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PROFESSOR: This is 8.333
Statistical Mechanics.
And I'll start by telling you
a little bit about the syllabus
before going through the
structure of the course.
So what I have written
here is a, kind of,
rough definition of statistical
mechanics from my perspective.
And the syllabus
is a guide to how
we are going to
approach this object.
So let's take a look at
the syllabus over there.
So the first thing that
you're going to focus on
is, what is it that
you're trying to describe?
And these are the
equilibrium properties
that are described
best through what
I've been doing
section one, which
has to do with thermodynamics.
You'll start today
with thermodynamics.
Basically, it's a
phenomenological approach,
so you essentially
look at something,
as kind of a black
box, without knowing
what the ingredients
are, and try
to give some kind of
description of how it's
function and properties change.
And these can be captured, for
the case of thermal properties
of matter through the laws of
thermodynamics, which we will
set out in this first section,
which will roughly take us
the first four
lectures of the course.
Then I said that
statistical mechanics
is a probabilistic approach.
So we need to establish what
the language of probability is.
And that can be the
topic for the second two
and half lectures of the course.
It is something that is less
physics-y, but since the topic
itself has to deal
with probabilities,
it is very important,
from my perspective,
to set out the language and
the properties of systems
that are described
probabilistically.
Separately, we'll devote a
couple of lectures to do so.
And in particular,
it is very important
that the laws of
probability, kind of,
simplify when you're
dealing with a large number
of variables, as captured,
for example, by what we
call the central limit theorem.
So you can see that the
third element of this course,
which is the law
of large numbers,
is inherent also to what
simplification you will
see in the section
on probability.
And feeds back
very much into how
statistical mechanics
is developed.
But then we said large
number of degrees of freedom.
So what are these
degrees of freedom?
Well, this is now taking
a different perspective.
For us in
thermodynamics, when you
look at the system
as a black box,
and try to develop laws
based on observations,
we say that well, from the
perspective of physics,
we know that this box
contains atoms and molecules.
And these atoms
and molecules are
following very specific laws,
either from Newtonian mechanics
or quantum mechanics.
And so if we know everything
about how atoms and molecules
behave, then we should
be able to derive
how large collections
of them behave.
And get the laws
of thermodynamics
as a consequence of these
microscopic degrees of freedom
and their dynamics.
And so that's what
we will discuss
in the third part
of the course that
is devoted to kinetic theory.
We will see that even at that
stage, it is beneficial to,
rather than follow individual
particles in a system,
to adopt a
probabilistic approach,
and think about densities,
and how those densities evolve
according to
Liouville's Theorem.
And what we will try
to also establish
is a very distinct
difference that
exists between
thermodynamics, and where
things are
irreversible and going
one direction, and Newtonian,
or quantum mechanics,
where things are
reversible in time.
And we'll see that really
it's a matter of adapting
the right perspective in
order to see that these two
ways of looking
at the same system
are not in contradiction.
So having established
these elements,
we will then finally
be in the place
where we can discuss
statistical mechanics in terms
of some postulates about
how probabilities behave
for systems that
are in equilibrium.
And how based on
those postulates,
we can then derive all
the laws of thermodynamics
and all the properties of
thermodynamics systems.
That they're ordained
while observations
and phenomenological
theories before.
Now initially, in
section four, we
will do that in the context of
classical systems-- description
of particles following
classical laws of motion.
And, again, as a
first simplification,
we will typically deal with
non-interacting systems,
such as ideal gas.
And make sure that we
understand the properties
of this important
fundamental system
from all possible perspectives.
Then in section five, we will
go on to more realistic systems
where there are interactions
among these particles.
And there are two ways to
then deal with interactions.
You can either go by the
way of perturbation theory.
We can start with ideal
system and add a little bit
of interaction, and see
how that changes things.
And we develop some elements of
graphical perturbation theories
in this context.
Or, you can take
another perspective,
and say that because of the
presence of interactions,
the system really adopts
a totally different type
of behavior.
And there's a perspective
known as mean field
theory that allows
you to do that.
Then see how the
same system can be
present in different
phases of matter,
such as liquids and gas,
and how this mean field
type of prescription
allows you to discuss
the transitions between the
different types of behavior.
Eventually, you will go on,
towards the last quarter
of the course, rather than the
classical description of matter
to a quantum description of the
microscopic degrees of freedom.
And we will see how the
differences and similarities
between quantum
statistical mechanics,
classical statistical
mechanics, emerge.
And just, historically,
of course,
radius macroscopic properties
of the matter, the black body
laws, or heat
capacities, have been
very important in
showing the limitations
of classical description
of matter, and the need
to have something else, such
as the quantum description.
We will not spend too much
time, more than three lectures,
on the sort of principles of
quantum statistical mechanics.
The place where quantum
statistical mechanics shows
its power is in dealing with
identical particles, which
classically, really
are kind of not
a very well-defined concept,
but quantum-mechanically, they
are very precisely defined,
what identical particles mean.
And there are two classes--
fermions and bosons--
and how even if there's no
interaction between them,
quantum statistics leads to
unusual behavior for quantum
systems of identical particles,
very distinct for fermions
and for bosons.
So that's a rough syllables of
how the course will be arranged
over the next 25, 26 lectures.
Any questions about what
we're going to cover?
OK, then let's go here.
So I will be
teaching the course.
My research is in
condensed matter theory
and statistical physics.
So this is a subject
that I like very much.
And I hope to impart
some of that love
of statistical physics to you.
And why it is an
interesting topic.
Lectures and recitations
will be conducted
in this room, Monday,
Wednesday, Friday.
And you ask, well, what does
it mean that both lectures
and recitations are here?
Well, for that you will have
to consult the timetable.
And that's probably
the most important part
of this web page.
And it tells you, for
example, that the first five
events for the
course are all going
to be lectures Monday,
Wednesday, Friday of next week.
And the first recitation will
come up on Monday, September
the 16th.
