Magnetization Using an Ising Model Based on a Search-Update-Feedback Cellular Automata (SCA) Approach

Electron spin, when sufficiently aligned, catalyzes macroscopic magnetization; this is probably a good summary of the 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 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 via the network view and lobe prominence observations as explained below.

: 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 "trigger | pause-flip spin state" 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. Toggle the "matrix pot versus network" control to switch between the matrix plot view and the network view; the latter shows the causal relationship between electron spin flips and macroscopic magnetization, and in no small way it quantifies the simulation. With respect to network spring electrical-embedding, a flattening of lobes suggests spin alignment randomization, while prominent lobing indicates preferential spin alignment (compare the second and third 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 view.

Perhaps the real beauty of SCA, programmatically at least, lies in the easy computational accessibility and tracking of the center cell (cc) or neighborhood in both time and space, and the rich modeling possibilities that emerge from that computational versatility. Mathematica nesting iterators easily locate cc/neighborhood in time, and any function like "newpos" in this code easily locates cc/neighborhood in space. Another major characteristic is SCA updating, which is neither synchronous nor sequential but 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. Thus SCA function well as complexity simulators through their random updating and their dynamic and versatile neighborhood overlay capabilities (in a sense SCA are like specialized computers). Further contributions to complexity simulation come by way of feedback, which is fundamental to SCA computing. Feedback cycles the information flow—the information continually updates throughout the active grid as usual, and yet it cycles at the same time—this is a picture of dynamic refresh. Recursion by default achieves this feedback, but there is an aspect of self-call recursion (giving intense feedback) inherent in SCA. Moreover, it is a bonus that recursion is to the discrete problem much the same as -order differentiation is to the continuous problem because the rich history of differentiation models may serve as boundaries which recursion models must not violate—both involve layers of nesting (iterative enfoldings for recursion and change in change change... for n^th-order differentiation), and both manifest some manner of curvature—the beauty and in some respects advantage of recursion is that in the high-speed, discrete computing setting, this curvature becomes a "curvature on the fly," so to speak; that is, recursion, and by association SCA through recursive feedback, allow one to simulate complex movement. It would seem to follow that certain things that differentiation approaches can model, recursion approaches should be able to model vastly better, and that which differentiation approaches cannot model, recursion approaches might well be able to model. Finally, the search for the next update location is, as suggested by the updating comments, achieved by way of a sort of random walk.

References

[1] M. R. Wehr, J. A. Richards, Jar., 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.