Video summary
Lattice models serve a dual purpose in statistical mechanics, extending beyond their traditional role in describing molecular positions and movements to also explain energy transfer within systems. By applying these models to thermal interactions, such as heat flowing from a hot block of metal to a colder one, we can visualize how energy redistributes among molecules without creating or destroying it. In this framework, the system's state is defined by assigning specific energy levels—such as zero or one unit—to individual molecules on either side of an interface. The likelihood of any particular arrangement, known as a macrostate, depends on its multiplicity, which represents the number of distinct microstates that satisfy those conditions using combinatorial mathematics like binomial coefficients.
The principle governing these energy distributions is rooted in probability and conservation laws: systems naturally evolve toward the most probable state, which corresponds to the highest multiplicity. For instance, when starting with all molecules on one side highly energized and none on the other, the system evolves by transferring energy units from high-energy molecules to low-energy ones until equilibrium is reached. However, intuition can be misleading; simply splitting total energy equally between two halves of a system does not always yield the most probable state if multiple energy levels are available. Instead, the highest multiplicity often occurs when energy is spread across different levels as well as locations, resulting in configurations where more molecules occupy lower ground states while fewer populate higher excited states, provided the total energy remains conserved.
Surprising counter-intuitive results emerge particularly when energy resources become scarce relative to the number of available particles. In scenarios with limited total energy, the most likely macrostate is not one of perfect evenness but rather a distribution where there are significantly more molecules in lower energy levels than in higher ones. This occurs because spreading out both the locations and the energy levels maximizes disorder without violating conservation constraints; for example, distributing six units of energy among twelve molecules yields a much higher multiplicity when four occupy the ground state, one occupies an intermediate level, and two are at different upper levels compared to other valid combinations. Ultimately, lattice models demonstrate that thermal equilibrium is not merely about equalizing temperature or location but represents the statistical dominance of highly disordered states where energy is optimally dispersed across all available degrees of freedom within the system's limits.
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[Music]
all right so we've seen how lattice
models can be useful in a number of
different circumstances but
we don't have to use lattice models just
to describe where molecules are and how
they move from place to place we can
also use lattice models for other
circumstances like how energy transfers
from one location to
from one molecule to another in a system
so this is useful for examples like
let's say
i have a block of material maybe a block
of metal doesn't really matter what it
is at some elevated temperature
that's in thermal contact let's say with
another
material that's at a colder temperature
so we know what's going to happen in
this system
is that
energy is going to be transferred in the
form of heat from the hotter system to
the colder system the one with high
temperature to the one with lower
temperature so
lattice models can help us understand
why that happens
and i'll point out that for the moment
i'm going to be fairly loose with the
terms energy and heat and temperature we
don't have definitions for those yet
we'll treat those a little more
precisely later on in the in the course
but right now i'll just use our
intuitive understandings of things being
hot things with high temperature
and
heat in the form of energy flowing from
hot to cold
all right so the lattice model that we
can use to understand this type of
behavior
let's make a lattice not for the
positions of our molecules but for the
energy that those molecules have let's
say that in my system
in my very oversimplified system
molecules can have an energy of zero or
they can have an energy of one they can
have a little energy or they can have
more energy
and let's say
i've got on the hot side to exaggerate
things i've got a few molecules that
have a lot of energy and on the colder
side
i've got
a few molecules that have less energy
so i'm using n equals six molecules
on each side of this system
and again i've just simplified the hot
and cold to having more energy or having
less energy one side of the system has
more energy than the other my diagram is
meant to indicate here are all the
individual molecules that have certain
amounts of energy six molecules have
energy one on the left side of the
system six molecules have energy zero on
the right side of the system
that's the microstate of the system i've
shown you each individual molecule and
how much energy it has on this energy
diagram
the macro state of the system would be
on
the left the description is all the
molecules all six of the molecules have
uh one unit of energy on the right all
six of the molecules have
zero units of energy
so we can
calculate the multiplicity calculate the
number of ways of
writing down a microstate that matches
the macro state i just described six on
the left with one energy
none with zero energy
on the right none with
one energy six with no energy
so the ways i have of choosing which six
of my molecules have one unit of energy
on the right that's just six choose six
i'm sorry on the left on the right side
how many ways are there of choosing
which zero of my molecules are in the
excited state and therefore which of
them are in the ground state that's six
choose zero
so as with the lattice models for
gas molecules or liquid molecules
we end up with these binomial or
multinomial expressions to describe the
multiplicity in this case for this very
distinct
macrostate where all the molecules are
excited on one side all the molecules
are low energy on the other side
six choose six is just one six to zero
is just one
there's only one microstate the one i've
drawn that matches that description
that is of course not the most
probable state of the system
we can write other microstates that are
more likely that have a higher
multiplicity
let me point out that
there's one additional constraint we
have to start paying attention to when
we're talking about energy transfer from
one place to another is that energy only
transfers i can't create energy i can't
give molecules more energy in the system
without that energy coming from
somewhere because energy is conserved so
if i want to
draw a different state of the system
where
i want to maybe elevate one of the