And the reason for that
is that the first problem
set will be due on the 18th.
And I will arrange for you to
have six recitations on the six
events that are before the due
dates of those problem sets.
Also indicated here is
the due dates, naturally,
of the problem sets.
And although I had indicated
that this will be handed out
tomorrow, the first
problem set is already
available on the
web, so you can start
going to take a look at that.
And eventually, also
on the web page,
will be posted the solutions.
And the first one
will be posted here.
Of course, it's
not available yet.
Surprisingly.
Once the due date has passed
on the date that is indicated,
the solutions will be posted.
Also indicated here is that
there will be various tests.
The first test will
fall on October 2.
And another time
and recitations will
take place are
prior to the tests.
So there is actually
three tests,
and there will be
three citations
that will take
place before that.
And, ultimately, at the end,
there will be a final exam.
It's date I don't know yet.
So I just randomly put it
on the Monday of the week
where the final
exams will be held.
And once the actual date is
announced by the registrar,
I will make sure that
I put the correct date
and place in this place.
OK, so that's the arrangement
of the various lectures
and recitations.
In addition to me,
the teaching staff
consists of Max [? Imachaov ?]
and Anton Goloborodko.
Anton is here, sitting
at that extreme corner.
Max, I believe, is now in Paris.
OK, both of them work
on biological systems
that use a lot of statistical
physics content to them.
So maybe they will tell you
some interesting problems
related to that in
the recitations.
At this point in time, they
have both set their office hours
to be Thursday, 3:00 to
5:00, in their lab, which
is close to where the
medical facilities are.
If you find that inconvenient,
you could potentially
change that, or you
could get in touch
with either the
TAs, or myself, when
you want to have
specific time to meet us.
Otherwise, I have indicated
my own availability
to be the half hours
typically after lectures
on Monday, Wednesdays,
and Fridays.
One other set of
important things to note
is how the course is organized.
So I already mentioned to you
what the syllabus of the course
is.
I indicated where and when
the lectures and recitations
are going to take place.
This is the web page that I
have been surfing through.
And all of the material will
be posted through the web page,
so that's where you have to go
for problem sets, solutions,
et cetera.
Also, grades.
And, in particular,
I have my own system
of posting the
grades, for which I
need a pseudonym
from each one of you.
So if you could all go
through this checking online,
indicate your name,
your email address,
and choose a pseudonym,
which I emphasize
has to be different
from your real name.
And if it is your real name,
I have to randomly come up
with something like "forgot to
put pseudonym" or something.
I cannot have your real
name followed by the grades.
OK?
I'll discuss anonymous
comments, et cetera, later on.
As you will see,
I will hand out,
through the web page, extensive
lecture notes covering
all of the material
that I talk about.
So in principle, you
don't need any textbooks.
You can refer to the notes and
what you write in the lectures.
But certainly, I--
some people like
to have a book sitting
on their bookshelf.
So I have indicated
a set of books
that you can put
on your bookshelf.
And hopefully consult
for various topics
at different stages.
And I will, through
the problem sets,
indicate what are
good useful chapters
or parts of these books
to take a look at.
Now how is the grade for
the course constructed?
An important part of it is
through this six problem sets
that we mentioned.
So each one of them
will count for 5%,
for a total of 30% going towards
the contribution of the problem
sets.
I have no problem with
you forming study groups,
as long as each person, at the
end, writes their own solution.
And I know that if you look
at around sufficiently,
you can find solutions from
previous years, et cetera,
but you will really
be cheating yourself.
And I really bring
your attention
to the code of honor that
is part of the MIT integrity
handbook.
We have indicated,
through the schedule page,
the timeline for this
six problem sets are due.
They are typically
due at 5:00 PM
on the date that is
indicated on that page,
and also on the problem set.
And the physics department
will set up a Dropbox,
so you can put the
problem set there,
or you can bring it to
the lecture on the date
that it is due.
That's also acceptable.
There is a grey area of
about a day or so sometime,
between when the problem set is
due and when the solutions are
posted.
If problem sets are handed
in during that gray area,
they will count
towards final 50%,
rather than full
towards the grade.
Unless you sort of write
to me a good excuse
that I can give
you an extension.
Now every now and then,
people encounter difficulties,
some particular week you are
overwhelmed, or whatever,
and you can't do the
particular problem set,
and they ask me for an
excuse of some form.
And rather than doing that,
I have the following metric.
That is, each one
of these problem
sets you will find that
there's a set of problems
that are indicated as optional.
You can do those problems.
And they will be graded
like all the other problems.
And in case, at
some later time, you
didn't hand in some
problem set, or you
miss half of the
problem set, et cetera,
what you did on these
optional problems
can be used to make up your
grade, pushing it, eventually,
up to the 30% mark.
If you don't do any of
the optional problems,
you just do the
required problem,
you will correctly you
will reach the 30% mark.
If you do every single problem,
including optional ones,
you will not get more than
30%, so the 30% is upper-bound.
So there's that.
Then you have the
three tests that
will be taking place during
the lecture time, 2:30 to 4:00,
here.
Each one of them will count
15%, so that's another 45%.
And the remaining 25%
will be the final exam
that will be scheduled
in the finals week.
So basically, that's the way
that the grades are made up.
And the usual definition of
what grades mean-- typically,
we have been quite generous.
I have to also indicate
that things will not
be graded on a curve.
So that's a MIT policy.
And there are some
links here to places
that you can go to if you
encounter difficulties
during the semester.
So any questions about the
organization of the course?
OK, so let's see
what else we have.
OK, course outline and
schedule, we already discussed.
They're likely to be
something that are unexpected,
or some things that
have to be changed.
Every now and then
there is going
to be a hurricane almost with
probability, close to one.
We will have a hurricane
sometime during the next month
or so.
We may have to postpone
a particular lectures
accordingly.
And then the information about
that will be posted here.
Currently, the only
announcement is
what I had indicated
to you before.
