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Magnetisation and Mean Field Theory in the Ising Model

by Dalton A R Sakthivadivel

Submission summary

As Contributors: Dalton A R Sakthivadivel
Arxiv Link: https://arxiv.org/abs/2102.00960v4 (pdf)
Date submitted: 2021-10-14 04:37
Submitted by: Sakthivadivel, Dalton A R
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
Specialties:
  • Statistical and Soft Matter Physics
Approach: Theoretical

Abstract

In this set of notes, a complete, pedagogical tutorial for applying mean field theory to the two-dimensional Ising model is presented. Beginning with the motivation and basis for mean field theory, we formally derive the Bogoliubov inequality and discuss mean field theory itself. We proceed with the use of mean field theory to determine a magnetisation function, and the results of the derivation are interpreted graphically, physically, and mathematically. We give a new interpretation of the self-consistency condition in terms of intersecting surfaces and constrained solution sets. We also include some more general comments on the thermodynamics of the phase transition. We end by evaluating symmetry considerations in magnetisation, and some more subtle features of the Ising model. Together, a self-contained overview of the mean field Ising model is given, with some novel presentation of important results.

Current status:
Editor-in-charge assigned


Author comments upon resubmission

The author thanks both reviewers for their attention to this manuscript, and the suggested revisions. These have been taken into account as described in author replies, and have been listed below as changes made.

List of changes

In this resubmission, the following revisions have been made:

Expanded commentary about
- Collective phenomena (Introduction)
- Fluctuations in MFT and statistical physics more broadly (Introduction)
- Order parameters (IIE)

Amendments to
- Discussion of renormalisation group and critical dimension (Introduction)
- Notation in some areas

All pursuant to reviewer comments. Additionally, a link to an interactive surface plot hosted by the author on GeoGebra has been included, for pedagogical value.


Reports on this Submission

Anonymous Report 2 on 2021-10-16 (Invited Report)

Report

I read the article again, as much objectively as possible. I am sorry to say that I found no reason to alter my decision of rejection.

Mean field theory starts with the basic assumption that the magnetic field acting at any site is the average magnetic moment. With this assumption, one can derive a transcendental equation for magnetic moment (Eq. (14) in the article) following a simple argument, as can be found in any standard text book. The author however arrives at this equation by looking for the value of magnetic moment which minimises the variational free energy. This alternative approach (i) uses “Bogoliubov inequality”, in addition to standard treatments; (ii) is lengthy (iii) adds to our understanding ONLY the knowledge that the conventional mean-field value of magnetic moment minimises the variational free energy. Although I understand that the philosophy of different scientists may be different, I strongly feel that there is no reason why the point (iii) could be helpful to any beginner (student) or researcher (in other areas of physics).

Hence, I stick to my decision of rejection.

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Anonymous Report 1 on 2021-10-16 (Invited Report)

Report

I am satisfied with the response of the author to my report and hence recommend the manuscript for publication.

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