...
Three
...
to
...
four
...
weeks
...
of
...
statistical
...
mechanics
...
lectures
...
are
...
planned.
...
Today
...
there
...
is
...
a
...
motivation
...
and
...
an
...
introduction.
...
There
...
is
...
a
...
book
...
that
...
can
...
serve
...
as
...
a
...
resource.
...
The
...
author
...
is
...
McQuarrie
...
and
...
studying
...
closely
...
Chapters
...
1-6
...
and
...
11
...
is
...
recommended.
Motivation
Statistical mechanics connects the microscopic to the macroscopic. It serves as a bridge between the two. Thermodynamics is blind to the microscopic world. For example, there are equations of state that are independent of atomistic interpretations.
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h1. Motivation Statistical mechanics connects the microscopic to the macroscopic. It serves as a bridge between the two. [Thermodynamics|http://en.wikipedia.org/wiki/Thermodynamics] is blind to the microscopic world. For example, there are equations of state that are independent of atomistic interpretations. {latex}\[ \left ( \frac {\partial E} {\partial V} \right )_{N,T}-T\left ( \frac{\partial P}{\partial T}\right )_{N,V}=-P\]{latex} |
Thermodynamics
...
does
...
provide
...
mathematical
...
relations
...
that
...
are
...
very
...
useful,
...
but
...
they
...
need
...
experimental
...
input.
...
Once
...
we
...
know
...
the
...
equation
...
of
...
state,
...
we
...
can
...
get
...
much
...
information,
...
but
...
thermodynamics
...
does
...
not
...
provide
...
microscopic
...
intuition.
...
Relations
...
in
...
thermodynamics
...
give
...
a
...
little
...
feeling
...
but
...
are
...
not
...
as
...
intuitive
...
as
...
in
...
statistical
...
mechanics.
...
There
...
is
...
a
...
need
...
for
...
experimental
...
input
...
when
...
using
...
thermodynamics.
...
In
...
contrast,
...
experiments
...
aren't
...
needed
...
when
...
applying
...
statistical
...
mechanics.
...
Assumptions
...
about
...
the
...
system
...
are
...
made
...
and
...
information
...
is
...
plugged
...
into
...
the
...
Schrodinger
...
equation.
Summary
Advantages of thermodynamics
- Provides mathematical relations with experimental input
- independent of atomistic interpretations
Drawbacks of thermodynamics
- Need experimental input
- No physical intuition
Statistical mechanics
Statistical mechanics starts from microscopic world. Assumptions are made about particles, and the Schrodinger equation is applied. A solid contains
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*Summary* Advantages of thermodynamics * Provides mathematical relations with experimental input * independent of atomistic interpretations Drawbacks of thermodynamics * Need experimental input * No physical intuition h2. Statistical mechanics [Statistical mechanics|http://en.wikipedia.org/wiki/Statistical_mechanics] starts from microscopic world. Assumptions are made about particles, and the Schrodinger equation is applied. A solid contains {latex} \[ 10^{23} \]{latex} |
vibrating
...
atoms.
...
Assumptions
...
are
...
made
...
about
...
the
...
interactions
...
between
...
particls
...
and
...
the
...
heat
...
capacity
...
is
...
derived.
...
Statistical
...
mechanics
...
starts
...
with
...
quantum
...
mechanics,
...
which
...
is
...
plugged
...
into
...
the
...
framework
...
of
...
statistical
...
mechanics
...
and
...
connected
...
to
...
thermodynamics.
...
Statistical
...
mechanics
...
provides
...
an
...
intuition
...
about
...
the
...
system,
...
and
...
it
...
is
...
possible
...
to
...
get
...
properties
...
that
...
may
...
not
...
have
...
been
...
as
...
intuitive.
...
It
...
is
...
possible
...
to
...
use
...
statistical
...
mechanics
...
without
...
experimental
...
input.
...
The
...
goal
...
of
...
statistical
...
mechanics
...
is
...
to
...
understand
...
and
...
predict
...
macroscopic
...
phenomena
...
from
...
microscopic
...
interactions.
...
This
...
provides
...
a
...
more
...
intuitive,
...
mechanistic
...
understanding
...
of
...
thermodynamic
...
quantities.
...
First-principles
...
calculations
...
are
...
possible.
...
These
...
calculations,
...
called
...
ab
...
initio
...
calculations
...
employ
...
the
...
