Hello, bonjour, salamatsizbe, dobry den,
welcome to the sixth tutorial of Statistical Mechanics: Algorithms and Computations,
from the Physics Department of Ecole Normale Superieure.
In this week's lecture and homework
action was all about the Levy quantum path
one-dimensional geometrical object
that are easy to sample
yet that carry a profound message.
They lend a geometrical interpretation to the Heisenberg uncertainty principle:
at high temperature and high energy
a particle is at position x with probability rho(x, x, beta)
the quantum path connecting x to x
has no fluctuations and the particle is really located at x.
Likewise, the density matrix
has no off-diagonal part Likewise, the density matrix
has no off-diagonal part
In contrast, at low temperature
the particle is uncertain in its position x
as the quantum path
fluctuates wildly from x to x, for beta being large
This is also mirrored in the large off-diagonal part of the density matrix
In this tutorial, we touch on the second pillar
of quantum statistical mechanics, besides the Heisenberg uncertainty principle
namely the indistinguishability of particles,
in our case bosons.
For our introduction, this week
and next week we will concentrate on the case of non-interacting ideal bosons.
This is not really a restriction
because non-interacting bosons
are the only system in all of physics that have a phase transition without interactions.
In fact, a curious statistical interaction
appears in a system of indistinguishable bosons
and Michael and Alberto will trace it down
in this week's tutorial.
Unfortunately Vivien cannot be with us for a couple of weeks
but he'll come back for our discussion of classical spin systems.
We will put our ideal bosons into a harmonic trap
and this same system has set the stage
for ground-breaking experiments in atomic physics,
where Bose-Einstein condensation has lead to a revolution
in experimental glass cells just like this one.
We will follow this revolution
you will follow this ground-breaking experiment
in your own path integral Monte Carlo calculation, using the path integral picture.
but before doing this, Michael and Alberto
will discuss Bose-Einstein condensation using wavefunctions and energy levels.
This is necessary in order to understand what it is all about
the Bose-Einstein condensation into the ground-state.
They will then discuss all of this
in a simple of a few trapped bosons.
Before putting many particles into a three-dimensional trap, let us first consider
a single particle.
This means you enumerate all single particles states with a given energy E.
In three dimensions, the harmonic trap has three spring constants
one in x, one in y and one in z direction, as shown in the picture here.
Just like last week
we set all the spring constants omega_x, omega_y, omega_z = 1,
so that the energies E_x, E_y, E_z
are equal to 0, 1, 2, 3, 4, ..
For simplicity, we set the ground-level energy equal to zero
and label all the states
by the energy E_x, E_y, E_z, so that they have the total energy E = E_x + E_y + E_z
Let us now write a little program
naive_single_particle.py
to enumerate all these states. Output of the program is shown here
For example here we have state 12
it is in the ground state in x
in the second excited states in y
and in the first excited state in z.
There are many states, but notice that
each tuple E_x, E_y, E_z
corresponds to a unique
single-particle quantum state of energy E
E = E_x + E_y + E_z
Let's have a look at them, there's one ground-state of energy zero
there are three states of energy one
there are six states of energy two
ten states of energy three
fifteen states of energy four, and so on..
Do you see a pattern?
So let us now compute the number of choices
E_x, E_y, E_z that give an energy three.
For a harmonic trap, the number of states with a given energy E
can be calculated explicitly, as shown here.
This expression means that for each choice of E_x
we can choose E_y as an integer between 0 and (E - E_x)
given E_x and E_y,
then the remainder E_z is fixed.
This means for each choice of E_x,
we have (E - E_x + 1) choices for E_y
This makes that we have
(E + 1) + (E) + (E - 1) + ..
.. + 1 choices
In total, this makes (E + 2) (E + 1) / 2 choices.
