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.