Schrodinger
...
equation
...
to
...
get
...
properties.
...
No
...
experimental
...
input
...
is
...
needed
...
and
...
parameters
...
are
...
not
...
fitted
...
to
...
the
...
solid.
...
Probability
...
theory
...
is
...
used
...
in
...
statistical
...
mechanics.
...
Examples
An example of improving mechanistic understanding is a better physical interpretation of entropy. Entropy is a state function, is additive, and its maximum defines equilibrium, but what does it mean physically? The concept of entropy was introduced, but we hope to gain an intuitive understanding through statistical mechanics. Satistical mechanics says that entropy is a measure of disorder. Other examples of improved physical intuition include the meaning of the second and third law.
There are other examples of fruit generated by statistical mechanics. It is possible to derive the heat capacity by the way atoms interact. The form of the heat capacity as temperature goes to zero and its asymptotic nature is derived. This is not possible with classical mechanics.
Another example includes derived the typical topology of a phase diagram. Topology refers to the material being liquid at high temperature, a disordered solid at moderate temperature, and ordered at low temperature. Entropy, heat capacity, and phase diagrams are all explained more intuitivelly and more mechanistically.
The ideal gas equation of state is derived from first principles caculations. Particles in a gas are assumed not to interact. It is possible to get explicit expressions for chemical potential depending on assumptions of interactions. A limitation of first-principles calculations is that the Schrodinger equation must be applied to more complicated systems.
A more complicated example of first-principles calculations is research of the lecturer. The lattice dynamics and thermodynamic properties of the \beta-Sn phase were calculated. Every atom can vibrate, and there need to be assumptions about the vibrations. With an interaction model, it is possible to derive thermodynamic properties. One assumption is that they vibrate independently. With correct assumptions, the experiments and calculations can match perfectly. The phase diagram can be derived, and it is possible to get information about entropy and volume.
Consider the different degrees of freedom in a system of CdMg. The configurational degrees of freedom are related to the ways to place atoms. This is the largest contribution. From this it is possible to derive the correct order in phase diagrams. Consider other degrees of freedom which can be decoupled from each other.
Summary
- Statistical mechanics starts from microscopic world
- Solid
Latex \[ N \approx 10^{23} \]
...
vibrating
...
- atoms
...
- -->
...
- interactions
...
- -->
...
- derive
...
- heat
...
- capacity
...
Has
...
the
...
goal
...
of
...
understanding
...
and
...
predicting
...
macroscopic
...
thermodynamic
...
phenomena
...
from
...
microscopic
...
interactions
...
- more
...
- intuitive
...
- mechanistics
...
- understanding
...
- enable
...
- first
...
- principle
...
- predictions
...
- (no
...
- experimental
...
- input)
...
Typical
...
topology
...
of
...
a
...
phase
...
diagram:
...
we
...
are
...
going
...
to
...
understand
...
through
...
entropy
...
why
...
the
...
phase
...
diagram
...
looks
...
the
...
way
...
it
...
does
...
- Liq
...
- at
...
- high
...
- T,
...
- disordered
...
- solution
...
- at
...
- intermediate
...
- T,
...
- ordred
...
- at
...
- low
...
- T
...
Deriving
...
the
...
...
...
- assuming no interactions
- chemical potential with explicit form for
Latex \[ \mu_0 \]
...
Silicon
...
vibrational
...
properties
...
from
...
interaction
...
model
...
+
...
stat
...
mech
...
+
...
thermo
...
Introduction
...
to
...
Statistical
...
Mechanics
...
We
...
are
...
going
...
to
...
use
...
...
...
.
...
Think
...
now
...
from
...
a
...
classical
...
point
...
of
...
view.
...
The
...
properties
...
and
...
equation
...
of
...
state
...
are
...
from
...
V,
...
N,and
...
T.
...
It
...
is
...
macroscopically
...
simple.
...
But
...
microscopically
...
we
...
are
...
dealing
...
with
| Latex |
|---|
}\[10^{23}\]{latex} |
particles
...
that
...
move
...
with
...
some
...
velocity.
...
Each
...
particle
...
moves
...
with
...
some
...
velocity,
...
and
...
there
...
are
...
position
...
and
...
momentum
...
vectors
...
associated
...
with
...
each
...
particle.
| Latex |
|---|
}\[ \overline{r}_i = (r_{xi},r_{yi},r_{zi}) \]{latex} {latex} |
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|---|
{html}<p>{html} |
| Latex |
|---|
\[ \overline{p}_i = (p_{xi},p_{yi},p_{zi}) \] |
...
