# Ising model - Which quantities (e.g. energy, pressure, magnetization) are observables? How are they measured?

This is a question about bridging reality to theory after attending a maths lecture on the ising model. I assume that the term "ising model" may be a bit ambiguous, but I think that we considered the simplest possible setup (see the footnote for more details).

I assume that we can not measure the microstate of the system (by knowing the microstate I mean that we know the spin of each lattice site). Since the energy, pressure and (total) magnetization were defined in terms of the microstate$$^1$$, it is not clear that they are observables. What quantities are observables (I may not have listed all base quantities) and how are they measured/defined in an experiment? What experiment(s) does the Ising model describe?

$$^1$$ Let $$\Lambda$$ be the set of all lattice sites, which is identified with a finite subset of $$\mathbb Z^d$$. In my lecture a microstate $$\omega$$ was defined as a function from $$\Lambda$$ to $$\{-1,1\}$$, $$E(\omega):=-\sum_{\{i,j\}:i\sim j}\omega_i\omega_j-h\sum_i\omega_i,$$ is its energy and $$M(\omega):=\sum_i\omega_i$$ its total magnetization. The pressure was defined in terms of the value of the partition function, which was defined in terms of the energy.

• The Ising model is a toy model. It doesn't refer to an actual physical system. We can solve the model numerically, in which case we know the microstate and we can calculate any observable we wish. The only difficulty with the model is the extrapolation to infinite size, which is numerically impossible. I also don't think we require knowledge of the microstate. The system is homogeneous, i.e. if we know the behavior of any one lattice point and the correlation functions, then we are done, aren't we? Am I missing something? Commented Apr 22, 2023 at 0:14
• "The system is homogeneous, i.e. if we know the behavior of any one lattice point and the correlation functions, then we are done, aren't we?" - AFAIU the Gibbs probability measure tells us that the probability of the system being in the energy-minimizing state converges to $1$ in the limit $T\to 0$. So I don't think that you can assume that all spins are aligned with the field. Commented Apr 22, 2023 at 7:26
• Since you mentioned the correlation functions: As you can see from the formula for the energy, we only considered the "trivial" case... Commented Apr 22, 2023 at 7:28
• Intuitively I am fairly sure that at T=0 the spins have to be aligned. You are correct, of course, that the real system can get stuck in a false ground state. Real life examples of that are glasses... including spin glasses. That's simply not the correct ground state solution. It can be ruled out even numerically fairly easily, though. All we have to do is to flip a couple of spins to find a slightly lower energy state. Commented Apr 22, 2023 at 13:41
• Like I said, this is a toy model. It doesn't exist in nature, so we don't have to concern us about actual measurements. In a real magnetic materials one can measure individual spins on the surface with magnetic force microscopy and similar techniques and bulk correlation functions were usually investigated with neutron scattering experiments, if I remember correctly. My solid state days were over 40 years ago and I hated them, so I can't guarantee that I am not talking nonsense here. There is a ton of experimental literature, though. Commented Apr 22, 2023 at 13:57

The Ising model is a lattice model that can be mapped to various physical systems:

• Spin systems: in the original implementation is a model for ferromagnetism

• Lattice gas: if we take spin-up to mean occupied site and spin down to mean "unoccupied site" it is a model for a lattice gas

• Binary solution: If we take spin-up to mean "component A and spin-down "component B", it is a model for a binary solution or alloy

In general the observables are quantities represented by the log of the partition function, its partial derivatives and various combinations of these quantities. Some examples are energy, pressure, chemical potential, heat capacity etc.

• Thank you for the overview (+1)! Regarding the last paragraph: Is the partition function an observable? Commented Apr 21, 2023 at 22:41
• The log of the microcanonical partition function is entropy. The log of the canonical partition function is free energy. They can be measured within an additive constant. Commented Apr 21, 2023 at 23:34
• More precisely, $\text{entropy}=k_B\ln\Omega$, $\text{free energy}=-k_B T \ln Q$, where $\Omega(E,V,N)$ and $Q(T,V,N)$ are the microcanonical and the canonical partition function, respectively. Commented Apr 21, 2023 at 23:38
• "They can be measured within an additive constant." - I am particularly interested in this. Could you please elaborate? Commented Apr 22, 2023 at 13:40
• Entropy, $S(T,P)$ is a function of $T$ and $P$. We calculate changes in entropy by $$\Delta S_{12}=\int_1^2\frac{dQ}{T}$$ where $Q$ is the heat during the process, assuming reversible conditions. The formula calculates differences, not absolute values. By fixing the zero of entropy at some $T_0$, $P_0$, we can tabulate "absolute" entropy just like we tabulate "absolute" elevation from reference state at seal level. Commented Apr 22, 2023 at 14:56