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I am working on the Metropolis-Montercarlo algorithm for the square lattice ising 2D. Im running the simulations for a given lattice size, running from low temperature to high temperature, and getting for example, the magnetization vs Temp of the lattice. I get a set of data points that, in my supossition, are adjustable by a sigmoid or the inverse tangent function. I know I just can't fit a dataset of points with a given function. I must have a model behind that function to justify fitting them. Is there any theory (mean field, onsager) that one gets the magnetization vs the temperature?

This are some of my data set, and the sigmoid fitting I tried.

On the Y axis, there is the Spin per lattice site and in the X axis is temperature.

2D square lattice. Size 7x7.

2D square lattice. Size 4x4.

Sorry for the Spanish figures, im on the phone and access the Before You ask, the fluctuations near $"T_c"$ are still there, even when i do a lot of iterations.

Thanks!

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Mean field and Onsager solution alone won't get you very far. The problem is that both results already embody the thermodynamic limit. So, any size effect, is lost forever. On the contrary, when one is trying to fit simulation data, they are certainly affected by size effects and neglecting them may be a serious problem.

In literature there are different ways to take care of finite size effects. Kurt Binder did a lot of work on this topic and looking for some of his pioneering papers could be useful. A concise idea of what can be done is illustrated in this paper. There you could see, one could explore the consequences of the so called scaling of thermodynamic quantities.

However, one has always to be careful to separate formulae which should be valid only at the thermodynamic limit, from formulae which take care of finite size effects.

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