The
...
vectors
...
are
...
of
...
dimension
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|---|
}\[ 3N \]{latex} |
,
...
and
...
there
...
is
...
enormous
...
complexity.
...
The
...
time
...
dependence
...
of
...
these
...
vectors
...
are
...
given
...
by
...
Newton's
...
equation
...
of
...
motion
...
for
...
some
...
given
...
boundary
...
condition.
...
One
...
condition
...
is
...
the
...
energy.
| Latex |
|---|
}\[ E = E_k + E_{pot} \]{latex} |
Quantum
...
mechanically,
...
everything
...
is
...
gained
...
from
...
the
...
wave
...
function,
...
including
...
the
...
state
...
of
...
the
...
system.
...
The
...
wavefunction
...
is
...
below
| Latex |
|---|
}\[ \Psi ( \overline{q} ,t ) \overline{q} \approx ( \overline{r} ,\sigma ) \]{latex} |
Consider
...
the
...
many
...
body
...
Schrodinger
...
equation.
...
Assume
...
stationary
...
conditions
...
and
...
solve
...
the
...
eigenvalue
...
problem
...
to
...
find
...
eigenvalues
...
and
...
eigenstates.
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|---|
} \[ \hat H \Psi = i \hbar \frac {d \Psi} {dt} \hat H \Psi_v = E_v \Psi_v \] {latex} |
There
...
are
...
rapid
...
fluctuations
...
between
...
states.
...
Classically
...
and
...
quantum
...
mechanically,
...
there
...
are
...
a
...
huge
...
number
...
of
...
degrees
...
of
...
freedom,
...
and
...
the
...
first
...
postulate
...
resolves
...
this
...
problem.
...
Summary
With a gas,
...
we
...
get
...
the
...
thermodynamics
...
properties
...
from
...
| Latex |
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\[equations of state|http://en.wikipedia.org/wiki/Equations_of_state](V, N, T) In a classical description each particle has a position {latex} \] |
In a classical description each particle has a position
| Latex |
|---|
\[ \overline{r}_i = (r_{xi},r_{yi},r_{zi}) \] {latex} |
and
...
momentum
| Latex |
|---|
} \[ \overline{p}_i = (p_{xi},p_{yi},p_{zi}) \] |
- Two vectors with dimension 3N
The time depenence of these vectors are given by Newton's equation of motion, for certain boundary condition
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\[ {latex} * Two vectors with dimension 3N The time depenence of these vectors are given by Newton's equation of motion, for certain boundary condition E = E_k + E\_{pot}(?) Quantum \] |
Quantum mechanically,
...
the
...
state
...
of
...
the
...
system
...
is given by
| Latex |
|---|
\[ given by\Psi(\overline {q},t) \] |
where
| Latex |
|---|
\[ \overline{q} ~\approx (\overline {r},\sigma} \] |
from
| Latex |
|---|
\[ \ from\hat H \Psi = i \hbar \frac{d \Psi}}{dt} < \] |
, (
| Latex |
|---|
\[ \hat H \] |
is Hamiltonian) Stationary conditions lead to
| Latex |
|---|
\[ /math>, (<math>\hat H</math> is Hamiltonian)Stationary conditions lead to <math>\hat H \Psi_v = E_v \Psi_v</math>The grand idea behind statistical mechanicsObservation time over which therodynamic quantities are mesured is very large compared to the time scale of molecular activity. Consider the momentum evolution as a function of time. The system does fluctuate but is near equilibrium. The average is a good description. The fluctuations are not seen at time scale <math>\Delta t</math>. Over <math>\Delta t</math> the particle has accessed many states. Much of phase space has been covered, and value of momentum is close to some average number.Thermodynamic properties are time averages of microscopic counterparts. There is a crucial connection between microscopic states and thermodynamics. Consider the thermodynamic energy, <math>U</math>. Write the energy explicitly through a classical description and quantum description.<center><br><math>U = \overline v \] |
The grand idea behind statistical mechanics
Observation time over which therodynamic quantities are mesured is very large compared to the time scale of molecular activity. Consider the momentum evolution as a function of time. The system does fluctuate but is near equilibrium. The average is a good description. The fluctuations are not seen at time scale
| Latex |
|---|
\[ \Delta t \] |
. Over
| Latex |
|---|
\[ \Delta t \] |
the particle has accessed many states. Much of phase space has been covered, and value of momentum is close to some average number.