Please check in
online indicating
that you are taking
this course, and what
your pseudonym is going to be.
OK?
I give you also the
option, I would certainly
welcome any questions that
you may have for me here.
Please feel free to interrupt.
But sometimes people, later
on, encounter questions.
And for whatever
reason, it may be
question related to the
material of the course.
It may be related to when
various things are due,
or it may be there is some wrong
notation in the problem set
or whatever, you can
certainly anonymously send
this information to me.
And I will try to respond.
And anonymous responses will be
posted and displayed web page
here.
Currently there is
none, of course.
And, finally,
something that-- OK,
so there's a web page where
the problems will be posted.
And I want to emphasize that
the web page where the solutions
are posted, you may see
that you cannot get to it.
And the reason would be that you
don't have an MIT certificate.
So MIT certificates are
necessary to reach the solution
page.
And also they are
necessary to reach
the page that is
devoted to tests.
And actually there is
something about the way
that I do these three in-class
tests that is polarizing.
And some people very
much dislike it.
But that's the way it
is, so let me tell you
how it's going to be conducted.
So you will have this one and
a half hour [INAUDIBLE] test.
And I can tell you, that
the problems from the test
will be out of this collection
that I already posted here.
So there is a
collection of problems
that is posted on this page.
And furthermore, the
solutions are posted.
So there's a version of
this that is with solution.
So the problems will be
taken from this, as well
as the problem sets that you
have already encountered--
[INAUDIBLE] solution is posted.
So if you are familiar
with this material,
it should be no problem.
And that's the way the first
three tests this will go.
The final will be, essentially,
a collection of new problems
that are variants of things
that you've seen, but will not
be identical to those.
OK, so where is my cursor?
And finally, as I
indicated, the grades
will be posted according
to your pseudonym.
So as time goes on, this
table will be completed.
And the only other thing to
note is that there's actually
going to be lecture notes
for the various materials,
starting from the
first lecture, that
will be devoted
to thermodynamics.
Any questions?
OK.
So let me copy that
first sentence,
and we will go on and
talk about thermodynamics.
The phenomenological description
of equilibrium properties
of microscopic systems.
And get rid of this.
One thing that I
should have emphasized
when I was doing the syllabus is
that I expect that most of you
have seen
thermodynamics, have done
in a certain amount of
statistical mechanics,
et cetera.
So the idea here
is really to bring
the diversity of
backgrounds that you have,
for our graduate students, and
also I know that our students
from other departments, more or
less in line with each other.
So a lot of these
things I expect
to be, kind of, review
materials, or things
that you have seen.
That's one reason that we kind
of go through them rapidly.
And hopefully,
however, there will
be some kind of logical
systematic way of thinking
about the entirety of them
that would be useful to you.
And in particular,
thermodynamics, you say,
is an old subject, and
if you're ultimately
going to derive the
laws of thermodynamics
from some more precise
microscopic description,
why should we go
through this exercise?
The reason is that
there is really
a beautiful example
of how you can
look at the system
as a black box,
and gradually, based
on observation,
build a consistent
mathematical framework
to describe its properties.
And it is useful
in various branches
of science and physics.
And kind of a more 20th century
example that I can think of
is, Landau's approach
to superconductivity
and superfluidity, where without
knowing the microscopic origin
of that, based on
kind of phenomenology,
you could write down
very precise description
of the kinds of things
that superconductors
and superfluids could manifest.
So let's sort of put
yourselves, put ourselves,
in the perspective of how
this science of thermodynamics
was developed.
And this is at the time
where Newtonian mechanics had
shown its power.
It can describe orbits of
things going around the sun,
and all kinds of other things.
But that description,
clearly, does not
apply to very simple things like
how you heat up a pan of water.
So there's some elements,
including thermal properties,
that are missing from
that description.
And you would like to
complete that theory,
or develop a theory, that
is able to describe also
heat and thermal properties.
So how do you go about that,
given that your perspective is
the Newtonian prescription?
So first thing to
sort of a, kind of,
parse among all of these
elements, is system.
So when describing
Newtonian mechanics,
you sort of idealize,
certainly you
realize that Newtonian mechanics
does not describe things
that we see in everyday world.
You kind of think
about point particle
and how a point particle
would move in free space.
And so let's try to do a similar
thing for our kinds of systems.
And the thing that is
causing us some difficulty
is this issue of heat.
And so what you can do
is you can potentially
isolate your system thermally
by, what I would call,
adiabatic laws.
Basically say that there's
these things, such as heat,
that goes into the system
that causes difficulty for me.
So let's imagine in
the same that I'm
thinking of the point
particle, that whatever
I have is isolated from
the rest of the universe,
in some kind of box that
does not allow heat transfer.
This is to be opposed with
walls that we would like
to eventually look at, which
do allow heat transfer.
Let me choose a different color.
Let's say green.
So ultimately, I want to
allow the exchange of whatever
this heat is in
thermal properties
to go and take place
with my system.
OK?
Now, this is
basically isolation.
The next element is to
wait for your system
to come to equilibrium.
be You realize that when
you, for example, start
with something
that is like this,
you change one of the walls
to allow heat to go into it.
Then the system
undergoes some changes.
Properties that you're
measuring are not
well-defined over some
period where these changes
taking place, but if
you wait sufficiently,
then they relax to
some new values.
And then you can start
making measurements.
So this is when
properties don't change.
And the key here is
observation time.
This is part of
the phenomenology,
because it is not precise.
I can't tell you how
long you have to wait.
It depends on the system
under consideration.
And some systems come
to equilibrium easily,
some take a long time.
So what are the properties
that you can measure?
Once things have settled down
and no longer change with time,
you can start to measure
various properties.
The ones that are very
easy for you to identify
are things that are associated
with mechanical work,
or mechanical properties.
And, for example, if you have
a box that contains a gas,
you can immediately see well,
what's the volume of the gas?
You can calculate
what pressure it
is exerting on its environment.
