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... Schneider¹ Frederick Cancer Research and Development Center, P. O. Box B, Building 144, Room 469, Frederick, MD 21702. email address: toms@alum.mit.edu

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... earlier!²

The range $F_c = \frac{2}{12}$ was determined by using the Rsim program. See [Stephens & Schneider, 1992].

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... pin.³

Alternatively, each vibrational mode of a molecule can be thought of as corresponding to a pin. The important point here is that there can be parts of a molecule consisting of groups of atoms which move almost independently from the other groups.

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... symmetric.⁴

The x and y axis projections of the polar coordinate point are independent and normally distributed: f(x) = e^-x2 and f(y) = e^-y2. The combined distribution is f(x, y) = e^-x2 e^-y2 =e^-r2, where x² + y² = r². This is circularly distributed. See [Schneider, 1991a] for further details.

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... energy?⁵

At this point, an anonymous reviewer asked why the problem was not phrased in terms of ``the resetting of the demon'' because this is said to be the important energy dissipating step by people concerned with the limits of computation [Leff & Rex, 1990]. To ``reset'' usually means to set the state of a binary device to 0. That requires choosing one state from two since one could alternatively ``reset'' to 1. Actually, ``resetting'' is a particular case of first adding energy to a device and then choosing a sub-state as the energy dissipates into the surroundings. This is what molecular machines do, but the term ``reset'' generally does not fit well with molecular machines operations. For example, molecular biologists never say that ribosomes ``reset'' themselves when they bind to their sites. It is more sensible to speak of them as making a choice or decision. Molecular biology forces us to use this more general paradigm.

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... decrease.⁶

If H_after = 0, equation (17) collapses. Unfortunately, this has fooled many authors into thinking that equation (16) gives the information. Because of equation (18), they then think that entropy is proportional to information. Thus a measure of ``disorder'' becomes associated with ``information'' and the confusion is complete. To avoid this widespread error, always measure information as a decrease in uncertainty.

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... law.)⁷

Landauer's statement that ``throwing away information requires dissipation'' [Landauer, 1991] contradicts this form of the Second Law. Suppose we have a binary device which can be either in a 0 or a 1 state. Let us define ``erase'' and ``reset'' to mean to set the device to 0, starting from either the 0 or 1 state. Since the device is stable in its initial state, whether 0 or 1, we must add some energy to it to alter that state. This is called ``priming'' the machine [Schneider, 1991a]. From the excited state, the device is guided by pre-set inputs to fall into the 0 state, and hence it must dissipate energy to the surroundings. For example, to change the state of a coin requires that we add some energy to rotate it. Then the coin can only come to rest again by giving up its rotational energy. (Do not be fooled into thinking that a coin in the 0 state need not go through this process because ``you'' can see it is in that state and save yourself the trouble. Remember: you are not involved in the actual operation of molecular machines, and even if you were, your seeing, thinking and acting require energy dissipation! Specifically, to know that the state is 0 requires a reading operation.) Thus this second step of erasure looks exactly the same as selecting a sub-state, which is equivalent to an information gain of the device, by the precise definition we use [Schneider et al., 1986,Schneider, 1991b]. This is not really so odd as it may seem, since we must do exactly the same thing to set the device to 1. ``Erasing'' requires the same process as filling a memory with whatever information patterns we choose. It is easy to demonstrate this--simply switch the labels on each flip-flop so that they all register zero. Thus ``erasing'' consists of two steps: priming and setting. Information in a memory is lost at the priming step, and not at the energy dissipation step. This is a direct restatement of the Second Law because energy must be absorbed by the memory to destroy the information there. The energy is then dissipated to set the device and gain different information. Landauer's confusion came about because he lumped the priming and the setting operations into a single step, and because he did not take care to rigorously define what he means by ``information''. We can avoid the confusion by always describing these steps explicitly, and by defining ``information'' as a difference in the uncertainty state function [Schneider, 1991b]. Similar arguments apply even when the two states are at different energy levels, as in a transistor [Mead & Conway, 1980].

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... conditions.⁸

The fundamental importance of equation (26) is not widely recognized. Authors often loosely use the phrase `` $k_{\mbox{\scriptsize B}}T \ln(2)$'' or even ``approximately $k_{\mbox{\scriptsize B}}T$''. This leads to two difficulties. First, using disconnected mathematical phrases has led people to overlook the idea that they are part of an important equation. Further, this equation has units: joules per bit. If one chooses base 10 for measuring choices, one obtains ${\cal E}\geq k_{\mbox{\scriptsize B}}T \ln(10)$ joules per digit. For making calculations, it is necessary to keep the units in mind.

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... configuration.⁹

These steps could be linked to save energy.

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... operation.¹⁰

Feynman only counted output gates.

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... Easterns''.¹¹

The originator of the method was E. M. Southern [Southern, 1975]; the later names are whimsical.

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