This corresponds to one choice for E=0
three choices for E=1, six choices for E=2 and so on
But let use now use a more systematic method
to compute the number of states with an energy E
Alberto will soon use it in a more involved context
The method consists in writing the number of states
as a free sum over E_x, E_y and E_z
with a condition that we implement by means of Kronecker delta functions
you simply sum up the states, as is shown in this equation here
where the Kronecker delta is defined as shown here
Because of the Kronecker delta function
only combinations of E_x, E_y, E_z
that sum up to E contribute to the number of states.
We can represent the Kronecker delta function by this integral
We can then enter this integral representation of the Kronecker delta function into the above sum
we then exchange sums and integrals
then we see that the sums have become independent
as is shown here
These are geometric sums that can be evaluated explicitly, as you see here
Finally, we can evaluate the remaining integral
using brute-force Riemann integration, as Alberto will demonstrate in a few minutes.
This numerical integration is all that we will need here
but those of you who feel at home in the complex plane
know of course that this integral can also be evaluated exactly
The substitution exp(i lambda) = z
leads to a complex contour integral that can be evaluated using the residue theorem.
and of course the result is (E + 2) (E + 1) /2 .
Now, let us do two things
first: in our list of single particle states
(shown right here) let us retain only the 35 states of lowest energies
these are states that have energy smaller or equal to four.
We now have a list of 35 states
and we can still look at them here on the screen
second thing: we now put five bosons into these states
Remember that the partition function of the physical system
is equal to the sum over all states
of the probability of each state
and this probability is given by the Boltzmann weight
exp(-beta E)
the partition function of our five bosons system
is then put together from all the different ways we have
to put the five bosons into the single particle states
For example, we might put our red particle into this state here
the blue particle into this state here
the yellow particle into this state here
the green particle into the state here, and finally the pink particle into this state here.
The sum of these possibilities is the partition function
something we can measure
While we should be careful not to forget any of the five-particles states
in our partition function
we are not allowed to overcount them either
The problem arises because bosons are identical particles
they don't just look the same, they are indistinguishable. There's no way to tell them apart.
This three states here are different for particles
that are distinguishable, for example by their color,
but for bosons they are one and the same.
In our partition function, we should count only one of them
this is one of the most profound insights in all of physics
it is due to Bose (in 1923, for photons)
and Einstein (in 1924, for massive bosons)
We can translate this great achievement of Bose and Einstein into a simple algorithm
in order to avoid overcounting many-particle states
we count only states that satisfies that the single-particle state of particle 0
is smaller or equal to the state of particle one
is smaller or equal to the state of particle two
is smaller or equal to the state of particle three
is smaller or equal to the state of particle four.
Out of our three states
we thus pick the last one. The other two have to be dropped from our partition function
Alberto will now leave this graphical discussion
and put our five bosons model with the 35 states
onto the computer.
But before doing so, please take a moment or two
to go over our discussion of single-particle states
and five-particle states, and to download and to run
our nice little program: naive_single_particle.py
that helped us to create these nice states that we have been discussing all along.
The ordering trick that was introduced by Michael
allows us to compute the bosonic partition function
with distinguishable particles
Remember
you put the particle 0 in a state between 0 and 34
then you put the particle 1
in a state between the state of the particle 0 and 34
then you put the particle 2 between the state of the particle 1
and 34, and so on..
This five-particles state has an energy
E given by this formula
and a statistical weight exp(-beta E)
this allows us to write the bosonic partition function as follows
But maybe you prefer to see this as an algorithm
Note the curious line
the multiplication of a list
with an integer factor k simply concatenates
the list k times
So that [1] * 3 is simply [1, 1, 1].
Here we compute the partition function
the multiple loops are really naive
but we can compute the number of states
which contribute to the partition function
It is a large number: 575757
but it is 90 times smaller
than the number of states for distinguishable particles
which is 35^5
can we compute this number? Of course
Let's look again at the bosonic states, and erase all information
We have 35 boxes
which means 34 inner walls
we can now put the 5 bosons inside the boxes
the number of states correspond
to counting how we can arrange 34 + 5 walls and bosons
taking into account that walls as well as bosons are indistinguishable.