So this is for a gas.
You could have, for
example, a wire.
If you have a wire, you
could calculate, rather than
its volume, its
length and the force
with which you are pulling it.
It could be something
like a magnet.
And you could put some kind
of a magnetic field on it,
and figure out what
the magnetization is.
And this list of mechanical
properties goes on.
But you know that that's
not the end of the story.
You kind of expect that there
are additional things that
have to do with thermal
aspects that you
haven't taken into account.
And as of yet, you don't
quite know what they are.
And you have to gradually
build upon those properties.
So you have a system.
These are kind of the
analogs of the coordinates
and potentially
velocities that you
would have for any
Newtonian particles.
A way of describing
your idealized system.
And then you want to find rules
by which these coordinates are
coevolving, or doing
things together.
And so for that you
rely on observations,
and construct laws
of thermodynamics.
All right so this is
the general approach.
And once you follow
this, let's say,
what's the first thing
that you encounter?
You encounter what is
encoded to the zeroth law.
What's the zeroth law?
The zeroth law is the
following statement,
if two systems-- let's call
them A and B-- are separately
in equilibrium with C--
with a third system-- then
they are in equilibrium
with each other.
This is sort of this
statement that the property
of equilibrium has the
character of transitivity.
So what that means, pictorially,
is something like this.
Suppose I have my two boxes,
A and B. And the third system
that we're calling C.
And we've established
that A and B are separately
in thermal equilibrium with C,
which means that we have
allowed exchange of heat
to take place between A
and C, between B and C,
but currently, we
assume nothing--
or we assume that B and C are
not connected to each other.
The statement of
the law is that if I
were to replace this red with
green, so that I have also
exchange that is going
on between A and B,
then nothing happens.
A and B are already
in equilibrium,
and the fact that you open
the possibility of exchange
of heat between them
does not change things.
And again, this is this
consequence of transitivity.
And one of its kind of
important implications
ultimately is that
we are allowed now
based on this to add one more
coordinate to this description
that you have.
That coordinate is the
analog of temperature.
So this transitivity really
implies the existence
of some kind of
empirical temperature.
And you may say well, this
is such an obvious thing,
there should be transitivity.
Well, I want to
usually give people
examples that transitivity
is not a universal property,
by as follows.
Suppose that within
this room, there
is A who wants to
go on a date with C,
and B who wants to
go on a date with C.
I'm pretty sure that it's not
going to be the property that A
wants to go through the
date with B. It's 20%.
All right.
So let's see how
we can ensure this.
So we said that if some system
has reached equilibrium,
it has some set of coordinates.
Let's call them
A1, A2, et cetera.
There's some number of them.
We don't know.
Similarly, here, we
have C1, C2, et cetera.
And for B, we have B1, B2.
Now what does it mean that I
have equilibrium of A and B?
The implication is that if
I have a system by itself,
I have a bunch of
possible coordinates.
I can be anywhere in
this coordinate space.
And if I separately
have system C,
I have another bunch
of coordinates.
And I can be anywhere in
this coordinate space.
But if I force A and B to come
together and exchange heat
and reaches equilibrium,
that is a constraint,
which means that this set of
coordinates of the two cannot
be independently varied.
There has to be some functional
relationship between them,
which we can, for example,
cast into this form.
So equilibrium, one
constraint, one kind
of mathematical relation.
Similarly, equilibrium
of-- this was A
and C, B and C-- would
give us some other function
BC of coordinates of B and
coordinates of C equal to 0.
OK?
So there is one
thing that I can do.
I can take this expression
and recast it, and write it
as let's say, the first
coordinate that describes
C is some other function,
which I will call big F
AC of coordinates of
A. And all coordinates
of C, except the first
one that I have removed.
Yes?
AUDIENCE: When you
put that function F
AC or all the coordinates of
A, and all the coordinates of C
being zero, is that
[? polynomy ?] always going
to be true for the first law,
or do you just give an example?
PROFESSOR: It will
be always true
that I have some set
of coordinates for 1,
some set of coordinates
with 2, and equilibrium
for the two sets
of coordinates is
going to be expressible in
terms of some function that
could be arbitrarily
complicated.
It may be that I
can't even write this,
but I have to graph it,
or some kind of place
in the coordinates space.
So what I mean is
the following, you
can imagine some
higher dimensional
space that is spanned
by As and the Cs.
And each point, where the
things are in equilibrium,
will be some-- you can put a
cross in this coordinate space.
And you can span
the various places
where equilibration takes place.
And you will have a
surface in this space.
And that surface potential you
can describe mathematically
like this.
OK?
And similarly, I can
do the same thing.
And pick out coordinate
C1 here, and write it
as F BC of A1, A2, and C2,
C3-- sorry, this is B1, B2.
Actually, this brings me, this
question, to maybe one point
that I should make.
That sometimes during this
course, I will do things,
and maybe I will
even say that I do
things that are physically
rigorous, but maybe not
so mathematically.
And one example of this is
precisely this statement.
That is, if you
give a mathematician
and there is a
function like this.
And then you pick one,
C1, and you write it
as a function of
all the others, they
say, oh, how do know that
that's even possible, that this
exists?
And generally, it isn't.
A simple example would
be A squared plus C
squared equals to zero,
then C's multiple valued.
So the reason this
is physically correct
is because if we set up
this situation, and really
these very physical
quantities, I
know that I dialed my
C1 to this number here,
and all of the other
things adjusted.
So this is kind
of physically OK,
although mathematically, you
would have to potentially do
a lot of [INAUDIBLE] to
arrive at this stage.
OK.
So now if I have to put all
of those things together,
the fact that I have
equilibrium of A and B,
plus equilibrium
of B and C, then
implies that there
is eliminating
C1 between these
two, some function
that depends on the coordinates
of A and coordinates of C,
starting from the second one,
equal to some other function.
And these functions
could be very different,
coordinates of B
and coordinates of C
starting from the second one.