Thermodynamic properties are time averages of microscopic counterparts. There is a crucial connection between microscopic states and thermodynamics. Consider the thermodynamic energy,
| Latex |
|---|
\[ U \] |
. Write the energy explicitly through a classical description and quantum description.
| Latex |
|---|
\[ >U = \overline{E} = \frac{1}{\Delta t} \int_{\Delta t} E(v(t),p(t))dt \] |
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|---|
{html}<p>{html} |
| Latex |
|---|
\[ U = \overline{E} |
...
= \frac |
...
{1}{\Delta t} \int_{\Delta t} \langle \Psi (\overline{q},t) \mid \hat H \mid \Psi(\overline{q},t) \rangle dt \] |
It is not easy to calculate the time dependence and practically impossible to calculate the time evolution of a quatum mechanical system or an N-body system. The major postulate of quantum mechanics connects time averages with something we can compute.
The time average is equal to the weighted average over all possible states a system can lie in for a given boundary condition. The average energy can be written as a weighted average over states. Associate each energy with a probability that the system is at that energy.
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\[ It is not easy to calculate the time dependence and practically impossible to calculate the time evolution of a quatum mechanical system or an _N_-body system. The major postulate of quantum mechanics connects time averages with something we can compute. The time average is equal to the weighted average over all possible states a system can lie in for a given boundary condition. The average energy can be written as a weighted average over states. Associate each energy with a probability that the system is at that energy. \overline{E} = \overline{E} = \langle E(t) \rangle = \int E(\vec r, \vec p)P(\vec r, \vec p)d\vec r d\vec p d\vec r d\vec p \] |
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|---|
{html}<p>{html} |
| Latex |
|---|
\[ P(\vec r, \vec p) = \mbox{probability density} \] |
The probability density is still a hard function of position and momentum
Summary
Observation time over which therodynamic quantities are mesured is very large compared to the time scale of molecular activity
- we take the average over time
- the crucial connection between micorscopic states and thermo
Thermodynamic properties are time averages of their microscopic counterparts
Classical description:
| Latex |
|---|
\[ The probability density is still a hard function of position and momentum *Summary* Observation time over which therodynamic quantities are mesured is very large compared to the time scale of molecular activity * we take the average over time * the crucial connection between micorscopic states and thermo Thermodynamic properties are time averages of their microscopic counterparts Classical description: U = \overline{E} = \frac{1} {\Delta t} \int_{\Delta t}} E(v(t),p(t))dtQuantum description: dt \] |
Quantum description:
| Latex |
|---|
\[ U = \overline{E} = \frac{1}{\Delta t} } \int\_ {\Delta t} \langle \Psi (\overline {q},t) \mid \hat H \mid \Psi(\overline{q} ,t) \rangle dt \] |
It
...
is
...
practically
...
impossible
...
to
...
calculate
...
the
...
...
...
of
...
an
...
N-body
...
system
...
!
...
Time
...
average
...
=
...
weighted
...
average
...
over
...
all
...
possible
...
states
...
that
...
a
...
system
...
can
...
be
...
in
...
(for
...
a
...
given
...
boundary
...
condition)
...
Classical:
| Latex |
|---|
\[ \overline {E}= \langle E(t) \rangle = \int E(\vec r, \vec p)P(\vec r, \vec p)d\vec r d\vec p \] |
where
| Latex |
|---|
\[ P whereP(\vec r, \vec p) \] |
is
...
the
...
probability
...
density.
Major
...
postulate
...
of
...
statistical mechanics
Sum over all states.
...
Multiply
...
by
...
the
...
energy
...
of
...
a
...
state
...
and
...
probability
...
of
...
being
...
in
...
that
...
state.
...
Sum
...
over
...
all
...
states
...
to
...
calculate
...
the
...
expected
...
value
...
of
...
energy.
...
This
...
is
...
done
...
instead
...
of
...
solving
...
Newton's
...
equation
...
of
...
motion
...
or
...
time
...
averaging. Given
| Latex |
|---|
\[ G GivenG_{\nu} \] |
we
...
can
...
calculate
...
everything.
| Latex |
|---|
\[ \overline{E} = \langle E(t) \rangle = \sum_v E_v P_v \] |
Go
...
after
...
the
...
probability
...
function.
