# «Found Phys (2013) 43:1384–1410 DOI 10.1007/s10701-013-9753-1 Brownian Computation Is Thermodynamically Irreversible John D. Norton Received: 21 June ...»

The two results match up close enough if we set P = 1 − η and approximate OP ≈ 1/(1 − P ) when P is very close to unity. However the result does not conform quite

**as well with Bennett’s ([1], p. 531) remark that:**

If all steps had equal dissipation, ε kT, the ﬁnal state occupation probability would be only about ε/kT. By dissipating an extra kT ln(3kT /ε) of energy during the last step, this probability is increased to about 95 %.

A ﬁnal stage probability of P = 0.95 corresponds to an odds ratio of OP = 20, so that the extra energy dissipated should be kT ln(20kT /ε). Compatibility would be restored if we assume a missing “+” in Bennett’s formula, for ln 20 = 3, so that

4 Thermodynamic Reversibility is Mistakenly Attributed to Brownian Computers The results of the last section can be summarized as follows. An n stage computation on a Brownian computer is a thermodynamically irreversible process that creates a minimum of k ln n of thermodynamic entropy (see Eq. (7a)). Additional thermodynamic entropy of k ln(1 + OP ) is created to complete the computation by trapping the computer state in a ﬁnal energy trap with probability odds OP (see Eq. (7b)). If we accelerate the computation by adding an energy gradient of ε per stage, we introduce further creation of thermodynamic entropy according to Eq. (7c). For a larger gradient, the thermodynamic entropy created grows linearly with the number of stages.

**While it is thermodynamically irreversible, a Brownian computer is routinely misreported as operating thermodynamically reversibly. Bennett [3] writes:**

A Brownian computer is reversible in the same sense as a Carnot engine: Both mechanisms function in the presence of thermal noise, both achieve zero dissipation in the limit of zero speed, and both are, in accordance with thermodynamic convention, presumed to be absolutely stable against structural decay (e.g., thermal annealing of a piston into a more rounded shape), despite their being non-equilibrium conﬁgurations of matter.

This misreporting is especially awkward since the case of the Brownian computer is

**offered as the proof of a core doctrine in the recent thermodynamics of computation:**

that logically reversible operations can be computed by thermodynamically reversible 1402 Found Phys (2013) 43:1384–1410 processes. Bennett ([6], pp. 329–331) reports that the “proof of the thermodynamic reversibility of computation [of logically reversible operations]” arose through the investigation into the biochemistry of DNA and RNA that culminated in the notion of the Brownian computer. Bennett ([6], p. 531) reports that the objection against thermodynamically reversible computation of logically reversible operations “has largely been overcome by explicit models.” He then cites the non-thermodynamic, hard sphere model of Fredkin and Toffoli; and “at a per-step cost tending to zero in the limit of slow operation (so-called Brownian computers discussed at length in my review article [2]).”

**5 Thermodynamically Reversible Processes**

Evidently, thermodynamically reversible processes can be hard to identify correctly.

The above misidentiﬁcation remains unchallenged in the literature. Hence it will be useful to review here just what constitutes thermodynamic reversibility and how it can be misidentiﬁed.

**5.1 What They Are**

The key notion in a thermodynamically reversible process is that all thermodynamic driving forces are in perfect balance. This traditional conception is found in the old text-books. Poynting and Thomson ([22], p. 264) give the “conditions for reversible working” that “indeﬁnitely small changes in the external conditions shall reverse the order of change.” They list these conditions as: bodies exchanging heat “never differ sensibly” in temperature; and that the “pressure exerted by the working substance shall be sensibly equal to the load.” It follows that “exactly reversible processes are ideal, in that exact reversibility requires exact equilibrium with surroundings, that is, requires a stationary condition.” This means that nothing changes, so there is no

**process. They then offer the familiar escape:4**

... we can approximate as closely as we like to the conditions of reversibility, by making the conditions as nearly as we like [to] those required, and lengthening out the time of change.

Planck ([21], §§71–73) gives an essentially similar account. He writes of pressures that differ “just a triﬂe,” “inﬁnitesimal differences of temperature” and “inﬁnitely slow” progress. The process consists of “a succession of states of equilibrium.” More

**fully:**

If a process consists of a succession of states of equilibrium with the exception of very small changes, then evidently a suitable change, quite as small, is sufﬁcient to reverse the process. This small change will vanish when we pass over to the limiting case of the inﬁnitely slow process...

We need only add to these classic accounts that generalized thermodynamic forces, such as those derived from (9) and which generalize the notion of pressure, must also be in balance.

