The interpretation of free energy as bit-erasure capacity

Our paper discussed in the previous blog post might prompt this question: Is there still a way to use Landauer’s principle to convert the free energy of a system to its bit erasure capacity? The answer is “yes”, which we can demonstrate with a simple argument.


Summary: The correct measure of bit-erasure capacity N for an isolated system is the negentropy, the difference between the system’s current entropy and the entropy it would have if allowed to thermalize with its current internal energy. The correct measure of erasure capacity for a constant-volume system with free access to a bath at constant temperature T is the Helmholtz free energy A (divided by kT, per Landauer’s principle), provided that the additive constant of the free energy is set such that the free energy vanishes when the system thermalizes to temperature T. That is,

    \[N = \frac{A}{kT} = \frac{U-U_0}{kT} - (S - S_0),\]

where U_0 and S_0 are the internal energy and entropy of the system if it were at temperature T. The system’s negentropy lower bounds this capacity, and this bound is saturated when U = U_0.


Traditionally, the Helmholtz free energy of a system is defined as \tilde{A} = U - kTS, where U and S are the internal energy and entropy of the system and T is the constant temperature of an external infinite bath with which the system can exchange energy.Here, there is a factor of Boltzmann’s constant k in front of TS because I am measuring the (absolute) entropy S in dimensionless bits rather than in units of energy per temperature. That way we can write things like N = S_0 - S.a   (I will suppress the “Helmholtz” modifier henceforth; when the system’s pressure rather than volume is constant, my conclusion below holds for the Gibbs free energy if the obvious modifications are made.)

However, even in the case of fixed bath temperature, we cannot naively use Landauer’s principle to divide the free energy by kT to get the erasure capacity. Indeed, the free energy definition of \tilde{A} above is only meaningful up to an additive constant, and most traditional results are about free-energy differences as the properties of the system change. More specifically, when the system evolves from one state to the next, the free energy difference tells us how much work must have been done by the systemOr on the system, depending on the sign.b  : W = \tilde{A}_{\mathrm{initial}} - \tilde{A}_{\mathrm{final}}.

In order to fix the absolute value of the free energy, we want to set it to zero when when the system has equilibrated to the same temperature T as the bath, i.e., when the system has ceased to be useful for powering erasures by exploiting either internal system resources or the system-bath differential. (Recall that generically the free energy is well defined even when the system isn’t internally thermalized and therefore doesn’t have a well-defined temperature.) Thus we want to define

    \[A = \tilde{A} - A_0 = (U-U_0) - kT(S - S_0),\]

where U_0 and S_0 are the energy and entropy of the system if it were thermalized to temperature T. (Note that U_0 and S_0 cannot be trivially inferred from macroscopic variables of the system when it’s in an arbitrary state since they depend on material properties like the heat capacity.) In particular, this free energy does not vanish when the system is in its ground state, which you might have thought if you took the common definition \tilde{A} = U - kTS to be meaningful in absolute value rather than just for differences. This makes sense because we can extract work (and hence perform bit erasures) when the system is hotter or colder than the bath.

With our new definition it’s easy to check directly from the 1st and 2nd law that Landauer’s principle can be used to determine the erasure capacity of the system-bath combination. When brought from some initial state to a final (bath-thermalized) state, the system’s entropy increases by S_0 - S. If N bit erasures are made on a memory tape, the tape’s entropy changes by -N bits. So the total amount of entropy pushed into the bath is (at least) N + S - S_0 by the 2nd law, requiring minimum energy (N + S - S_0) kT. The latter quantity lower bounds U_0-U, the system’s internal energy change, by the 1st law. Re-arranging that bound defines the erasure capacity: N \ge A / kT.

To see this achieved explicitly in the case of a system initially thermalized to some different temperature, just

  1. insert a Carnot engine between the system and the bath;
  2. insert a bit eraser (Szilard engine) that uses the resulting work to pump entropy from the memory tape to the systemOr from the memory tape to the bath — it doesn’t matter.c  ; and
  3. run the engine and eraser until the system thermalizes to the bath temperature T.

Footnotes

(↵ returns to text)

  1. Here, there is a factor of Boltzmann’s constant k in front of TS because I am measuring the (absolute) entropy S in dimensionless bits rather than in units of energy per temperature. That way we can write things like N = S_0 - S.
  2. Or on the system, depending on the sign.
  3. Or from the memory tape to the bath — it doesn’t matter.
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