Yes?
AUDIENCE: Do you need A
and C on the left there?
PROFESSOR: A and C,
and B and C. Thank you.
OK?
So this, everything that
we have worked out here,
really concerns
putting the first part
of this equation in
mathematical form.
But this statement--
in mathematical form--
but given the first part
of this statement, that
is, the second part
of the statement, that
is that I know that if I remove
that red and make it green,
so that heat can exchange, for
the same values of A and B,
A and B are in equilibrium, so I
know that there is a functional
form.
So this is equilibrium
of A and B, implies
that there is this function
that we were looking at before,
that relates
coordinates of A and B
that constrains the
equilibrium that
should exist
between A and B. OK.
So the first part of the
statement of the zeroth law,
and the second part of the
statement of the zeroth law
can be written mathematically
in these two forms.
And the nice part
about the first part
is that it says that the
equilibrium constraint that I
have between A's and
B's can mathematically
be kind of spread out into some
function on the left that only
depends on the
coordinates of A, and some
function on the right that only
depends on coordinates of B.
So that's important,
because it says
that ultimately the
equilibration between two
objects can be
cast mathematically
as having some kind
of a function--
and we don't know anything about
the form of that function--
that only depends on
the coordinates of A.
And equilibration
implies that there exists
some other function
that only depends
on the coordinates
of the other one.
And in equilibrium, those
two functional forms
have to be the same.
Now, there's two ways of getting
to this statement from the two
things that I have
within up there.
One of them is to choose some
particular reference system
for C. Let's say
your C is seawater
at some particular
set of conditions.
And so then, these
are really constants
that are appropriate
to see water.
And then you have
chosen a function that
depends on variables of A, of
course of some function of B.
Or you can say,
well, I can replace
this seawater by something else.
And irrespective
of what I choose,
A and B, there by our definition
in equilibrium with each other,
no matter what I did with C. Or
some various range of C things
that I can do, maintaining this
equilibrium between A and B.
So in that perspective,
the C variables
are dummy coordinates.
So you should be able
to cancel them out
from the two sides
of the equation,
and get some kind
of a form like this.
Either way, what
that really means
is that if I list all of
the coordinates of, say, A,
and I put it, say, in
equilibrium with a bath that
is at some particular
temperature,
the coordinates of A
and B are constrained
to lie in some particular
surface that would correspond
to that particular theta.
And if I have
another system for B,
the isotherm could have
completely different for that
data.
But any time A and B
are in equilibrium,
they would be lying
on the isotherm that
would correspond
to the same fate.
Now, again, a mechanical
version of this,
that is certainly
hopefully demystifies
any mystification that may
remain, is to think a scale.
You have something
A on this scale.
And you have something
C on this scale.
And the scale is balanced.
You replace A with B, and
B and C are in balance,
then you know that A and B are
in balance with each other.
And that implies
that, indeed, there
is a property of the objects
that you put on the balance.
You can either
think of it as mass,
or more appropriately,
the gravitational force
that they experience,
that they need.
The thing is balanced.
The forces are equal.
So it's essentially
the same thing.
Now having established
this, then you
want to go and figure
out what the formula is
that relates the property
that needs to be balanced,
which is maybe the mass or the
gravitational force in terms
of density, volume, et cetera.
So that's what you
would like to do here.
We would like to be able
to relate this temperature
function to all the other
coordinates of the system,
as we're going to.
Any questions?
Yes?
AUDIENCE: So how did [INAUDIBLE]
that there is the isotherm?
Is it coming from the
second conclusion?
PROFESSOR: OK.
So what we have said is
that when two objects are
in equilibrium, this
function is the same between.
So let's say that we
pick some kind of a vat--
like, it could be a lake.
And we know that
the lake is so big
that if you put something
in equilibrium with that,
it's not extracting too much
heat or whatever from the lake
can change its
temperature function.
So we put our system
in equilibrium
with the lake, which is
at some particular fixed
value of theta.
And we don't know what theta
is, but it's a constant.
So if I were to fiddle
around with the system
that I put in the lake--
I change its volume,
I change its length,
I do something--
and it stays in equilibrium
with the length,
there is some function
of the coordinates
of that system that
is equal to theta.
So again, in general,
I can make a diagram
that has various coordinates
of the system, A1, A2, A3.
And for every combination
that is that this theta,
I will put a point here.
And in principle, I can
vary these coordinates.
And this amounts
to one constraint
in however many
dimensional space I have.
So if we span some
kind of a surface--
so if you're in three
dimension, there
would be a two
dimensional surface.
If you're in two
dimension, there
would be a line that would
correspond to this constraint.
Presumably, if I change
the lake with something
else, so that theta
changes, I will
be prescribing some other curve
and some other surface in this.
Now these are surfaces
in coordinate space of A.
In order to be in equilibrium
with an entity at the fixed
value of theta, they prescribe
some particular surface
in the entire coordinate space
and they're called isotherms.
OK?
Actually, let's
state that a little
bit further, because
you would like
to give a number to
temperature, so many degrees
Celsius, or Fahrenheit,
or whatever.
So how do you do that?
And one way to do that is to
use what is called the ideal gas
temperature space--
the ideal gas scale.
So you need some
property at this stage
in order to construct
a temperature scale.
And it turns out that a
gas is a system that we
keep coming back
to again and again.
So as I go through the various
laws of thermodynamics,
I will mention
something special that
happens to this law for
the case of this ideal gas.
And actually, right
now, define also
what I mean by the ideal gas.
So we said that a
gas, in general, you
can define through
coordinates P and V.
So if I put this
gas in a piston,
and I submerge this piston,
let's say in a lake,
so that it is always at whatever
temperature this lake is.
Then I can change
the volume of this,
and measure what the pressure
is, or change the pressure,
and figure out
what the volume is.
And I find out that there is
a surface in this space where
this curve that corresponds
to being equilibrium
with this [? leaves-- ?] this
is the measure of the isotherm.
