Negative Critical Correlation Length Exponent (Nu)

In summary, the speaker is seeking input on their project involving a mathematical model of a statistical system undergoing a critical transition. They introduce a gradient to investigate the critical point and calculate the correlation length exponent which is found to be negative. This is possible but it could also indicate that they have not reached the critical point yet.
  • #1
mhsd91
23
4
Hi everyone,
I've encountered a curious problem I just can't figure out, and any input would be much appriciated!This is a personal project I'm working on, and as far as I know, there is no one else working on exactly the same. However, the computational study of critical phenomena is quite popular and I hope some of you can enligthen some apsects about the critical correlation exponent for me:

I have a mathematical (Monte Carlo) model of a statistical system on an [itex]L \times L[/itex] square grid, which undergoes a (critical) transition when adjusting some dimensionless parameter [itex]\tau[/itex]. Some really crude testing shows that the system is DISlocalized for [itex]\tau<0.6[/itex] and localized for [itex]\tau>0.6[/itex].

Then, to more accuratley investigate the critical point [itex]\tau_c[/itex], somewhere around 0.6, I've introduced a gradient in [itex]\tau[/itex] across my system, which effectivley creates a propagating front I may analyse. The simulations enables me to calculate/estimate [itex]\tau_c[/itex] as one can actually observe the front becoming localized at some point, and then by repeating this for multiple system sizes, I extrapolate the sys. size to infinity (finite size scaling) and find [itex]\tau[/itex]. And here comes my problem:

I try to deduce the correlation length exponent [itex](\nu)[/itex] of the transition. It is easy to find in litterature (e.g. considering percolation, the Ising Model, or similar systems) that this critical exponent (at the transition point) is related to the emerging front by

[itex]
w \sim L^{-a},\quad a=\frac{\nu}{1+\nu},
[/itex]

as the system size goes to infinity. Here, [itex]w[/itex] is the width if the front (equal to the correlation length of the system [itex]\xi[/itex] at the critial point), where we calculate [itex]w[/itex] as the standard deviation of the front on the lattice:

[itex]
w = \sqrt{\frac{1}{n}\sum^{n}_{i=1} (x_i-\bar{x}_{front})^2},
[/itex]

where [itex]x_i[/itex] are the lattice points in the front, and [itex]\bar{x}_{front}[/itex] their mean value. For multiple system sizes, I plot [itex]w[/itex] against [itex]L[/itex] and find a nice power law and a straight line from [itex]\log (w)[/itex] against [itex]\log (L)[/itex] resulting in

[itex]
a \approx 1.87
[/itex]
[itex]
\nu = \frac{a}{1-a}\approx -2.15
[/itex]

This is my issue: I've never encountered a negative critical length exponent before. I don't know if that should even be possible? My first guess is that I'm just close, but not close enough to the critical point such that the exponent doesn't behave properly. However, If this was the case, I would expect the straight line of the log-log plot to be slightly curved. The plot is shown below.

estimate_nu.png


I may provide more details, but this is quite comprehensive already, and I didn't want to make the post too long to read (even though I might already have failed on that point). Anyways, I hope someone is willing to discuss this matter with me, regardless of their expertise in critical phenomena. In my experience, any discussion is always better than none.
 
Last edited:
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  • #2
Thanks in advance!It is possible for the correlation length exponent to be negative. This is because the correlation length is a measure of the size of a region in which correlations are present, and it can be negative if the correlations are anti-correlated, meaning that when two points are close together, they have an opposite effect on each other than when they are far apart. This kind of behavior is seen in some systems, such as the Ising model. However, it is also possible that you simply have not reached the critical point yet. If you are not at the critical point, then the correlation length exponent may not behave properly. For example, if you are too far away from the critical point, then the correlation length may not diverge, and the exponent will not be meaningful.
 

Related to Negative Critical Correlation Length Exponent (Nu)

1. What is the Negative Critical Correlation Length Exponent (Nu)?

The Negative Critical Correlation Length Exponent (Nu) is a measure of the rate at which a physical system approaches criticality. It is typically used in the field of statistical mechanics to describe the behavior of systems undergoing phase transitions.

2. How is the Negative Critical Correlation Length Exponent (Nu) calculated?

The Negative Critical Correlation Length Exponent (Nu) is calculated by analyzing the scaling behavior of a physical system as it approaches a critical point. This involves measuring the correlation length of the system and fitting it to a power law function, with the exponent representing the Nu value.

3. What does a negative Nu value indicate about a system?

A negative Nu value indicates that the system is undergoing a continuous phase transition. This means that as the temperature or other control parameter is changed, the system smoothly transitions from one phase to another without any abrupt changes.

4. How is the Nu value related to the critical temperature of a system?

The Nu value is directly related to the critical temperature of a system. As the temperature approaches the critical temperature, the Nu value will decrease. At the critical temperature, the Nu value will become zero, indicating that the system is in a critical state.

5. Can the Nu value be used to predict the behavior of a system?

Yes, the Nu value can be used to predict the behavior of a system near its critical point. It can provide insight into the type of phase transition that will occur and the critical exponents that describe the behavior of the system. However, it is important to note that the Nu value is an average value and may not accurately predict the behavior of individual particles or molecules within the system.

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