The number 575757
is then 39 ! / 5 ! / 34 !
We can easily extend
the program naive_bosons_trap.py
counting the number of particles that are in the ground state
In this state we have zero particles in the ground state
here we have three particles in the ground state
and here we have one particle in the ground state
we can implement this in naive_boson_trap.py
using the count of zeros in the list state
We can compute the average number N0
and the condensate fraction
considerably larger numbers of particles and states can be achieved if we introduce
the occupation number associated to a single particle state
In our scheme of 35 states
instead of drawing colored particles, we just indicate that the state 0 is occupied by n0 particles
the state 1 is occupied by n1 bosons and so on
So for example this state can be written with these occupation numbers
and the energy associated to any state can be written as follows
the partition function of the system writes as follows
This expression is equivalent to what we have computed in naive_boson_trap.py
but it is even more complicated
instead of the 575757 states
here we have 6^35 terms
However now we have a bunch of sums and a delta constraint
does this ring a bell with you?
This is exactly what was considered by Michael 10 minutes ago
So we can use his providential representation of the Kronecker delta function
Note: the sums are geometrical sums, so that we can compute them
and of course each sum depends only on the energy of the state and not any more on the occupation number
we have to think two minutes to understand if we have to consider
the finite geometrical sum or the infinite geometrical sum
and this is also done in the fact-sheet which is associated to this tutorial
In any case, we can now arrive to an expression of the partition function which is a simple one-dimensional integral over the variable lambda.
This expression for the partition function allows us to compute all thermodynamic quantities
as the mean energy or the condensate fraction, which is shown here
we can compute this integral numerically, using a simple discretization
and moreover the cut-off of the energy E_max which was fixed to 4
can now be increased as much as we like, and we can also increase the number of particles.
For N=5 and E_max=4
we recover exactly the same results as in naive_boson_trap.py
Here we show the plot of the condensate fraction
as a function of the rescaled temperature
for a system of N bosons in a harmonic trap
Here we see that all particles are in the ground states at very low temperatures
this is a simple consequence of Boltzmann statistics
At zero temperature all the particles populate the ground state
the Bose-Einstein condensation is something else
it means that a finite fraction of the system
is in the ground-state for temperatures which are
much higher than the gap between the gap between the ground-state and the first excited state, which is one, in our system.
You see here that this fraction goes to zero for a rescaled temperature T / N^(1/3) = 1
this means that the critical temperature for the Bose-Einstein condensation
grows like N^(1/3) for a harmonic trap.
This is a much higher temperature than 1, for large system
Clearly this observation can be confirmed by explicit calculation
so you can really believe us when I say that the critical temperature
grows like N^(1/3) for a harmonic trap
Finally take a moment to download, run and modify
naive_boson_trap.py
that studied the bosonic statistics and the Bose-Einstein condensation
for a problem of five bosons, that was really too small
but we came up with some idea
actually a one-dimensional integration, that allowed us to go much further.
In conclusion, in this tutorial 6 of Statistical Mechanics:
Algorithms and Computations, we made our first step in the studying of quantum indiscernability
the second pillar of quantum statistical mechanics, next to the Heisenberg uncertainty principle
Five bosons in a trap allowed us to shed light on the very essence of quantum statistics
The same model also allowed to understand something about Bose-Einstein condensation
when all of a sudden a finite fraction
of particles populate the single-particle ground state
In a trap this happens at higher and higher temperature as we increase the particle number
Next week we will take up the study of bosons and Bose-Einstein condensation
in the language of density matrices and path integrals
and you will produce your own path integral Monte Carlo calculation
to produce Bose-Einstein condensation in a situation very similar to the experimental one that takes place in this glass cell.
So in the meantime have fun with homework session 6
and see you again next week in session 7 of Statistical Mechanics: Algorithms and Computations
for a study of Bose-Einstein condensation.