When a Brownian particle is released into a liquid, its resulting exploration of the accessible volume is driven, thermodynamically speaking, by an unbalanced osmotic pressure, as Einstein argued in his celebrated analysis of 1905. Hence it is a thermodynamically irreversible expansion. Correspondingly, when a Brownian computer is set into its initial state and then allowed to explore the accessible computational space, the exploration is a thermodynamically irreversible process.

5.2 How We Might Misidentify Them There are many ways we may come to misidentify a thermodynamically irreversible process as thermodynamically reversible.

Inﬁnite Slowness Is not Sufﬁcient to identify Thermodynamic Reversibility While thermodynamically reversible processes are inﬁnitely slow, the converse need not hold. Sommerfeld ([23], p. 17) gives the simple example of an electrically charged capacitor that can be discharged arbitrarily slowly through an arbitrarily high resistance.

While the process can be slowed indeﬁnitely, it is a thermodynamically irreversible conversion of the electrical energy of the capacitor into heat. The driving voltage is not balanced by an opposing force. A simpler example is the venting of a gas at high pressure into a vacuum through a tiny pinhole. The process can be slowed arbitrarily, but it is not thermodynamically reversible since the gas pressure is unopposed.

Reversibility of the Microscopic or Molecular Dynamics not Sufﬁcient to Assure Thermodynamic Reversibility One cannot discern thermodynamic reversibility by afrming the reversibility of the individual processes that comprise the larger process at the microscopic or molecular level. They may be reversible, in the sense that they can go either way, while the overall process is itself thermodynamically irreversible.

As a general matter, any thermodynamically irreversible process may be reversed by a vastly improbable ﬂuctuation. That possibility depends upon the microscopic reversibility of the underlying processes.5 A pertinent example is the thermodynamically irreversible expansion of a one molecule gas. Its momentary, microscopic behavior is reversible. To see this, consider a one molecule gas suddenly released into a large chamber ﬁlled with ﬁxed, oddly shaped obstacles. If we simply attend to the molecule’s motion over a brief period of time, while it undergoes one or two collisions, we would see mechanically reversible motions, as illustrated in Fig. 9.

5 For isothermal, isobaric chemical reactions, the relevant generalized force is the chemical potential μA = (∂GA /∂nA )T,P, where GA = E + P V − T S is the Gibbs free energy of reagent A and nA the number of moles of A. In dilute solutions, μA = μA0 + RT ln[A] for R the ideal gas constant, μA0 the chemical potential at reference conditions and [A] the molar concentration. While each chemical reaction is reversible at the molecular level, the term RT ln[A] contributes an entropic force, so that a chemical reaction will be thermodynamically irreversible if the concentrations of the reagents and products are not constantly adjusted to keep them at equilibrium levels.

1404 Found Phys (2013) 43:1384–1410

**Fig. 9 Microscopic reversibilityof a thermodynamicallyirreversible expansion**

However that would mislead us. We need to attend to the initial low entropy state of conﬁnement of the one molecule gas; and its ﬁnal high entropy state in which it explores the larger volume freely in order to recognize the thermodynamically irreversible character of the expansion.

Precisely the same must be said for both Brownian motion and a Brownian computer. They are both initialized in a state of low thermodynamic entropy; and then expand in a thermodynamically irreversible process to explore a larger space. At any moment, however, their motions will be mechanically reversible. We would be unable to tell whether we are observing their development forward in time or a movie of that development run in reverse. To separate the two, we would need to observe long enough to see whether the time evolution carries the system to explore the larger accessible space or whether it carries it back to its initial state of conﬁnement.

Bennett ([2], p. 912) reports that a Brownian computer “is about as likely to proceed backward along the computational path, undoing the most recent transition, as to proceed forward.” Similarly Bennett and Landauer ([4], p. 54) report for the Brownian computer that “[i]t is nearly as likely to proceed backward along the computational path, undoing the most recent transition, as it is to proceed forward.”6 This sort of reversibility is insufﬁcient to establish thermodynamic reversibility.

Tracking Internal Energy Instead of Thermodynamic Entropy is Insufﬁcient to Identify Thermodynamic Reversibility A thermodynamically reversible process is one in which the total thermodynamic entropy Stot = Ssys + Senv remains constant, where Ssys is the thermodynamic entropy of the system and Senv that of the environment.

Thermodynamically reversible processes must be identiﬁed by tracking this entropy.

They cannot be identiﬁed by tracking internal energy changes.

What confounds matters is that we often track thermodynamic reversibility by means of quantities that carry the label “energy,” such as free energy F = E − T S.

These energies are useable this way in so far as they are really measures of thermodynamic entropy adapted to special conditions. For example, Brownian computers 6 I believe the “nearly” refers to the small external force they add corresponding to the energy ramp of Sect. 3.5 above.

