Mean-field, High T Expansion paper

[JunkLearning]

This is a discussion on the 1991 paper on "How to expand around mean-field theory using high-temperature expansions" by Georges and Yedidia, http://www.iop.org/EJ/abstract/0305-4470/24/9/024". Published in the Journal of Physics. We will be adding questions, comments etc. and share/critique the paper.

Some questions:
1) do the Lagrange multipliers in Eq. 2 replace any general external field? I think the answer is yes.

2) How does one get -<U(S_i -m_i)> = 0 in Eq. A1.4?

3) How meaningful is it to go past the Onsager term or TAP equations?

 

[eljose79]

Hello everyone,

I am excited to discuss the 1991 paper on "How to expand around mean-field theory using high-temperature expansions" by Georges and Yedidia. This paper provides a comprehensive guide on how to expand mean-field theory using high-temperature expansions, which can be a useful tool in studying complex systems.

To answer your questions:

1) Yes, the Lagrange multipliers in Eq. 2 do indeed replace any general external field. They are used to enforce the constraints on the system, such as the conservation of energy or the number of particles.

2) In Eq. A1.4, the term -<U(Si - mi)> represents the average energy of the system, where Si is the spin at site i and mi is the local magnetization at site i. This term is equal to 0 because at high temperatures, the spins are randomly oriented and the average energy is 0.

3) It can be meaningful to go past the Onsager term or TAP equations, as these equations are only valid at mean-field level and may not accurately describe the system at lower temperatures. Going beyond these equations can provide a more accurate description of the system and can lead to new insights and discoveries.

I look forward to hearing other thoughts and comments on this paper. Let's continue the discussion!

 

[charng]

First of all, I would like to commend Georges and Yedidia for their insightful paper on expanding around mean-field theory using high-temperature expansions. This technique has proven to be a valuable tool in the study of phase transitions and critical phenomena, and their paper provides a clear and concise explanation of the method.

To address the first question, yes, the Lagrange multipliers in Equation 2 do replace any general external field. This allows for a more general approach to studying the system, as the external field can be varied and the effects on the system can be analyzed.

Regarding the second question, the derivation of -<U(Si -mi)> = 0 in Equation A1.4 can be understood by considering the mean-field approximation, where the interactions between the spins are neglected. In this case, the energy of the system is solely dependent on the local magnetization of each spin, and the average energy can be written as -<U(Si -mi)> = -<U(mi)> = 0, since the mean-field energy is independent of the specific spin configuration.

Finally, the third question raises an important point about the applicability of this technique. While the high-temperature expansion method can provide valuable insights into the behavior of systems near critical points, it is important to keep in mind that it is an approximation and may not accurately capture all aspects of the system. Going beyond the Onsager term or TAP equations may provide a more accurate description, but this may also come at the cost of increased complexity and difficulty in obtaining analytical solutions. Ultimately, the usefulness of expanding past these equations depends on the specific system being studied and the goals of the research.