QMC for harmonic bosons Post your questions on this hwk in the blog at the end of the page. If you hand your Hwk by e-mail, PLEASE write the file in the format HW7_YOUR-NAME and send it to G. Roux.

Introduction

In this exercise, as in Tutorial 7, we consider N non-interacting bosons in an harmonic trap. We focus on the boson energy using a simple direct-sampling Quantum Monte Carlo algorithm.

Thermodynamics of N non-interacting bosons

We shall use the notations of the Tutorial 7.

A- The energy of non-interacting bosonic particles

-1- Consider the following expression for the mean energy ⟨E⟩ at inverse temperature β

Using the recursion relation (1) of Tutorial 7, write the mean energy ⟨E⟩ in terms of the partition functions Zn and zn and their derivatives with respect to β, ∂Zn/∂β and ∂zn/∂β.

-2- We now chose a harmonic trap, such that the energy levels are En = n in each of the three spatial directions. The (three-dimensional) single particle partition function zk writes:

From this expression, find a relation between ∂zk/∂β and zk.

-3- Inspired by the algorithms of Tutorial 7, write a program to compute the mean energy by using a recursion relation on the pair (ZN,∂ZN/∂β). Plot the mean energy as a function of the reduced temperature T٭ for different values of N. Consider a sufficiently large range of reduced temperatures. Comment on the behavior as a function of N. Identify and give an explanation for the the high-temperature asymptotics. In particular, what is the advantage of defining T٭ ? Where does the N1/3 in the definition of T٭ come from?

Important remark: at fixed N and high temperature, you may obtain irrelevant results. This is due to an overflow in the computation of the Zn's. You either have to check that no overflow occurs, or, better, find a cure by changing the normalizations of the partition functions.

B- The condensate fraction of non-interacting bosonic particles

-1- Consider the partition function W≥k of N bosons with ≥k of them in the ground state (energy 0). Show that

-2- Consider the partition function Wk of N bosons with precisely k of them in the ground state. Show in details that

-3- Deduce from those results the probability π(N0) of having N0 bosons in the ground state in terms of the partition function.

-4- The condensate fraction, which is the mean value ⟨N0⟩ of the number N0 of bosons in the ground state, writes

Using the results of the previous questions, show that

-5- Modify the program you've written in the previous section so as to include a computation of ⟨N0⟩. Plot this quantity as a function of the reduced temperature T٭ for different increasing values of N. Comment.

QMC for harmonic bosonsPost your questions on this hwk in the blog at the end of the page.If you hand your Hwk by e-mail, PLEASE write the file in the format HW7_YOUR-NAME and send it to G. Roux.

## Introduction

In this exercise, as in Tutorial 7, we considerNnon-interacting bosons in an harmonic trap. We focus on the boson energy using a simple direct-sampling Quantum Monte Carlo algorithm.## Thermodynamics of N non-interacting bosons

We shall use the notations of the Tutorial 7.

## A- The energy of non-interacting bosonic particles

E⟩ at inverse temperatureβE⟩ in terms of the partition functionsZnandznand their derivatives with respect toβ, ∂Zn/∂βand ∂zn/∂β.En=nin each of the three spatial directions. The (three-dimensional) single particle partition functionzkwrites:zk/∂βandzk.ZN,∂ZN/∂β). Plot the mean energy as a function of the reduced temperatureT٭ for different values ofN. Consider a sufficiently large range of reduced temperatures. Comment on the behavior as a function ofN. Identify and give an explanation for the the high-temperature asymptotics. In particular, what is the advantage of definingT٭ ? Where does theN1/3 in the definition ofT٭ come from?Important remark: at fixedNand high temperature, youmayobtain irrelevant results. This is due to an overflow in the computation of theZn's. You either have to check that no overflow occurs, or, better, find a cure by changing the normalizations of the partition functions.## B- The condensate fraction of non-interacting bosonic particles

W≥kofNbosons with ≥kof them in the ground state (energy 0). Show thatWkofNbosons with preciselykof them in the ground state. Show in details thatπ(N0) of havingN0 bosons in the ground state in terms of the partition function.N0⟩ of the numberN0 of bosons in the ground state, writesUsing the results of the previous questions, show that

N0⟩. Plot this quantity as a function of the reduced temperatureT٭ for different increasing values ofN. Comment.## References

SMAC part 4.2.3 - 4.2.6[Print this page]