EXPLORABLES

by Dirk Brockmann

This explorable illustrates one of the most famous models in statistical mechanics: The Ising Model. The model is structurally very simple and captures the properties and dynamics of magnetic materials, such as ferromagnets. It is so general that it is also used to describe opinion dynamics in populations and collective behavior in animal groups.

Press Play and keep on reading.

This is how it works

In the Ising model \(N\) interacting spins (think of little magnetic rods) \(s_n\) with \(n=1,...,N\) are placed on a lattice (here a two-dimensional one). Each pixel in the display corresponds to a single spin.

Each spin can be in one of two orientation states, so \(s_n=\pm 1\) colored black (\(+1\)) and white (\(-1\)), respectively. We can imaging spins either point "up" or "down" for example. Spins interact with their eight nearest neighbors on the lattice. Spins prefer to align their orientation with that of their neighbors and avoid states with opposite sign. In the Ising model this is captured by an energy \(H_{nm}\) between neighboring spins \(m\) and \(n\):

\[ H_{nm}= - \frac {1}{2}\, J\, s_n s_m \]

where \(J\) is a positive constant. So, if \(s_n=s_m\) (aligned spins) the pair-energy is \(H_{nm}=-J/2\) whereas when \(s_n=-s_m\) (spins of oppositve orientation) \(H_{nm}=+J/2\). Therefore, if in a pair of spins of opposite orientation one spin "flips" the energy is lowered by an amount \(J\). The total energy of the system is given by:

\[ H = - \frac{1}{2}J\sum_{\left< n,m\right>} s_n s_m. \]

where the \(\left< n,m\right>\) denotes summing over all neighboring spins on the lattice.

Dynamics

Now we let the system evolve in time. First the system is randomly initialized, each spin having equal odds for each orientation. At each time step a random spin is chosen. The spin is virtually flipped and the expected change in energy \(\Delta H\) is computed.

If the \(\Delta H\leq 0\) the flip is always accepted, the spin is actually flipped, and the energy decreases by \(\Delta H\). Continuing this way the system will eventually decrease its internal energy until no more energy decreasing flip occurs.

If, on the other hand, \(\Delta H>0\) the spin is only flipped with a probability of

\[ P=\exp (-\Delta H / T). \]

The parameter \(T\) is the temperature of the system. This means that if \(T\ll \Delta H\) the spin is almost never flipped, state changes that increase the energy are rarely accepted. If, however, \(T\gg \Delta H\) we have \(P\approx 1\) so effectively any state change is accepted.

Try this

By default the temperature is set to zero. So only states that do not increase the energy are accepted. If you press play, the system approaches a state in which uniform regions of uniform orientation emerge with meandering boundaries. The energy drops quickly to a minimum. Eventually only one orientation will prevail. The magnetization of the system defined by

\[ M = \frac {1}{N} \sum _n s_n \]

will eventually either approach \(+1\) or \(-1\).

Now increase the temperature (slider) to the maximum. In this regime any spin flip is accepted and the pattern is more mixed. The total energy \(H\) and magnetization \(M\) both equilibrate at zero. Now slowly decrease the temperature. Eventually a critical point is reached at which broad regions of uniform orientation emerge with occasional superimposed noise.

Breaking the symmetry

There's an additional slider that controls an external field \(B\). The spins also try to align their orientation with this external field and the total energy is

\[ H = - \frac{1}{2}J\sum_{\left< n,m\right>} s_n s_m - B \sum _n s_n. \]

If you increase the external field a bit, spins are a slightly more likely to flip into the \(+1\) state. This breaks the symmetry of the system and, even if the external field is only small, below the critical temperature the system is likely to go into the corresponding uniform state.

In the Ising model spins can only point into two opposite directions. A very similar and more complex systems is the XY-model that you can explore with the Explorable If you ask your XY.

A poem for Alex Vespignani

This explorable is dedicated to my great idol, my hero, my brother in spirit, to the one and only Alex Vespignani who advertises and praises Complexity Explorables like no other. As a token of my appreciation I composed the following poem for him:


Gronk sulky funk anook daroom,
clunz brooked tooluz far dunk!
oolard the froozed thru vuzz and oom,
eeporzle fnuck a zunk.



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