Found Phys (2013) 43:1384–1410 1405 implement isothermal processes while in thermal contact with an environment with which they exchange no work. Hence, if we have a thermodynamic process parameterized by λ so that d = d/dλ, then the constant entropy condition of thermodynamic reversibility for a computer “comp” in a thermal environment “env” is 0 = dStot = dScomp + dSenv = dScomp − dEcomp /T = −dFcomp /T.

It corresponds to constancy of the free energy Fcomp of the computer.

Tracking internal energies gives the wrong result for Brownian computers. The thermodynamically irreversible n stage expansion of the Brownian computer is a constant energy process. The ﬁnal energy trap could be replaced by a trap stage with a large volume Vtrap = Ntrap V of accessible conﬁguration space. Then the ﬁnal trapping can also be effected without any change of internal energy. The odds for the computer state being in the trap are OP = P /(1 − P ) = Ntrap /n. Using (7a), the total thermodynamic entropy created is Stot = Scomp + Senv = k ln(n + Ntrap ) = k ln n + k ln(1 + Ntrap /n) = k ln n + k ln(1 + OP ) which agrees with the thermodynamic entropy creation of the energy trap (7b).

Bennett ([1], p. 531) introduced a small energy gradient in order to bring some “positive drift velocity” into Brownian computation. As we saw in Sect. 3.2 and Eq. (11), without it, no average speed can be assigned to Brownian motion. However it is also unnecessary for completion of the computational processes. They proceed as does any diffusion process, with progress increasing with the square root of time.

That means the computation will take longer to complete. Since temporal efﬁciency is not the issue, there seems no point in including a superﬂuous source of thermodynamic irreversibility.

In assessing the thermodynamic efﬁciency of the Brownian computation of logically reversible functions, Bennett and Landauer do not track thermodynamic entropy. Rather they track the wrong quantity, energy. Bennett writes of energy “dissipated,” both as the energy ε per step and in the trap energy or “latching” energy Etrap.

See Bennett ([1], p. 531; [2], pp. 915, 921). Bennett and Landauer ([4], pp. 54–56)

**write of energy “expended” or “dissipated”:**

A small force, provided externally, drives the computation forward. This force can again be as small as we wish, and there is no minimum amount of energy that must be expended in order to run a Brownian clockwork Turing machine.

and The machine can be made to dissipate as small an amount of energy as the user wishes, simply by employing a force of the correct weakness.

If the energy ε per stage is made arbitrarily small, the change of internal energy E of the Brownian computer will also become arbitrarily small. However it would be an elementary error to confuse that with the operation of the computer becoming thermodynamically reversible, so that no net thermodynamic entropy is created; or to confuse it with the change in free energy F = E − T S becoming arbitrarily small.

1406 Found Phys (2013) 43:1384–1410 For one must also account for the “T S” term in free energy. For a Brownian computer driven by an energy gradient of ε per stage, the free energy change is given by (8c).

As we saw in Sect. 3.5, it reverts to the value −kT ln n when ε becomes arbitrarily small.

Finally, I will mention another confusion here, although it has only played an indirect role in the misidentiﬁcation of Brownian computation. It is common to assign an additional thermodynamic entropy of k ln 2 to a binary memory device merely if we do not know the datum held in the device. As I have argued in Norton ([15], §3.2), this additional assignment is incompatible with standard thermodynamics. If one persists in using it, one will misidentify which are processes of constant thermodynamic entropy and thus which are thermodynamically reversible. Thus Bennett ([6], p. 502) describes erasure of a cell with “random data” as “thermodynamically reversible,” but one with “known data” as “thermodynamically irreversible.” Since this literature uses the same erasure process in both cases, it follows that whether a process is thermodynamically reversible depends on what you know. That is incompatible with thermodynamic reversibility as a (near) balance of physical forces. They will balance independently of what we know. To rescue these claims, we need to rebuild thermodynamics with new notions of entropy and reversibility. Ladyman et al.

[11] have tried to construct such an augmented thermodynamics. Norton ([17], §8) explains why I believe their efforts have failed.7

**6 Relation to Landauer’s Principle**

Brownian computation is normally limited to logically reversible operations, so that the accessible phase space forms a linear channel. If it is applied to logically irreversible operations, the accessible phase space becomes branched, possibly exponentially so. This branching has been associated with Landauer’s principle of the entropy cost of information erasure. I have argued elsewhere [15, 17] that, even 50 years after its conception, the principle is at best a poorly supported speculation.8 None of the attempts to demonstrate it have succeeded. Can Brownian computation ﬁnally provide the elusive justiﬁcation? We shall see below that the principle gains no support from Brownian computation.

**7 Bennett ([6], p. 329) writes:**