Now the ideal gas law is that
when I go to the limit there,
V goes to infinity,
or P goes to zero.
So that the whole thing
becomes very dilute.
No matter what
gas you put here--
whether it's argon,
oxygen, krypton,
whatever-- you find that in this
limit, this shape of this curve
is special in the sense
that if you were to increase
the volume by a factor
of two, the pressure
will go down by a
factor of two, such
that the product
PV is a constant.
And again, this is
only true in the limit
where either V goes to
infinity or P goes to 0.
And this is the property
that all gases have.
So you say OK, I will use that.
Maybe I will define what
the value of this product
is to be the temperature.
So if I were to replace
this bath with a bath that
was hotter than this product
for the same amount of gas,
for the same wire,
would be different.
And I would get a
different constant.
You maybe still want a number.
So what you say is
that the temperature
of the system in degrees
Kelvin is 273 times 16.
The limit of PV, as V goes
to infinity of your system,
divided by the limit
of the same thing
at the triple point
of water [? iced. ?]
So what does that mean?
So you have this
thing by which you
want to measure temperature.
You put it in contact
with the system-- could
be a bath of water,
could be anything, yes?
You go through this
exercise and you
calculate what this product is.
So you have this product.
Then what you do is
you take your system,
and you put it in a case
that there are icebergs,
and there's water,
and there will
be some steam that will
naturally evaporate.
So you calculate the
same product of PV
in this system that
is the triple point
of ice water, et cetera.
So you've measured one product
appropriate to your system,
one product appropriate
to this reference
point that people have set, and
then the ratio of those things
is going to give you the
temperature of the system
that you want to measure.
This is clearly a very
convoluted way of doing things,
but it's a kind of
rigorous definition of what
the ideal gas
temperature scale is.
And it depends on this
particular property
of the diluted gases that the
production of PV is a constant.
And again, this number
is set by definition
to be the temperature of the
triple point of the ice water
gas.
OK.
Other questions?
All right.
So this is now time to
go through the first law.
Now I'll write again
this statement,
and then we'll start to
discuss what it really means.
So if the state of an otherwise
adiabatically isolated system
is changed by work,
the amount of work
is only function of
initial and final points.
OK.
So let's parse what
that means and think
about some particular example.
So let's imagine that we
have isolated some system.
So that it's not
completely boring,
let's imagine that
maybe it's a gas.
So it has P and V as
some set of coordinates.
Let's say that we put some
kind of a spring or wire in it,
so we can pull on it.
And we can ask how
much we pulled,
and what is the
length of this system.
Maybe we even put
a magnet in it,
so we have magnetization
that we can measure
if we were to pass
some kind of a current
and exert some kind
of magnetic field.
So there's a whole
bunch of coordinates
that I'm completely familiar
with from doing my mechanics
courses and
electromagnetic courses.
So I know various ways to do
mechanical work on this system.
So the system is
otherwise isolated,
because I don't really know how
to handle this concept of heat
yet, but I certainly have no
problems with mechanical work.
And so what I do is, I imagine
that it is initially isolated.
What I do is therefore, I
have some-- in this case,
six dimensional
coordinate space.
I'm will only draw
two out of the six.
And I start some initial
point, let's call it I.
And then I start doing various
types of things to this.
I could, for example,
first pull on this,
so that the length
changes, changes
the current, put
pressure so that
the volume changes, et cetera.
At the end of this story,
I'm at some other point
that I will call F.
Now I could have performed
this change from the initial
to the final state
through one set of changes
taking place one
after the other.
But maybe I will change that,
and I will perform things
in a different way.
So there's path number 1.
Then there's path number 2.
And there's huge number
of different paths
that I can, in
principle, take in order
to change between the
initial and final points
by playing around with the
mechanical coordinates that
describe the system.
I always ensure that,
initially, I was in equilibrium,
so I could know exactly what the
value of these parameters are.
And finally, I wait until
I have reached equilibrium.
So again, I know
what this things are.
And I know what
mechanical work is.
And I can calculate
along each one
of these paths, the
net amount of work.
The work is delivered
in different ways--
through the magnetic
field, through the pulling
of the spring, to the
hydrostatic pressure,
et cetera-- but
ultimately, when I add up
all of the increments
of the work,
I will find that all of them
will give you the same delta
W, irrespective of the path.
OK.
Now this reminds me
of the following,
that if I'm rolling a
ball on top of a hill.
And there is no friction.
The amount of work
that I do in order
to take it from here to here,
irrespective of the path
that I take on the
hill, it really
is a function of the
difference in potential energy
between the final
and initial points.
So it's the same type of thing.
Rather than moving these
coordinates on a hill,
I am moving them in
this set of parameters
that thermodynamically
describe the system,
but I see the same
thing that I would
see in the absence of friction
for rolling a ball of the hill.
And immediately, I
would deduce here
that there is this
potential energy,
and the amount of
work that I have
to do to roll this off
the hill is the difference
between the potential energy
between the two points.
So here, similarly, I would
say that this delta W--
I define it to be the difference
between some function that
is called the internal
energy-- that depends
on the final set of coordinates.
And there's a whole
bunch of them.
Think of them in some
pictorial context.
Minus what you have initially.
So in the same sense that
the zeroth law allowed
me to construct some
function of coordinates that
was relevant to
thermal equilibrium,
the first law allows me
to define another function
of coordinates, which
is this internal energy.
Of course, the internal energy
is the remnant of the energy
that we know in
mechanical systems
to be conserved quantity.
And this is the
statement of fact.
So far, nothing surprising.
Now the real content
of the first law
is when we violate
this condition.
So essentially, what we do is
we replace the adiabatic walls
that were surrounding our
very same system with walls
that allow the exchange of heat.
And I do exactly the
same set of changes.
So maybe in one
case I stretch this,
and then I change the
volume, et cetera.
I do exactly the
same set of changes
that I did in this case,
I try to repeat them
in the presence of
diathermic walls,
go from the same initial
state to the same final state.
