
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\[ \mbox{Distribution A} \] |
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\[ w(A) = \frac{4!}{0!4!} \] |
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<P>Â </P>{html} |
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\[ w(A) = 1 \] |
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<P>Â </P>{html} |
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\[ \mbox{Distribution B} \] |
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\[ w(B) = \frac{4!}{1!3! |
...
} \] |
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<P>Â </P>{html} |
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\[ w(B) = |
...
4P_1 = \frac{1}{4} \left ( \frac{1 \cdot 0 + 4 \cdot 1 + 6 \cdot 2 + 4 \cdot 3 + 1 \cdot 4}{1 + 4 + 6 + 4 + |
...
1} \right ) \] |
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<P>Â </P>{html} |
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\[ P_1 = \frac{1}{2 |
...
} \] |
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<P>Â </P>{html} |
Distribution of
| Latex |
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\[ w(a) \] |
The term
| Latex |
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\[ w The termw(a) \] |
is
...
the
...
number
...
of
...
permutations
...
for
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a
...
particular
...
distribution.
...
As
...
the
...
number
...
of
...
systems
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increases,
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or
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asAincreases,
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the
...
distribution
...
becomes
...
more peaked.
Consider the probability.
| Latex |
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\[ peaked. !W%28a%29_versus_a.PNG! !W%28a%29_versus_a_--_large_A.PNG! Consider the probability. P_{\nu} = \frac{1}{A} \frac{ \sum_{\overline {a}} \omega (\overline{a}) a_{\nu} (\overline{a} ) }{ \sum_{\overline {a}} \omega (\overline{a}) } (\overline{a}) } \] |
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\[ P_{\nu} \approx \frac{ \frac{1}{A} w ( \overline{a}^*) a_{\nu}^*}{w ( \overline{a}^*) } \] |
| Latex |
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\[ P_{\nu} \approx \frac{a_{\nu}^*}{A} \] |
(Equation
...
1)
Look at the distribution that maximizes
| Latex |
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\[ w Look at the distribution that maximizesw ( \overline{a}) \] ) |
,
...
the
...
permutation
...
number. To get
| Latex |
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\[ a To geta_{\nu}, \] |
, maximize
| Latex |
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\[ w maximizew ( \overline{a} ) \] |
subject
...
to
...
the
...
constraints
...
below.
| Latex |
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\[ \sum_{\nu} a_{\nu} - A = 0 \] |
| Latex |
|---|
\[ \sum_{\nu} a_{\nu}E_{\nu} - \epsilon = 0 \] |
Use Lagrange multipliers, and maximize
| Latex |
|---|
\[ Use Lagrange multipliers, and maximize\ln w ( \overline{a} ) \] |
in
...
order
...
to
...
be
...
able
...
to use Stirling's approximation.
| Latex |
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\[ use Stirling's approximation. \frac{\partial}{\partial a_{\nu}} \left ( \ln w ( \overline{a} ) - \alpha \sum_k a_k - \beta \sum_k a_k E_k \right ) = 0 \] |
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\[ w ( \overline{a} ) = \frac{A!}{\pi_k a_k!} \] |
| Latex |
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\[ \ln w ( \overline{a}) = \ln A! + \left ( - \ln \pi_k a_k! \right ) = \ln A! - \sum_k \ln a_k! Use \] |
Use Stirling's
...
approximation as
| Latex |
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\[ as A \] |
and
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the
...
occupation
...
number,
| Latex |
|---|
\[ a_k \] |
,
...
go
...
to
...
infinity.
| Latex |
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\[ \sum_k \ln a_k! = \sum_k \left( a_k \ln a_k - a_k \right ) \] |
| Latex |
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\[ = \sum_K a_K \ln a_K - \sum_K a_K \] |
| Latex |
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\[ = \sum_K a_K \ln a_K - A \frac{\partial}{\partial a_{\nu}} \left ( \ln A! - \sum_k a_k \ln a_k + A - \alpha \sum_k a_k - \beta \sum_k a_k E_k \right ) = 0 \] |
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\[ \left ( a_v \to x \mbox{ , } \ln A! - \sum_k a_k \ln a_k + A \to -a_v \ln a_v \mbox{ , } \alpha \sum_k a_k \to \alpha a_v \mbox{ , } \beta \sum_k a_k E_k \to \beta a_v E_k \right ) - \ln a_v - 1 - \alpha - \beta E_{\nu} = 0 \] |
[
The term
| Latex |
|---|
\[ a The terma_{\nu}^*is the occupation number that maximizes the expressione^{- \] |
is the occupation number that maximizes the expressione
| Latex |
|---|
\[ \alpha '} \cdot e^{-\beta E_{\nu}} \] |
, where
| Latex |
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\[ where\alpha ' = \alpha + 1. use\] |
. Use constraints
...
to
...
determine
...
the
...
