Introduction to the Ising Model

Introduction to the Ising Model

In class, we examined thermodynamic equilibria and phase behavior and linked them to the statistical mechanical concepts involving sampling over the many microstates of an ensemble. The Ising Model simulation will illustrate the microscopic view in greater detail. At the macroscopic level, we describe our system by coarse-grained variables, such as the thermodynamic quantities N,V, T, etc, but at the microscopic level, we can distinguish the range of different microstates that make up the ensemble. As such, the Ising Model is used to understand the microscopic origins of phenomena such as critical behavior (spontaneous broken symmetry, divergent correlation lengths, divergent susceptibilities such as the heat capacity), or for first-order phase transitions, nucleation and growth, and domain coarsening.

Perhaps the most studied model in statistical mechanics, the Ising Model is a simplified model for describing ferromagnetism, but also describes liquid-gas transitions as well as mixing-demixing transitions in binary alloys. One of the deep insights emerging from the past few decades of statistical mechanical research is that these three systems (among others) show the same universal critical behavior, and so the study of one model tells us general information relevant to all three systems.

In your writeup, record your observations, notes, answers to questions, and any independent experiments you decide to conduct.

Introduction to Monte Carlo simulation

The simulation method we use to investigate and visualize the Ising model is called "Monte Carlo simulation". In contrast to Molecular Dynamics (MD), which numerically implements Newton's laws of motion, Monte Carlo (MC) is concerned only with generating microstates with their probabilities as found in an equilibrium ensemble. Often, the ensemble is the canonical ensemble, and so temperature is held fixed, while microstates are sampled according to their Boltzmann weight.

Furthermore, only the configurations (positions) are sampled, neglecting the momentum degrees of freedom (since the momenta obey the Maxwell-Boltzmann distribution, and are uncorrelated with the spatial configurations). This means that MD is generally to be used when seeking dynamical quantities (such as diffusion rates, etc.), while MC (like MD also) can be used to study equilibrium systems. The advantage of MC is that since microstates are not sampled according to Newtonian time evolution, it may be possible to reach equilibrium much quicker than through "natural" dynamics.

To help visualize the difference between MC and MD, click on this page of Java applets, or the local mirror server. The top pair of applets under "Particle Dynamics" shows an MC simulation of hard disks next to an MD simulation of Lennard-Jones (Van der Waals interaction like) disks. Read the descriptions and try both, and record what you observe to be differences between the two simulation methods. Although the two systems do not have exactly the same interaction potential between disks, they are qualitatively very similar, and both are very good descriptions of the packing of atoms in a fluid.

Questions: Which of the two simulated systems above reaches equilibrium faster? If MC and MD are used to simulate the same interaction potential (say LJ disks), would we be able to tell whether the simulation is MC or MD just by looking at a single snapshot?

In the MC simulation of hard disks, the microstates evolve in (simulation) "time" by attempting random changes to the current microstate. For instance, a hard disk is selected at random, and then displaced by a small random vector, and this "new trial microstate" is then either accepted (if there are no overlaps) or rejected (if there are overlaps).

Exploration the Ising Model

Just below the two "Particle Dynamics" simulations is a MC simulation of the Ising Model, which has two types of squares: spin up (green) and spin down (black). There is an interaction energy between neighboring spins that favors two spins pointing in the same direction (either green next to green, or black next to black). At high temperatures, this like-like energy is overwhelmed by thermal energy, and the grid randomizes between green and black, while at low temperatures, the energetic interaction dominates, and we see large domains forming, which eventually take over the whole grid (macroscopically settling into one of two coexisting phases).

For the Ising model, MC evolution occurs by picking a spin (a square), and attempting to generate a new trial state by flipping that spin from green to black or vice versa. Depending on the change in energy of the system due to spin flipping, the new trial state is either accepted or rejected. The probability of acceptance is chosen such that the steady state distribution of microstates is that given by the equilibrium ensemble. The enforcement of this last condition is sometimes called detailed balance.

There are two controls on the Ising applet: temperature and field. (Leave the menu tabs on Square Lattice and Ferromagnet.) The field controls the energetic bias for black or green, and is initially set on the neutral setting of 0.0 field. The temperature is initally set at the critical temperature Tc = 2.2691. Below the critical temperature, the system on a macroscopic scale prefers to be predominantly one or the other of the two colors (black or green). The black and green "phases" are separated by a first-order phase transition (like a liquid and its vapor below its critical point). Above the critical temperature, the system will be a mixture of black and green.

Try this: Leaving the field at zero, vary the temperature with the slider bar above and below the critical temperature. Try dropping the temperature slowly. Try dropping it quickly. What phenomena do you observe? What behaviors do you see that are analogous to the liquid-gas transition? At high temperatures, is it energy or entropy that drives the phase to be random? At low temperatures? Explain.

The behavior seen is characteristic of an interacting system, and is to be contrasted with the comparatively uneventful behavior of an ideal (non-interacting) system. The fact that each spin interacts only with its nearest-neighbors is a feature shared with the polypeptide molecule undergoing hydrogen bonding. Another feature that is shared is that of nucleation. In the Ising Model, this can be seen at temperatures below the critical temperature, when there is a barrier between converting the green phase to the black phase (and vice versa). This barrier will exist even if there is a field on that biases black spins. Occasionally, because the system is constantly fluctuating, a patch of black spins may form a large enough cluster that it will continue to grow and turn the entire system from green to black.

Question: In what ways is this like cooling water below 0°C, and waiting for ice to nucleate?

Try this: With the field initially at zero, drop the temperature down to about 1.0. Wait for the system to become completely one color (say green). If this takes too long, you can nudge it along by carefully controlling the field. Once it is one color (green), we are ready to try to nucleate a cluster which will turn the color back to the other color (black). To do this, turn the field back to the other side of zero, to a magnitude of no more than 0.6. What do you see? Repeat this the same temperature, and at a different temperature (a different field may be required).

Nucleation behavior is seen when the system changes from one (metastable) state to the preferred equilibrium state, but in doing so, it requires the formation of a "nucleus" through random fluctuations, that then allows further rapid growth.

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