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Derivation of ${\cal E}_{min}$ from the Second Law of Thermodynamics

Instead of a coin, the thermodynamic system we will consider is a single molecular machine. The Second Law of Thermodynamics must apply here, if it is to apply at all [McClare, 1971]. Therefore we may write the Clausius inequality [Castellan, 1971,Weast et al., 1988,Atkins, 1984]:

$$dS \geq \frac{dq}{T} .$$ (13)

That is, in a small volume that exactly encloses the molecular machine, if a small amount of heat energy dq enters the volume, then the entropy of the molecular machine must increase (dS) by at least $\frac{dq}{T}$, where T is the absolute temperature in $\mbox{K}$.

Molecular machines operate at one temperature [Schneider, 1991], so T is a constant and we may integrate (13) to obtain:

$$\Delta S \geq \frac{q}{T}$$ (14)

where q is the total heat entering the volume. By substituting (11) into (14) and rearranging, we obtain a relationship between the information R and the heat q:

$$k_{\mbox{\scriptsize B}}T \ln(2) \leq \frac{- q}{R} \;\;\;\;\; \mbox{(joules per bit)} .$$ (15)

The interpretation of this equation is straightforward. There is a minimum amount of heat energy:

$${\cal E}_{min}= k_{\mbox{\scriptsize B}}T \ln(2) \;\;\;\;\; \mbox{(joules per bit)}$$ (16)

that must be dissipated (negative q) by a molecular machine in order for it to gain R = 1 bit of information. More energy than ${\cal E}_{min}$ could be dissipated for each bit gained, but that would be wasteful. This derivation, which consists of definitions and simple rearrangements, shows that (15) and (16) are just restatements of the Second Law under isothermal conditions.