# Fluctuations of observables for free fermions in a harmonic trap at finite temperature

### Submission summary

 As Contributors: Aurélien Grabsch · Gregory Schehr Arxiv Link: http://arxiv.org/abs/1711.07770v2 (pdf) Date accepted: 2018-02-15 Date submitted: 2018-02-09 01:00 Submitted by: Grabsch, Aurélien Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Quantum Physics Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

We study a system of 1D noninteracting spinless fermions in a confining trap at finite temperature. We first derive a useful and general relation for the fluctuations of the occupation numbers valid for arbitrary confining trap, as well as for both canonical and grand canonical ensembles. Using this relation, we obtain compact expressions, in the case of the harmonic trap, for the variance of certain observables of the form of sums of a function of the fermions' positions, $\mathcal{L}=\sum_n h(x_n)$. Such observables are also called linear statistics of the positions. As anticipated, we demonstrate explicitly that these fluctuations do depend on the ensemble in the thermodynamic limit, as opposed to averaged quantities, which are ensemble independent. We have applied our general formalism to compute the fluctuations of the number of fermions $\mathcal{N}_+$ on the positive axis at finite temperature. Our analytical results are compared to numerical simulations. We discuss the universality of the results with respect to the nature of the confinement.

### Ontology / Topics

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Published as SciPost Phys. 4, 014 (2018)

Please find enclosed the revised version of our paper

"Fluctuations of observables for free fermions in a harmonic trap at finite temperature"

Our manuscript was reviewed by two referees.

Both of them wrote very positive reports and suggested the publication of our paper in SciPost. Their main concern was about the
universality of our results, beyond the case of the harmonic trap which is exactly solved in the present manuscript. We have added a new
section (5.5) and extended the discussion of Appendix A to address the question of universality.

We hope that you will receive positively our resubmission.

### List of changes

- We have added a section (5.5) to discuss the universality of our results with respect to the choice of the potential;

- We have generalised the discussion of Appendix A about the variance of the total number of particles in the grand canonical ensemble to other types of potentials (not only harmonic);

- We have corrected the typos pointed out by referee 2;

- We have slightly changed the title and abstract to make them more intelligible for non specialists (we replaced "linear statistics" by "observables").