Lagrange multipliers
...
and
...
determine
...
the
...
probability.
| Latex |
|---|
\[ a_{\nu}=e^{-\alpha '} \cdot e^{-\beta E_{\nu} \] |
| Latex |
|---|
\[ \sum_{\nu} a_{\nu} = A \] |
| Latex |
|---|
\[ \sum_{\nu} e^{-\alpha '} \cdot e^{-\beta E_{\nu} = A \] |
| Latex |
|---|
\[ e^{\alpha '} = \frac{1}{A} \sum_{\nu} e^{-\beta E_{\nu} The probability of being in a certain state\nucan be calculated, and it is still in terms of the second Legrange multiplier. Plugging back into Equation 1: } \] |
The probability of being in a certain state
| Latex |
|---|
\[ \nu \] |
can be calculated, and it is still in terms of the second Lagrange multiplier. Plugging back into Equation 1:
| Latex |
|---|
\[ P_v = \frac{a_{\nu}^*}{A} = \frac{A}{\sum_{\nu} e^{-\beta E_{\nu}}} \cdot \frac{e^e{-\beta E_{\nu}}}{A} = \frac{e^{-\beta E_{\nu}}}{\sum_{\nu} e^{-\beta E_{\nu}} \] |
Partition Function
The denominator is the partition function,
| Latex |
|---|
\[ h2. Partition Function The denominator is the partition function,Q = \sum_{\nu} e^{-\beta E_{\nu} \] |
.
...
It
...
tells
...
us
...
how
...
many
...
states
...
are
...
accessible
...
by
...
the
...
system. Determine
| Latex |
|---|
\[ Determine\beta \] |
,
...
a
...
measure
...
of
...
thermally
...
accessible
...
states
...
.
...
Look
...
at
...
how
...
the
...
partition
...
function
...
connects
...
to
...
macroscopic
...
thermodynamic
...
variables. Find
| Latex |
|---|
\[ \beta \] |
and find
| Latex |
|---|
\[ \overline{E} \] |
| Latex |
|---|
\[ Find\betaand find\overline{E} \overline{E} = \sum_{\nu} P_{\nu} E_{\nu} \overline{E} = \frac{\sum_{\nu} E_{\nu} e^{-\beta E_{\nu}}}{Q} \] |
Consider the average pressure. The pressure for one microstate is
| Latex |
|---|
\[ p Consider the average pressure. The pressure for one microstate isp_{\nu}. \] |
.
| Latex |
|---|
\[ p_{\nu} = \frac{-\partial E_{\nu}}{\partial V} \] |
| Latex |
|---|
\[ p_{\nu} = \sum_{\nu} P_{\nu} p_{\nu} \] |
| Latex |
|---|
\[ p_{\nu} = \frac{ -\sum_{\nu} \left ( \frac{\partial E_{\nu}}{\partial V} \right ) e^{-\beta E_{\nu}}}{Q} \] |
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 |
|---|
\[ w h2. Summary In the case of a canonical ensemble, the energy of the entire ensemble is fixed. Each state is equally probable, and there is degeneracy. The probability is a function of how many ways to get the distribution. The distribution with the most permutation is the most probable. The graph can become very peaked. Once the distribution is known, do a maximization ofw(\overline{a}) \] |
.
...
Use
...
Lagrange multipliers
...
and
...
two
...
constraints. The term
| Latex |
|---|
\[ a The terma_{\nu}^* \] |
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 |
|---|
\[ \beta \] |
is a measure of how many states are thermally accessible.
| Latex |
|---|
\[ The term\betais a measure of how many states are thermally accessible. \overline{a} * \] |
Consider
...
every
...
possible distribution
| Latex |
|---|
distribution\{[ a_i \} = \overline{a} \] |
(consistent
...
with
...
boundary
...
conditions)
...
- For
...
- each
...
- distribution,
...
- consider
...
- every
...
- possible
...
- permutation
...
- The
...
- number
...
- of
...
- ways
...
- to obtain
isLatex \[ \overline{a} \]
Latex \[ \omega (\overline{a}) = \frac{ A! }{ a_1! a_2! a_3! ..... a_v! } = \frac{ A! }{ \Pi_v a_v!}
...
, where\]
is the number of systems in stateLatex \[ a_v \]
.Latex \[ v \]
Probability
| Latex |
|---|
\[ Pv</math> is the number of systems in state <math>v</math>.<br>Probability<math>P_v = \frac{\overline{a_v}}{A} = \frac{1}{A}\frac{ \sum_{\overline a} \omega (\overline{a}) a_v (\overline{a} ) }{ \sum_{overline a}}\omega (\overline{a}) } \] |
Averaging
| Latex |
|---|
\[ a_v \] |
over all possible distributions.
| Latex |
|---|
\[ </math>Averaging <math>a_v</math> over all possible distributions.<br><math> \w(\overline a) \] |
| Latex |
|---|
</math><math> \[ w(\overline a) \] |
is very peaked around a specific distribution
Increase
| Latex |
|---|
\[ A \] |
and
| Latex |
|---|
\[ </math> is very peaked around a specific distributionIncrease <math>A</math> and <math>\omega{\overline a}</math> becomes more peaked<br><math>a_v^*</math>To get <math>a_v^*</math>, maximize<math> \] |
becomes more peaked
| Latex |
|---|
\[ a_v \] |
To get
| Latex |
|---|
\[a_v \] |
, maximize
| Latex |
|---|
\[ w (\overline a)</math> subject to constraints. *Find the partition function, <math>Q</math> \] |
subject to constraints.
Find the partition function,
| Latex |
|---|
\[ Q \] |