...
Plug
...
in,
...
and
...
get
...
average
...
quantities.
...
Get
...
values
...
from
...
first
...
principles.
Summary
| Latex |
|---|
\[ *Summary* \overline {E}= \langle E(t) \rangle = \sum_v E_v P_vWithP_v(the probability v \] |
With
| Latex |
|---|
\[ P_v \] |
(the probability function),
...
we
...
can
...
calculate
...
everthing!
| Latex |
|---|
\[ \overline{E} </math> \] |
can be plugged into thermodynamic equations.
Simple Example
Calculate the average number of students asleep given it is Wednesday morning and a particular lecturer is teaching. This is a function of time. Consider the state of each student. The probability is given below in a time dependent equation and in an equation given by approximating with an average quantity,
| Latex |
|---|
\[ P can be plugged into thermodynamic equations.Simple ExampleCalculate the average number of students asleep given it is Wednesday morning and a particular lecturer is teaching. This is a function of time. Consider the state of each student. The probability is given below in a time dependent equation and in an equation given by approximating with an average quantity, <math>P_{\mbox{sleep}}</math>, which is the average probability of falling asleep at any time.<center><br><math>\overline \] |
, which is the average probability of falling asleep at any time.
| Latex |
|---|
\[ \overline{N}_{\mbox{students asleep}} = \frac{1}{90} \int_{0}^{90} n_{sleep} (t) dt \] |
| Latex |
|---|
\[ \overline{N} \_{\mbox{students asleep}} = \int\_ {0}^{50} P_{sleep} dn \] |
| Latex |
|---|
\[ P_{sleep} dn \overline{N}_{\mbox{students asleep}} = \int_{0} ^ {50} 0.1 dn \] |
| Latex |
|---|
\[ \overline{N}_{\mbox{students asleep}} = 5 \] |
In statistical mechanics, there is a gamble that there is a good average number. In a wildly flutuating system, such as with a change in teacher, the boundary conditions change, and the probability would also change. Stationary conditions are assumed, and
| Latex |
|---|
\[ P In statistical mechanics, there is a gamble that there is a good average number. In a wildly flutuating system, such as with a change in teacher, the boundary conditions change, and the probability would also change. Stationary conditions are assumed, andP_{\mbox{sleep}} \] |
is
...
a
...
very
...
simple
...
example.
...
It
...
is
...
possible
...
to
...
take
...
a
...
snapshot
...
and
...
work
...
backwards
...
to
...
estimate
...
probability.
...
Summary
Let's
...
say
...
we
...
want
...
to
...
calculate
...
average
...
number
...
of
...
students
...
asleep,
...
given
...
the
...
boundary
...
condition
...
that
...
it
...
is
...
a
...
90
...
minute
...
class.
| Latex |
|---|
\[ \overline{N}_{\mbox{students asleep}} = \frac{1}{90} \int_{0}^{90} n_{sleep} (t) dt = \int_{0}^{50} P\_ {sleep} dn = \int_{0}^{50} 0.1 dn = 5 \] |
where
| Latex |
|---|
\[ P whereP_{sleep} \] |
is the average probability for being in a sleep stateClosing RemarksThere are two ways to remove time dependence that are not connected.
A review of quantum mechanics is next. Slides were shown in lecture, and below are a few remarksMechanical system is determined by a wavefunction
| Latex |
|---|
\[ </math> is the average probability for being in a sleep stateClosing RemarksThere are two ways to remove time dependence that are not connected.<p> </p>A review of quantum mechanics is next. Slides were shown in lecture, and below are a few remarksMechanical system is determined by a wavefunction <math>\Psi (\overline {q} , t)</math>Energy takes on discret valuesDegeneracy is common. There are many <math>\Psi</math> with same <math>E</math> \*Calculations are much easier in many-body problems when the Hamiltonian is summed up and some interactions are assumed to be de-coupled. When decoupled, the energy becomes a sum over particle energies. \] |
. Energy takes on discret values
Degeneracy is common. There are many
| Latex |
|---|
\[ \Psi \] |
with same
| Latex |
|---|
\[ E \] |
- Calculations are much easier in many-body problems when the Hamiltonian is summed up and some interactions are assumed to be de-coupled. When decoupled, the energy becomes a sum over particle energies.