molecules on the right hand side
from the ground state up to the excited
state that energy has to come from
somewhere one place it can come from the
only place in this system it can come
from is by dropping one molecule down to
the ground state on the other side of
the system so here's a microstate
where i've got
five
excited state molecules on the left side
only one on the right side
the total energy of this system is
conserved notice i had a total of six
units of energy in this system i've got
five on the left one on the right in
this system energy
is six and this system energy is still
six the energy is conserved i haven't
created or destroyed energy i've just
moved it from this molecule to this
molecule
the
multiplicity in this case
my macro state would be among the six
molecules on the left
i need to choose which five of them
are in the excited state and on the
right side i choose which of my six
molecules
which one of them are in the excited
state
so my multiplicity now
six choose five is six six choose one is
also six
i've got a multiplicity of 36
so
not surprisingly based on the examples
we've seen so far
this macro state is more probable than
this macrostate but of course this is
not the most probable macrostate i can
draw
another one
i can draw several more i can promote
two molecules or three or four any
number of molecules i want as long as i
pay for them with the appropriate d
motions over on the other side
it will not surprise you
when i tell you that
the
most probable macro state
the one with the highest multiplicity
is going to be the one where half of my
molecules are in the excited state on
the left half of them are in the excited
state on the right
the multiplicity in this case
of my six molecules on the left i have
to choose which three
are in the upper state so six choose
three
on the left
likewise 6 choose 3
on the right
let's work those factorials out so 6
choose 3 is 6 times 5 times 4
times 3 2 1 over
let's go ahead
so that's 6 factorial over 3 factorial 3
factorial
these
this 3 factorial cancels this 3
factorial that factor of 6 cancels this
factor of 6
and what i've got is
20 for 6 choose 3.
so 20 times 20.
so do this times another 6 choose 3
and i end up with
20 times 20 is 400.
so the multiplicity of this microstate
is 400 that's greater than this
multiplicity if i wrote out all the
possibilities that would be the state
the macro state with the highest
multiplicity so like i said it's not
surprising to learn that the lattice
model
helps us confirm our intuition that the
most likely state the most random state
the one with the highest multiplicity
the highest probability will be the one
where the energy is equally distributed
across both halves of the system i've
conserved energy there's still a total
of
six
units of energy
in this system but now half that energy
is on the left half that energy is on
the right
however
the lattice model can help us understand
a few things that are not quite as
intuitive so let's take a case that's
one step more complicated
let's take
a system with three
energy levels rather than two so let's
take a very similar example when i start
out
i've got six molecules all the way up
here in the most excited state on the
left
and on the right
none of them all none of the six
molecules have any energy so i again
have n equals 6 molecules on each side
now i have a total of 12 units of energy
that i need to conserve 6 molecules
times 2 is 12 units of energy
the multiplicities
very similar in this case but now since
i have three different states the macro
state the way i describe the macro state
of the system is on the left i've got
six molecules i don't care which ones
but six in the
energy state two none here none here
so my multiplicity is going to be 6
choose 6 0 0.
it's going to be a multinomial
coefficient rather than a binomial
coefficient that's multiplying the
multiple the the multinomial coefficient
on the right side which would be of my
six molecules i need zero in the upper
state zero in the middle state and all
six of them in the ground state
those multinomial coefficients six
factorial over six factorial zero
factorial zero factorial that's just
one
again the most
separated the most ordered state again
has a multiplicity of one
if we
conserve and well
we'll skip
the the step-by-step demotion of
molecules
one at a time
you might say all i need to do is get
half of the energy from the left over to
the right if i take these six molecules
and drop them each
into the energy equals one state
losing six units of energy in order to
pay for
elevation of these six molecules
up to that same state
that's the case where i've split the
energy equally between the left half and
the right halves of the system
the macro state would be all six
molecules in the equals one state on the
left all six molecules in the equals one
state on the right
my multiplicity is going to be
of my molecules of my six molecules zero
in the upper state six in the middle
state zero in the lower state
on the left
same thing on the right
and again those multiplicities those
multinomial coefficients 6 factorial
over 6 factorial gives me 1.
so that's perhaps surprising
i've equalized the energy on the left
and the right and somehow it's no more
likely than this state
if you think about it the right way it's
not surprising at all this is just as
ordered a state just as few ways of
drawing this
microstate that matches my macrostate
all of them in the middle
as there are of draw of writing this
macrostate
a much more probable state a much
state with much higher multiplicity
would be
for my six molecules let's put
two of them up here
two of them here to them here
so i've spread the molecule spread the
energy out not only among the left and
right halves of the system but i've
spread the molecules out between
different energy levels as well
so
same thing on the left same thing on the
right i've i've made the system much
more disordered by spreading the
molecules out between more states
notice that the energy is still
conserved
every molecule in this state has an
energy of two so i've got one two three
four molecules with energy of two so
that's eight energy units for the
molecules in the upper state only four
energy units for the molecules in the
middle state so the total energy eight
plus four is twelve
so essentially i can get from this state
to that state by promoting two molecules
and dropping two molecules same thing
promote two drop two
the multiplicity
of that third state that i've drawn
macro state would be 2 in the upper 2 in
the middle 2 in the lower on the left
and the same on the right so my
multiplicity is going to be 6 choose 2 2
2
multiplied by another 6 choose 2 2 2.
those factorials if i take
so 6 factorial over 2 factorial 2
factorial 2 factorial
i won't work those out
but after the cancellation that happens
there each one of these factorials 6
times 5 times 4 times 3 times 2 divided
by 8 that works out to be 90.
