Theory of fluctuations in disordered systems

In summary, the conversation is discussing the use of symmetry in an interaction term in a document by Pierfrancesco Urbani. The specific question is which symmetry is correct when using the interaction term < f(na,nb) f(nc,nd) f(ne,nf) >, with the suggestion that c = d and b = f, but it is unclear based on previous pages. Another reference is mentioned, which also discusses the use of symmetry in the interaction term. The specific symmetry used is symmetrizing Eq. (153) with respect to the exchanges a ↔ b, c ↔ d, e ↔ f, ab ↔ cd, ab ↔ ef, cd ↔ ef, but it is still unclear why the exchange of b
  • #1
giulio_hep
104
6
TL;DR Summary
In the computation of the dynamic exponents from the
cubic expansion, I'm asking clarifications and a clear explanation about the interaction term and what are the symmetries in the monomials
I'm reading the https://www.phys.uniroma1.it/fisica/sites/default/files/DOTT_FISICA/MENU/03DOTTORANDI/TesiFin26/Urbani.pdf at paragrph 4.6.2 "The interaction term".

They write a right hand side:

< f(na,nb) f(nc,nd) f(ne,nf) >

and they want to use a symmetry, for example they assume that <na3ndnf2> is equal to <na3nb2nc>

It looks like c = d and b = f at first sight, but which is the correct symmetry really? I can't find an explanation in the previous pages: any idea? Thanks
 
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  • #2
Indeed my doubt is somehow reinforced from what I read in "Static replica approach to critical correlations in glassy systems" (same authors, among which again Pierfrancesco Urbani and this year's Nobel Prize, Giorgio Parisi) ref. 12A540-22 paragraph "C. Expression of λ in HNC", page 23, where it is written:

Here we have again to symmetrize Eq. (153) with respect to the exchanges
a ↔ b, c ↔ d, e ↔ f, ab ↔ cd, ab ↔ ef, cd ↔ ef
because these have been used explicitly to derive Eq. (97).

Again, while the exchange of c with d would make sense for symmetry of f(.,.), my intuition was also able to get the exchange of ab with ef, but it is still a mystery for me to understand the exchange of just b with f... or (put together) of df ↔ cb
 

Related to Theory of fluctuations in disordered systems

1. What is the Theory of Fluctuations in Disordered Systems?

The Theory of Fluctuations in Disordered Systems is a branch of physics that studies the behavior of systems with disorder or randomness, such as disordered materials, glasses, or spin glasses. It aims to understand how fluctuations, or small changes, in the system's properties can affect its overall behavior.

2. How does disorder affect the behavior of a system?

Disorder can significantly alter the behavior of a system. In ordered systems, the behavior is predictable and follows well-defined laws, but in disordered systems, the behavior can become more complex and unpredictable due to the presence of random fluctuations. These fluctuations can lead to different outcomes and make it challenging to accurately predict the behavior of the system.

3. What are some real-world applications of the Theory of Fluctuations in Disordered Systems?

The Theory of Fluctuations in Disordered Systems has many practical applications, including in the study of materials with disorder, such as glasses, polymers, and biological systems. It is also used in fields like statistical physics, condensed matter physics, and materials science to understand the behavior of complex systems and develop new materials with specific properties.

4. How do scientists study fluctuations in disordered systems?

Scientists use various theoretical and experimental techniques to study fluctuations in disordered systems. These include statistical mechanics, computer simulations, and experimental methods such as X-ray scattering and nuclear magnetic resonance. By combining these methods, scientists can gain a better understanding of the behavior of disordered systems and make predictions about their properties.

5. What are some current challenges in the Theory of Fluctuations in Disordered Systems?

One of the main challenges in this field is to develop a comprehensive theory that can accurately describe the behavior of all types of disordered systems. Another challenge is to understand the role of fluctuations in phase transitions, where a system undergoes a sudden change in its properties. Additionally, the development of new experimental techniques and computational tools is crucial for advancing our understanding of fluctuations in disordered systems.

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