Understanding Critical Exponents and Their Application in Systems

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In summary, the conversation discusses critical exponents and how they can be extrapolated from charts using log-log fitting. It is explained that this procedure works by assuming a linear relationship between the logarithms of two variables, with the slope being equal to the critical exponent. However, this method only holds true near the critical point and may not be accurate if the data is not precise or if the system being studied is too small. The person asking the question is encouraged to provide more information and seek help from others on the topic.
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Talker1500
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Hi,

I've been reading about critical exponents and how they're related in any system. I've seen how, for example, several exponents can be extrapolated from charts using a log-log fitting.

I would like to know how this procedure works exactly, I know it's a silly question but I've been trying to use it to estimate some exponents in the Ising model and I get impossible results, so any help would be appreciated.

Thanks
 
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You question is a bit vague, so let me give two comments that may or may not help:

1) The reason critical exponents are inferred from a log-log plot of A and B is purely mathematical: The assumption is that A = a * B^E, where E is the critical exponent and "a" some proportionality constant. If you take the log of sides of the equation you straightforwardly arrive at log(A) = E * log(B) + log(a). That is, the logarithms of A and B are linearly related with a slope equal to the critical exponent E. So you plot the data, fit the best line through them, and read off E from that.

2) The critical relations hold true only very close to the critical point, a unique state in the phase diagram. If you relate A and B away from the critical point, you do not expect A = a*B^E to hold in the first place. So it is not expected that plotting log(A) vs. log(B) would result in a linear relation.

Oh well, a third comment:
3) As always in physics there is the possibility that your data are not sufficiently accurate or that you are in a regime where the expected relation does not hold true very well (that 2nd point is related to my comment 2, but a bit more general). For example, if you got your data from computer simulations, apart from your simulation being buggy you might have not reached the required accurancy (e.g. not simulated long enough) or have simulated an Ising system that is too small to show the behavior of the "real" Ising system, which is infinitely large (interesting keyword, but somewhat advanced: Finite-size scaling).

If these comments do not help you and you want to devote serious time to the extraction of critical exponents in the Ising model, I recommend you add an explanation of exactly what you were trying to do, what results you got, and why you think they are wrong. In principle, the task should be sufficiently basic for someone on PF to be able to discuss the issue with you.
 

FAQ: Understanding Critical Exponents and Their Application in Systems

1. What are critical exponents and why are they important?

Critical exponents are numerical values that describe the behavior of a system as it approaches a critical point. They are important because they provide insight into the properties of a system at criticality, such as its phase transitions and critical behaviors.

2. How are critical exponents determined experimentally?

Critical exponents can be determined experimentally through various techniques such as measurements of physical quantities, analysis of scaling laws, and numerical simulations. These methods allow for the determination of the critical exponents and their values can be compared to theoretical predictions.

3. What is the role of critical exponents in understanding phase transitions?

Critical exponents play a crucial role in understanding phase transitions as they describe the behavior of a system at criticality. They can provide information about the order of the phase transition, the universality class of the system, and the critical behaviors near the transition point.

4. Can critical exponents be applied to different types of systems?

Yes, critical exponents can be applied to various types of systems, including physical, chemical, and biological systems. They are universal values that describe the behavior of a system at criticality, regardless of the specific properties of the system.

5. How can understanding critical exponents be useful in practical applications?

Understanding critical exponents can be useful in various practical applications such as designing materials with specific properties, predicting and controlling phase transitions in industrial processes, and studying complex systems in physics and biology. They provide valuable insights into the behavior of systems at criticality and can aid in the development of new technologies and materials.

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