So the initial
state is the same.
The final state, I
postulate to be the same.
And what is observed
is that, in this case,
the diathermic walls--
which allow heat exchange--
that the amount of work
that you have to do
is not equal to the
change in internal energy.
OK?
Now you really believe that
energy is a good quantity.
And so at this stage, you
make a postulate, if you like,
it's the part of the corollary
of the first law that
allows you to define
exactly what heat is.
So you're gradually defining
all of the things that
were missing in the
original formulation.
You define this heat
input to the system
to be the difference in
energy that you expected,
minus the amount
of work that you
did in the presence
of these walls.
OK?
Yes?
AUDIENCE: If we need the first
law to the point of heat--
PROFESSOR: Yes.
AUDIENCE: --how did we define
[? adiabatic ?] used in--
PROFESSOR: Right.
AUDIENCE: --first law
and the zeroth law?
PROFESSOR: As I said,
it's an idealization.
So it's, in the same sense
that you can say well,
how would you define
Newtonian mechanics
that force is proportional to
mass times acceleration, what
is the experimental
evidence for that?
You can only do that really
when you go to vacuum.
So what you can do
is you can gradually
immerse your particle into
more and more dilute systems,
and see what is the limiting
behavior in that sense.
You can try to do
something similar here.
You can imagine that
you put your system
in some kind of a glass,
double glass, container,
and you gradually pump
out all of the gas
this is between the two of them.
So that, ultimately,
you arrive at vacuum.
You also have to
mirror things, so there
is no radiation
exchange, et cetera.
So gradually, you can try
to experimentally reach
that idealization
and see what happens.
But essentially, it
is certainly correct
that any statement that I
make about adiabatic walls
is an isolation.
But it's an isolation
in the same sense
that when you think
about the point
particle in Newton's laws.
OK?
Let's go a little bit
further with this.
So we have that in
differential form,
if I go from one point
in coordinate space
that describes the
system in equilibrium,
where energy's defined
to a nearby point,
I can calculate what the value
of the change in this energy
function is.
And I can say that
there is a quantity, dE,
that depends on a whole
bunch of coordinates that
define the system.
And what the first law
says is that if you
try to operationally make
this change from one point
to another point, you have to
supply work through the system,
or you have to supply
heat through the system.
And we write them in this form.
And what this D
and d bar-- and we
will encounter this many times
in future also and define
them better-- is that E
is a function of state.
It depends on where you are
in this parameter space.
So in the same sense that maybe
you have a function of x and y,
could be like x
squared plus 3x y,
I can define what dE is
in terms of the x and y.
So that's where I
have this quantity.
But dW and dQ depend on
precisely how this system
was made to go
from here to here.
And you can sort of go between
how much contribution to dE
comes from here or from
there, by certainly say,
changing the
properties of the walls
from being adiabatic to
being diathermal, et cetera.
So these quantities really, as
opposed to this quantity that
depends on stage,
these quantities
depend on path, the
conditions by which you
implement a particular change.
Now there is a desire, and
it's very important for us
to actually construct
what this function is.
You want to, sort of, know
what the energy function
for, let's say, a
mechanical system is.
In a potential, you want
to know what the form of E
is and then you can do
a lot of things with it.
So how do we construct that
for our thermodynamics system?
Again, we can
idealize things and go
to processes that are
so-called, quasi-static.
Which effectively means
slow, or slow enough,
to maintain equilibrium.
And the general
idea is, suppose I
wanted to calculate what the
potential energy of a spring
is, or what the potential energy
of a particle rolling on a hill
is, well, one way that I could
do that is I could, let's say,
pull on this sufficiently
slowly, so that as I'm pulling
this by a certain amount,
the spring does not
start to vibrate.
And so the force that
I'm exerting externally,
is really the force that
is internal to the spring.
If I really push it
to rapidly, the spring
will start to oscillate.
There's really no relation
now between the external force
that, let's say, is in uniform
value and the internal force
that is oscillating.
I don't want to do that
because when that happens
I don't know where I am
in this coordinate space.
I want to do things
sufficiently slowly
so that I go from here
to here here-- every time
I am on this plane that defines
my properties in equilibrium.
If I do that, then I can
calculate the amount of work,
and for the case
of the spring, it
would be the force
times extension.
And so we generalize
that, and we
say that if I have multiple ways
of doing work on the system,
there is the analog of
the change in length
of the spring-- that is the
displacement of the spring,
if you like-- multiplied by
some generalized force that
is conjugate to that.
And indeed,
mechanically, you would
define the conjugate
variables by differentiation.
So if you, for example,
know the potential energy
of the spring as a
function of its length,
you take a derivative and
you know what the force is.
So this is essentially
writing that relationship
for the case of the
spring, generalize
to multiple coordinates.
And we can make a table of
what these displacements are
and the corresponding
coordinates for the types
of systems that we are
likely to encounter.
So what is x?
What is j?
And the first thing
that we mentioned
was the case of the wire.
And for the wire we can-- the
displacement of the length
is important, and the
conjugate variable
is the force with which
you are pulling on this.
In the first problem set, you
will deal-- in the first test
preparation, you will deal
with the case of film.
And for the film, what you
do is you change the area,
or if you have a
balloon, you can
blow on the balloon, and the
surface area of the balloon
changes.
And there's a corresponding
conjugate variable
to that which is called
the surface tension.
This is essentially the same
thing, going from one dimension
to two dimension.
And if I were to go
one dimension higher,
in the case of the gas, I
have the volume of the box.
And I went through this
just for the notation
that the hydrostatic
pressure of the work
is defined to be
minus p w-- minus PBV.
Sorry.
pW is minus PVV, as opposed to,
say, the force of the spring,
that is FDL.
It's just, again,
matter of definition
and how we define the
sign of the pressure.
And for the case of the magnet
that we also briefly mentioned,
we have here MDB.
