Path Dependence

Like most geeks my age, I have owned a number of electronic calculators during my lifetime. My favorite was a Hewlett-Packard model 25 that employed reverse Polish notation, had dozens of scientific functions, and was mildly programmable. It was also nearly indestructible. And it had just the right amount of key click.

On the evidence, a lot of modern product engineers don't understand key click. The calculator that replaced my HP-25 was an HP-41C, which retained RPN, had considerably more functions, and had much more programmability. It also had a liquid crystal display that consumed considerably less power than the LED display of the HP-25. Offsetting all this was the fact that it was much less physically substantial and had poorer key click.

The microwave my wife just installed above our new stove has no key click at all. In fact, it has no buttons. You have to feel around on the touch pad to find just the right spot to press to activate the desired function. When you do find the right spot, there's a significant chance that the function will register twice. This usually means starting over, given the rather significant difference between cooking popcorn for 2 minutes 30 seconds versus 23 minutes. A calculator keypad that acted like this would be useless.

Key click is the tactile sensation that tells you that you have pressed the key. You feel a click as the key goes down, and another as the key comes back up. If the key is properly designed, it reliably registers exactly one press of the key for each click-unclick, which provides important tactile feedback when you are trying to enter a long string of digits accurately. It helps ensure that you don't miss a digit by pressing the key too lightly, or inadvertently enter a digit twice by "bouncing" the key.

Valuable as it is for calculator keypads, key click isn't an absolute requirement for keyboards. The Dell keyboard I'm presently typing on doesn't have any noticeable key click, but it is still reasonably satisfactory. It still supplies the most essential benefit of key click, which is hysteresis.

Hysteresis means that the finger pressure necessary to put the key into the "pressed" state is considerably greater than the pressure at which the key returns to the "unpressed" state. This means you have to let up on the key quite a bit before it becomes "unpressed", and then must press down quite a bit before it becomes "pressed" again. There is no twilight zone of "almost pressed" where slight changes in pressure flip the state of the key back and forth rapidly. The importance of this should be obvious. Good key click is simply a way to achieve hysteresis electromechanically.

Hysteresis also describes a number of other physical phenomena characterized by

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path dependence. It is a sometimes annoying characteristic of iron-core transformers, for example. When the electric current in the coils is flowing in one direction, it generates a magnetic field that is concentrated in the iron core. Ideally, when the current is reversed, the magnetic field should reverse as well — but the iron core "remembers" its previous magnetization, so the magnetic field reversal lags slightly behind the current reversal. If the current simply drops to zero, the core remains slightly magnetized. Electrical engineers usually hate these kinds of phase shifts, though they are occasionally useful. As a result, high-frequency transformers generally use ferrite cores that minimize hysteresis.

Path dependence itself is a broader concept. Anyone who has taken a course in thermodynamics has dealt with path dependence. It is a major concern in material modelling, where one must make a distinction between state variables and history variables.

State variables are variables that depend only on the current density and internal energy of a material. Pressure and temperature are state variables. They do not depend on how the material got to its current state. By contrast, the cumulative work done by the material, or the cumulative heat flow between the material and its surroundings, depend very much on how the material got to its current state. Each path from point A to point B leaves the material with the same pressure and temperature at point B, but the work done along the way, or the total heat flow into the material, is path-dependent.

Physicists noted that, while the heat flow was path-dependent, the ratio of heat flow to temperature, integrated over the path, was not path-dependent. This quantity received the name entropy and was recognized as a new state variable, but its significance was not immediately understood. It was some time before physicists came to understand that the entropy was a measure of the internal disorder of the system.

Physicists also came to recognize that the entropy could be increased by processes other than heat flow at a well-defined temperature. Consider an insulating container with a thin membrane down the center, gas on one side of the membrane, and vacuum on the other. If the membrane is ruptured, the gas rushes into the vacuum half of the chamber. If the gas is ideal, its temperature remains unchanged, but its volume has doubled. This implies an increase in entropy, even though no heat has flowed into the chamber. This seeming paradox is resolved by the realization that expansion into a vacuum is an irreversible process, and the heat flow over temperature formula for entropy only applies to reversible processes.

Some quantities cannot unambiguously be classified as either state or history variables. Ideally, the shear stress in a solid depends only on the final strain state, and not the path by which the strain state was reached. But it is a theorem of equilibrium thermodynamics that an equilibrium state cannot sustain shear stresses — only hydrostatic pressure. It follows that there must always be some slight path dependence in the shear stress, manifest as a slow creep in the material that eventually relieves the shear stress. This kind of slow creep underlies a lot of plate tectonics.

In fact, all history variables disappear when a material reaches thermodynamic equilibrium. Since all materials tend towards thermodynamic equilibrium, we are driven to the conclusion that all history is transitory and will eventually be forgotten.

But the process can be very slow. There is a popular urban legend that stained glass in some medieval cathedrals has devitrified; that is, the glass, which is essentially a supercooled liquid, has slowly crystallized, this being its equilibrium state. My understanding is that devitrification is a myth, though it is true that some glass has slowly flowed so that the panes are thicker at the bottom than at the top. A few hundred years is not long enough for stained glass to devitrify, though obsidian, which is natural glass produced by volcanoes, does devitrify over periods of thousands of years.

Likewise, Steven Hawkings recently lost a bet on the subject of information loss in black holes. Hawkings theorized that all history variables except mass and electric charge disappear in a black hole. That is, black holes are path-independent. It now appears that some of the information in a black hole takes a very long time to
disappear, consistent with the laws of quantum mechanics and with the behavior of history variables in more conventional systems.

I have discussed a form of political path dependence. The outcome of a Presidential election is not wholly defined by which candidate is elected. It also has a strong dependence on the manner in which the election took place.

This is true of human culture in general. The state of a society is not determined wholly, or even primarily, by the circumstances in which it finds itself. There is considerable dependence on the history of the society.

This makes it all the more unfortunate that history is so badly taught in our schools. I suspect that, if I took a poll of my readers, many would report that their high school's football coach was also their history teacher. I'm not entirely opposed to school sports, but it's not been my experience that there was a lot of overlap between the students who were good at sports and the students who were good at scholarly pursuits. There are exceptions, of course, including my own brother, one of the honors graduates in my high school class, and Steve Young, Esq. But almost every rule has exceptions.

I was fortunate to have had a freshman history teacher who was not associated with the sports program. He was a former Marine captain who took his history seriously. He didn't say a lot about Vietnam, other than that losing more than 10% of your men in a single engagement was grounds for relief. (This naturally led to some unkind, and almost certainly unwarranted, speculation about how he came to be our high school history teacher.) But he taught a lot of German and Russian history, and did it well. His approach to Russian history was "Know your enemy." This was in 1977, after all.

We need more of that kind of history teacher.

© 11 September 2004 by Kent G. Budge