- #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

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.

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.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.

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