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13 "ising"- Ising Model in Two Dimensions
"ising" simulates the Ising model in two dimensions. The magnetic field
and temperature are given and the spin orientations of a 30x30 array of
Ising spins are displayed as the system evolves in time after changes in
the applied field and/or temperature. The magnetization vs. evolution time
of the system is displayed graphically, and several parameters (field,
temperature, magnetization, and mean square fluctuation of the
magnetization) are available in numerical form for quantitative analysis.
A number of qualitative phenomena associated with phase transitions are
clearly illustrated. The first pair of frames shows the Ising lattice (red
spins are "up" and blue ones "down") at two temperatures with zero applied
field. On the left the temperature is high and the spins are assigned up
and down orientations at random. On average, an up spin is as likely to
have any neighbor up as down. On the right, the temperature has been
reduced to close to (but above) the critical temperature. (Tc) The spins are
now arranged in clusters of up and of down spins. A down spin is much more
likely to have another down spin as neighbor than up. The size of the
clusters increases as the critical temperature is approached.
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Ising spins at high T Ising spins at T near Tc
The program also prints a graph of magnetization versus time for a number
of cycles of the Monte Carlo simulation. The next pair of figures gives
these graphs for the same two temperatures for 300 cycles of the
calculation. Note the different scales for the two graphs. Near the
critical temperature (on the right) the magnitude of the magnetization
fluctuations is much larger than at the high temperature (critical
fluctuations); and the fluctuations take a much longer time to die away
(critical slowing down) when the temperature is near the critical
temperature.
M(t) at high T M(t) at T near Tc
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In the following movie the system is equilibrated in a small magnetic field
at a temperature below the critical temperature. The field is then
reversed and the movie started. The initial state is now NOT the
equilibrium state since most of the spins are oriented against the new
value of the applied field. Thermal fluctuations in the spins are evident,
small regions (nuclei) of reversed (favorably oriented) magnetization form
and disappear. Finally a fluctuation produces a nucleus of reversed
magnetization of size greater than the critical size and it grows until the
sample is (nearly) fully magnetized in the reverse direction. The magnetic
field is reversed again and the process repeated. The first of the two still frames shows a blue magnetization with small fluctuations to red, the orientation parallel to the applied field. The nucleus at the center of the field is about the size of the critical nucleus. In the second frame, this nucleus has grown and the whole field will soon be magnetized red.
movie (228K) . . . . . . .
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The final graphs give the magnetization versus time for another sequence showing nucleation. In the zoomed version on the left you see small fluctuations in the stable magnetization, with zero applied field, from time zero to 700. At time 700 a reverse field is applied, the magnetization is metastable and the fluctuations become larger. The system remains in the metastable state until about time 1200, when a critical nucleus has formed and the magnetization rapidly reverses as seen in the full scale graph at the right.
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