# Tag Info

17

My question is, when I run a simulation with $N$ particles and I track the Hamiltonian per particle $(H/N)$ and the magnetization per particle $\left(\sum _i s_i /N\right)$, with $K$ values going from $0.1$ to $0.7$ in increments of $0.1$, how do I spot the region of the critical coupling constant? There has to be a signature of the critical point that is ...

14

As Anyon correctly pointed out, there is no phase transition at finite temperature in 1D. In 2D there are a number of different ways to identify the phase transition (I'm assuming you're using Monte Carlo). You could directly look at the magnetization, but a more reliable signature is the magnetic susceptibility, $\chi_m (K)$, which is strongly peaked around ...

11

A locally interacting system displaying a continuous phase transition belongs to a universality class that is determined solely by the system symmetries and dimensionality. Drawing from Wikipedia's list (itself mostly based on Ódor's paper) and this answer from Physics SE, here's a partial list of universality classes and critical exponents: \begin{array}{|...

8

I think Anyon's and taciteloquence's answers are perfect. I just want to add an emphasis on the following fact that frequently leads to confusion for beginners. The formal definition of the magnetization $$m = \frac{\sum_i s_i}{N}$$ has a symmetry that $\mathrm{Prob}[m=+m_0]=\mathrm{Prob}[m=-m_0]$, since the energy of a particular ...

5

Off the top of my head, I can think of two reasons to be concerned about simulations near a phase transition, although I'm only really well-qualified to talk about one of them in detail. The one I know less about is finite size effects. There are many cases where the limited size of the system prevents it from behaving as it would at the thermodynamic limit. ...

5

Enthalpy and entropy are the temperature dependent terms in free energy from which enthalpy is a dominant term mainly at lower temperature and entropy is dominant at higher temperature (because randomness is more at higher temperature). This temperature effect can be considered by utilizing the heat capacity at constant pressure (Cp). Phonopy code is there ...

4

I felt that it's important to note that not all universality classes are determined solely by spacial dimension and symmetry, even when the interactions are local. The easiest model that exemplifies this is probably the Ashkin-Teller model in 2D. It has a continuously varying (thus infinite variety of) critical exponents (universality classes) depending on ...

4

Thermo-Calc's DGM is the normalized parallel tangent driving force per mole of components. To be able to compute the driving force of a phase, it must be set to "dormant" status so that it cannot become stable, but the driving force can still be computed. Here's an example macro of how this could be done in Thermo-Calc: @@ These commands can be ...

4

Thermodynamics There is no problem with simulating a phase transition. There is only a problem with calculating thermodynamic averages during a phase transition. Short simple answer: I don't know anyone who has ever had a use for a property that corresponds to part one phase, part another phase. There are plenty of needs for the properties of mixed phase ...

3

This seems to be a typo in the paper. As they mention in the introduction, austenite is the "hot" phase of this compound, so it should form on heating martensite. In their experiments, they heat samples with varying compositions to determine at what temperature the austensitic transformation starts/finishes. The martensite temperatures should ...

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