Electron spin, when sufficiently aligned, catalyzes macroscopic magnetization; this is probably a good summary of this Demonstration. In ferromagnetic material, a quantum-mechanical mechanism, the exchange interaction, predominantly forces electron spin alignment. The spin-discretization inherent in the Ising model together with the Q2R rule employed by the SCA are used to model this mechanism. The electrons are unpaired outer valence electrons of adjacent atoms, and the ferromagnetic material containing them is assumed to be below its Curie temperature; moreover, it is assumed that there is no external magnetic field acting upon the material.

In the energy diagram just below,

is some zero-referenced Fermi level of the ferromagnetic material—in the Demonstration this critical bulk (macroscopic) potential energy is realized at magnetization, and has energetically moved down from 0 potential energy (the initial lattice is comprised of a random spin-align orientation, as in the thumbnail and first snapshot, which is consistent with 0 potential energy and no net magnetization; isdd =0.5 here), and commensurately up from

potential energy. When

is realized, electron-spin aligning activity (which is spin magnetic moment growth equal to

, shown in the plot label) ceases, because nature's potential energy dynamics definitive,

, has been entirely satisfied; the equation states that the forces act in such a way as to lower potential energy. In practice, and in accord with the equation, electrons lower their electrostatic potential energy by aligning their spins, because this maximizes the separation between them as a consequence of the repulsive electrostatic forces acting in the aligned configuration. In turn, this maximized outer electron separation that pervades the lattice gives rise to the beautiful electron spatial symmetry and order characteristic of ferromagnetic material. Once this maximum electron separation is forced,

is satisfied, and ideally spin-magnetic moment growth ceases because

cannot be intrinsically driven any lower toward saturation. Attaining this stable magnetic moment is realized as a form of macroscopic magnetization—that is why electron spin stabilization is this Demonstration's magnetization arbitrator. A plot label

reading of

is perfectly stationary magnetic moment growth; the Demonstration considers

readings of

to be the condition for macroscopic magnetization—this window was determined by lobe prominence observations consequent to quantifying the matrix plot simulation as explained below.

**Bulk Material Energy Diagram**: spin-magnetic moment growth—expansion or contraction

0: random electron spin alignment (ideally,

here)

: magnetization (ideally,

here)

: saturation-complete electron spin alignment (ideally,

here)

For more physics information, see the Wikipedia entries for

antiferromagnetism,

Curie temperature,

diamagnetism,

exchange interaction,

ferromagnetism,

ferrimagnetism,

Ising model,

magnetic moment,

magnetism,

Néel temperature, and

paramagnetism.

The control "animate magnetization" initiates and pauses the local energy checks that either flip or keep unchanged a given electron spin state—in this check, when the automaton senses that flipping a given electron spin state conserves local (automaton neighborhood) energy, the spin flips to its opposite state. Use the "quantify magnetization" control to switch between the matrix plot simulation and the network quantification; the latter shows the causal relationship between electron spin flips and macroscopic magnetization and it quantifies the simulation. With respect to spring electrical-embedding, a flattening of lobes suggests spin alignment randomization, while prominent lobing indicates preferential spin alignment (compare the first and second snapshots). Watch also for lobe interplay, which reflects subtle spatial changes in magnetization; lobe interplay would be consistent with electron spin aggregation in the matrix plot. Use the "reset" control to change the lattice size and initial spin densities.

**Search-Update-Feedback Cellular Automata (SCA) Background**Active cell (AC) and neighborhood (NHD): The SCA NHD can take any shape—curvilinear or hybrid—that satisfies the objective at hand, and the AC need not be a neighborhood center cell.

Perhaps the real beauty of SCA lies in the easy computational accessibility and tracking of the AC or NHD in both time and space, and the rich modeling possibilities that emerge from that computational versatility.

*Mathematica* nesting iterators easily locate AC/NHD in time (see for example the Demonstration "2D Heat Diffusion, an SCA Approach"), and most any method locates AC/NHD in space (this code locks in the spatial possibilities; other methods have been employed; SCA search speed is of consideration here).

Updating: SCA updating is neither synchronous nor sequential, it is random, the motivating assumption being that nature is not perfectly synchronous or sequential in its updating. SCA are very natural in that regard in many applications, owing to the random updating driven by recursion that itself forces near-environment updating for as long as the environment will allow updating to persist.

Complexity and information feedback: By complexity is meant that, be it dynamic or static, artificial or real, which is not tractable by human cognition alone (by alone is meant without the aid of reduction as in SCA quantification or otherwise). Early insights into SCA computing seem to indicate that information feedback and complexity are intimately related, certainly in artificial systems that utilize information feedback computing. This simulated complexity manifests by way of persistently feeding back instantaneous system state information to the given system's operational rules, which mechanism (the feedback) is central to SCA computing. SCA feedback cycles information flow—the system state information randomly updates throughout system-space, practically represented by some SCA active grid, and yet it cycles at the same time, undergoing dynamic refresh in this way. Recursion by default forces the feedback, but there is an aspect of self-call recursion (giving intense feedback) inherent in SCA (see section 2 of the code). In light of the role of recursion, it is a major bonus that recursion is to the discrete problem much the same as

-order differentiation is to the continuous problem (both involve layers of nesting and both manifest some manner of curvature), because the rich history of differentiation models provides some starting-point-guidance for SCA rule construction and establishes boundaries within which recursive-feedback models must compute.

[1] M. R. Wehr, J. A. Richards, and T. A. Adair III,

*Physics of the Atom*, Reading, MA: Addison–Wesley, 1978.

[2] B. Chopard and M. Droz,

*Cellular Automata Modeling of Physical Systems*, Cambridge: Cambridge University Press, 2005.