90 squared works out to be 8 100.
so
significantly more likely to find the
system in this state than in either of
these two states
and again what we've discovered is it's
not necessary just to spread the energy
out between
the left and right halves of the system
but also spread the energy out between
different energy levels as well
as one final example that's going to
teach us something else surprising about
the way energy spreads throughout the
system let's do one more example a lot
like this one
again i'll take an energy level diagram
with three different energy levels zero
one and two
now though i'm gonna make energy a
little less
available a little more scarce so i'll
continue with six molecules on each half
of the system but now i'm only going to
have six units of energy to go around
instead of 12. so initially
let's put
the six molecules on the left in the
equals one state on the right let's put
them in the equal zero state
so
maybe ask yourself what you think is
going to happen
in this system this system
has a multiplicity of 6 choose 0 6 0
and on the right side 6 choose 0 0 6
which is going to work out to be 1
as usual
in this ordered state
you can ask yourself what you think the
most likely distribution of these
energies is going to be in the state
that turns out to be most probable go
ahead and pause the video and write down
the state you think is going to be
most likely i'll wait a couple seconds
so if you chose
this microstate
clearly that's got a higher multiplicity
this microstate is sorry this macrostate
this microstate is no good for this
system that one doesn't conserve energy
if i were to write down
two molecules here two molecules here
two molecules here same thing on the
left as the right
that's going to be a state with a very
high multiplicity
but
it violates the conservation of energy
that that state
uses 12 units of energy and i can't get
there because i only have six units of
energy i'd have to promote more
molecules than i can afford to pay for
so
that's no good i can't use that state
if we go back to our approach of just
one at a time
lifting some molecules up and paying for
those
with some other molecules that fall down
so my macrostate is
three in the equals one state three in
the equal zero state on both sides of
the system
that's spread the energy out
considerably more than where i started
it's got the same amount of energy on
the left and the right that's beginning
to sound pretty good
the multiplicity
so what have i got i've got six choose
three
so if i stick to the same order i've
been using it six choose
zero three three
times six choose zero three three
so that's the same calculation i did
right here six choose three
that works out to be twenty times twenty
is four hundred
you might have chosen that as the most
likely distribution of of energies among
these different states certainly more
probable than the one we started with
it's less it's we got a lower
multiplicity than this one but that
one's illegal we're not allowed to use
that one but it turns out there's one
that's even better than this
distribution
suppose i consider
the following one
suppose instead of three in the ground
state three in the middle state and none
up top
let's take
four in the ground state one in the
middle state one in the upper state
so the way i've generated this state for
example i can drop one molecule from the
middle to the bottom and simultaneously
promote one up to the top or i can just
add up the energies and say
zero plus one plus two that's a total of
three units of energy on the left
here's another three units of energy on
the right
so there's certainly a legal state a
valid state
when i calculate
the multiplicity of that state
among the six particles on the left
i've got one in the upper one in the
middle four in the ground state
multiply that by another six choose one
one four
those factorials
six factorial over four factorial
is 30 6 times 5 is 30 divided by another
2 factorial gives me
did i do that right
6 times 5 is 30 divided by 2 is 15.
um
yes
so
no no factor 6 factorial divided by 4
factorial is just 6 times 5 is 30. 30
times 30 is 900
that's the right
math multiplicity in that case turns out
to be larger than the multiplicity in
this case
right that may be a little surprising
this is in fact
the macro state
4 in the ground state one in the middle
one in the upper state that's the macro
state with the highest multiplicity if i
try all the different possible
combinations this is the one that turns
out to be highest multiplicity and
therefore
most likely the surprising thing is i
don't have a perfectly even distribution
of energy throughout the system i've got
more in the ground state than in the
upper states
and that turns out to be because energy
is relatively scarce in the system so
it's going to be useful to understand in
a more general way how can we predict
what is the most likely distribution of
the energies between the systems at the
moment
let's just be happy that we've got a way
of using lattice models to describe
something like energy transfer as well
as motion of molecules around the system
like an expansion of a gas or mixing of
fluids
and and notice along the way that the
results we get are perhaps a little
surprising counter-intuitive not quite
what we would have uh predicted if we uh
used a relatively naive expectation for
whether the energy is going to flow so
we'll look in more detail at how to
predict the most likely distribution of
the energies in a system like this
but first we're going to have to take a
little side trip and in order to
calculate these multiplicities for
systems with numbers of molecules bigger
than just a handful of molecules like
the six we've considered here we're
going to start having to do factorials
of numbers larger
than it's convenient to do calculations
with so that's our next topic is to to
think about how to do factorials of very
large numbers