QMC+for+harmonic+bosons

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 * Homework 07: QMC for harmonic bosons **
 * Don't hesitate to ask questions and make remarks on ** this page.

=Introduction= In this exercise, as in Class Session 07, we consider //N// non-interacting bosons in an harmonic trap. We focus on the energy of those bosons using a simple direct-sampling Quantum Monte Carlo algorithm. =Thermodynamics of N non-interacting bosons= We take the notations of Class Session 07.

A- The energy of non-interacting bosonic particles
math \langle E\rangle = -\frac 1{Z_N} \frac{\partial Z_N}{\partial\beta} math > From the recursion relation (1) of Class Session 07, write the mean energy ⟨//E//⟩ in terms of the partition functions //Z n // and //z n // and their derivatives with respect to //β//, ∂//Z n ///∂//β// and ∂//z n ///∂//β//. math z_k= \Big(\frac{1}{1-e^{-k\beta}}\Big)^3 math > From this expression, find a relation between ∂//z k ///∂//β// and //z k //.
 * -1- Consider the following expression for the mean energy ⟨//E//⟩ at inverse temperature //β//
 * -2- We now chose a harmonic trap the such that the energy levels are //E n // = //n// in each of the three spatial directions. The (three-dimensional) single particle partition //z k // writes:
 * -3- Inspired by the algorithms of Class Session 07, write a program to compute the mean energy by using a recursion relation on the pair (//Z N //,∂//Z N ///∂//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 with //N//. Identify and give an explanation for the the high-temperature asymptotics. In particular, what is the advantage of defining //T// ٭, where does the //N// 1/3 of its definition come from?


 * Important remark**: at fixed //N// and high temperature, you //may// obtain irrelevent results. This is due to an overflow in the computation of the //Z n //'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
math W_{\geq k}= Z_{N-k} math math W_{k}= \left\{\begin{array}{ll}W_{\geq k}-W_{\geq k+1} & \text{if}\quad k Using the results of the previous questions, show that math \langle N_0\rangle = \frac 1{Z_N}\sum_{p=0}^{N-1} Z_p math
 * -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 //W k // of //N// bosons with precisely //k// of them in the ground state. Show in details that
 * -3- Deduce from those results the probability //π//(//N// 0 ) of having //N// 0 bosons in the ground state in terms of the partition function.
 * -4- The condensate fraction, which is the mean value ⟨//N// 0 ⟩ of the number //N// 0 of bosons in the ground state, writes
 * -5- Modify the program you've written in the previous section so as to include a computation of ⟨//N// 0 ⟩. Plot this quantity as a function of the reduced temperature //T// ٭ for different increasing values of //N//. Comment.

=References=
 * Borrmann P., Franke G. (1993) Recursion formulas for quantum statistical partition functions, Journal of Chemical Physics 98, 2484–2485
 * Landsberg P. T. (1961) Thermodynamics with quantum statistical illustrations, Interscience Publishers
 * **SMAC part 4.2.3 - 4.2.6**

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