The one thing that you
can see is the trend
that all of these
quantities, if you
make the size of your system
twice as big, these quantities
will get proportionately bigger.
So they're called extensive.
Whereas the force and all the
other quantities are intensive.
So what we've established
is that if we focus only
on these types of
transformations, that
don't include heat, we can
relate the change in energy
directly to dW.
And the dW, we can write
as the sum over i Ji dxi.
Now in order to really
get dE, in general,
I have to add to this dQ.
And so really the
question that we have is,
is there some kind
of analog of dW
that we can write down for dQ?
And if you think about it,
if you have two springs that
are in equilibrium, so
that the thing does not
go one way or the other,
the force exerted from one
is the same as the force
exerted on the other.
So in equilibrium, mechanical
equilibrium forces,
pressures, et cetera,
are generally the same.
And we've established that when
you are in thermal equilibrium,
temperatures are the same.
So we have a very good
guess that we should really
have temperature appearing here.
And then the question that
we'll sort of build on
is what is the analog
of the quantity
that we have to put here.
Let me finish by telling
you one other story that
is related to the ideal gas.
I said that for every
one of these laws,
we can kind of come up with
a peculiar feature that
is unique to the case
of the ideal gas.
And there is a property
related to its energy
that we will shortly explore.
But let me first say what the
consequence of these kinds
of relations for
measurable quantities
that we can already
try to deduce are.
So one thing that you
can relate to heat,
and properties of particular
material, is the heat capacity.
So what you can do is you
can take your material
and put some amount
of heat into it.
And ask, what is the
corresponding change
in temperature?
That would be the heat capacity.
Now this bar that
we have over here,
tells me that this quantity--
which we will denote on C--
will depend on the path
through which the heat is
added to the system.
Because we've established that,
depending on how you add heat
to the system, the
change that you
have in the coordinate
space is going
to be distinct potentially.
Actually, we establish
the other way
that for a given change
in the coordinate system,
the amount of the
Q depends on path,
so they're kind of
equivalent statements.
So if I'm dealing with a gas,
I can add the heat to it,
at least among many
other possibilities,
in two distinct ways.
I can do this at constant
volume or at constant pressure.
So if I think about
the coordinate space
of the gas, which
is PV, and I start
from some particular
point, I can either
go along a path like
this, or I can go along
the path like this, depending
on which one of these quantities
I keep fixed.
So then in one
case what I need is
the change in Q
at constant v, dT.
In the other case, this
change in Q, a constant P, dT.
Now a consequence
of the first law
is that I know that dQ's
are related to dE minus dW.
And dW for the gas is minus PdV.
So I can write it
in this fashion.
Divided by dT, and in one
case, done at constant V,
and in the other case,
done at constant p.
The distinction
between these two paths
immediately becomes clear,
because along the paths where
the volume is kept
constant, there
is no mechanical work that is
done along the this path, yes?
And so this contribution is 0.
And you have the
result that this
is going to be related
to the change in energy
with temperature at
constant V, whereas here you
have the change in energy,
in temperature at constant P,
plus P dV by dT at constant P.
So there is some additional
manipulations of derivatives
that is involved, but
rather than looking
at that in the general case,
I will follow the consequence
of that for a particular
example, which
is ideal gas expansion.
So as this is an observation
called Joule's experiment.
I take a gas that is
adiabatically isolated
from its environment.
I connect it, also maintaining
adiabatic isolation
to another chamber.
And initially all
of the gas is here,
and this chamber is empty.
And then I remove this, and
the gas goes and occupies
both chambers.
Observation, there
is some temperature
initially in this system
that I can measure.
Finally, I can measure the
temperature of this system.
And I find that the two
temperatures are the same.
In this example,
since the whole thing
was adiabatically
isolated, that the Q is 0.
There is no mechanical work
that done on the system,
so delta W is also 0.
And since delta is
the same, I conclude
that delta E is the
same for this two cases.
Now, in principle,
E is a function
of pressure and volume.
And pressure and volume
certainly changed very much
by going from that
place to another place.
OK, so let's follow that.
So E we said is a function
of pressure and volume.
Now since I know that
for the ideal gas,
the product of pressure
and volume is temperature,
I can certainly exchange
one of these variables
for temperature.
So I can write,
let's say, pressure
to be proportional to
temperature over volume.
And then rewrite
this as a function
of temperature and volume.
Now I know that in this
process the volume changed,
but the temperature
did not change.
And therefore, the internal
energy that did not change
can only be a function
of temperature.
So this Joule expansion
experiment immediately
tells me that while internal
energy, in principle,
is a function of P
and V, it is really
a function of the
product of P and V,
because the product
of P and V, we
established to be
proportional to temperature.
OK?
So now if I go and look
at these expressions,
I can see that these, dE by
dT, and irrespective of V and P
is the same thing, because E
only depends on temperature.
And since V is-- we know that PV
is proportional to temperature,
a constant P, dV by dT
is the same as V over T.
And so if I look at the
difference between those two
expressions, what I find
is that this part cancels.
This part gives me the value
of this product, PV over T,
which we said is a constant.
And it is certainly depends
on the amount of gas
that you have.
And so you can pick a
particular amount of gas.
You can experimentally
verify this phenomenon,
that the difference of heat
capacity along these two paths
is a constant.
That constant is the same
for the same amount of gas
for different types of
argon, krypton, et cetera.
And since it is
proportional, eventually
to the amount of matter,
we will ultimately
see that it can be set to
be the number of particles
making up the gas in some
constant of proportionality
that we will identify later
in statistical physics
is the Boltzmann
parameter, which
is 1.43 times 10 to the
minus 23 or whatever it is.
All of this depends
partly on the definition
that you made of temperature.
What we will do next
time is to review
all of this, because I've went
through them a little bit more
rapidly, and try to identify
what the conjugate variable is
that we have to put
for temperature,
so that we can write the form of
dE in a more symmetric fashion.
Thank you.
