Bryce Kellogg
In his lectures on computation, Richard Feynman discussed a concept for a two state device which he used to demonstrate the possibility of doing certain computations without expending any energy at all. He used this idea, and a physical example of two axis aligned compass needles, to discuss a copy operation that dissipated zero energy. In this project I used standard magnetic dipole equations to analyse Feynman's concept. While his initial argument was not very rigorous, I found that he was basically correct with the minor caveat that the resulting operation is an inverting copy instead of a normal copy as described.
Given a potential well similar to the one shown in Figure 1 we can see that there are two stable states at the two global minimums and that such a device will not change from one state to the other spontaneously. If we wish to copy the (unknown) state of a bi-stable device into another with a known state, as shown in Figure 2 we do so by first moving the model close to the device to be copied into. This will lower one of the two stable states' potential. If we then remove the potential barrier in the middle, the state will change to the model state. This process is illustrated in Figure 3.
Figure 1: proposed bi-stable potential well.
Figure 2: copier and model potential wells.
Fgiure 3: the copy operation.
Feynman's physical example of this was two compass needles positioned near each other such that their magnetic fields interact. He goes on to claim that clearly the two compass needles will not be stable when they are not axis aligned and that there are only two stable states when they are axis aligned as seen in Figure 4. When we want to copy, we apply a magnetic field to the copier, which removes the potential barrier between the two stable states, and move the model towards the copier. This should result in manipulations of the potential like those seen in Figure 5. Thus we have a copy operation.
Figure 4: Two compass needles, only axis aligned configurations are allowed.
Figure 5:Feynman's proposed potential well for axis aligned compass needles.
I tested and verified Feynman's concept of a bistable compass needle storage device by modelling the compass needles as point dipoles and using Equation 1 to compute the total potential energy of the system of compass needles for every combination of angles for the two compass needles. Analysis was done in MATLAB.
We start by modelling the just two dipoles to confirm that there are indeed only two stable states. Figure 6 shows 3D and side views of the resulting potential energy. It can clearly be seen that there are only two stable states corresponding to both needles at either pi or zero compared to the horizontal and that the side plot (corresponding to axis alignment) closely matches the potential energy profile suggested by Feynman.
Figure 6: potential for all angle configurations.
Next we look at the copy operation itself. We begin by moving the model close to the copier and see in Figure 7 and Figure 8 that one of the potential wells gets deeper as the model approaches.
Figure 7: The model has moved closer.
Figure 8: Even closer...
Now that the model has affected the copier potential well, we need to remove the potential barrier to allow the copier to change state. This is done by applying an external magnetic field as shown in Figure 9 and Figure 10.
Figure 9: External magnetic field is applied.
Figure 10: External magnetic field increases in strength.
We can now see in Figure 11 that the path to the desired state is clear and the needles will assume the lowest energy state that corresponds with our model. One thing to note is that in these experiments the model had a configuration of while the lowest potential of the copier is now , opposite that of the model. This makes sense when you think about the configuration for the four compass needles. This makes the resulting operation an inverting copy. Below I show animated plots that show the copy operation.