«Found Phys (2013) 43:1384–1410 DOI 10.1007/s10701-013-9753-1 Brownian Computation Is Thermodynamically Irreversible John D. Norton Received: 21 June ...»
When truly random data (e.g. a bit equally likely to be 0 or 1) is erased, the entropy increase of the surroundings is compensated by an entropy decrease of the data, so that the operation as a whole is thermodynamically reversible....When erasure is applied to such [nonrandom] data, the entropy increase of the environment is not compensated by an entropy decrease of the data, and the operation is thermodynamically irreversible.
To interpret these remarks, one needs to know that Bennett tacitly assumes an inefﬁcient erasure procedure that also creates k ln 2 of thermodynamic entropy that is passed to the environment.
8 For other critiques of Landauer’s principle, see Maroney  and Hemmo and Shenker (, Chaps. 11– 12). For historical perspective, see Bennett .
Found Phys (2013) 43:1384–1410 1407
6.1 The Principle
Bennett (, p. 501) describes it as:
Landauer’s principle, often regarded as the basic principle of the thermodynamics of information processing, holds that any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment.
He then asserts a converse:
Conversely, it is generally accepted that any logically reversible transformation of information can in principle be accomplished by an appropriate physical mechanism operating in a thermodynamically reversible fashion.
6.2 Computing Logically Irreversible Operations
The simplest instance of logical irreversibility is erasure. An n stage erasure program applied to a single memory cell has two computational paths. One takes the cell, initially holding 0 to the end state, holding 0; the other takes a cell initially holding 1 to the end state 0. This logical branching backwards from the 0 end state is implemented in a Brownian computer as backward branched channels in the accessible phase space, as shown on the top left in Fig. 10.
While we may initialize the program to run on a cell holding, say, 0, when the computer state diffuses through the accessible phase space, it will also enter the other branch. This increases the accessible conﬁguration space from nV to 2nV and that will lead to a corresponding increase in thermodynamic entropy creation. The analyses of Sect. 3 still apply since they depend only on the accessible volume of phase
That is, there is an increase of thermodynamic entropy creation due to exploration of the additional branches of k ln 2.
Figure 10 shows the more general case in which the program uses the same n stages sequentially to erase an N cell memory device, holding binary data.
The volume of conﬁguration space accessible is
6.3 Failure of the Connection to Landauer’s Principle One might be tempted to see some sort of vindication of Landauer’s principle in this entropy increase. It is not there.
The lesser problem is that expression (14) is the wrong formula. The Landauer limit for erasure of memory cells with binary data is k ln 2 per cell; that is N k ln 2 for an N cell device. For large N, (14) approaches (N + 1)k ln 2.
The main problem is that nothing in the logical irreversibility of the erasure operation necessitates the creation of the entropy of (14). Rather, it is an awkward artifact of Brownian computation that it unnecessarily makes accessible volumes of phase spaces associated with unintended branches of the computation. In this regard it is akin to the category of failed proofs of Landauer’s principle listed in Norton (, §3) as “proof by thermalization.” Those proofs thermalize a memory device, thereby introducing an unnecessary thermodynamically irreversible expansion and then misreport the thermodynamic entropy created as a necessity of erasure.
The issue with Landauer’s principle is to determine which operations can be carried out by thermodynamically reversible computations and which cannot; and to specify how much thermodynamic entropy the latter must create. Brownian computation is driven by thermodynamically irreversible processes. Hence it is the wrong instrument to use. That some Brownian computation creates some amount of thermodynamic entropy is no basis for determining that another device, operating in a thermodynamically reversible way, cannot do better.
Thermodynamic entropy is always created in Brownian computation. Its extent depends only on the volume of phase space into which the computation expands and not on whether the operation computed is logically reversible. Consider a logically reversible operation that chains (2N +1 − 2) operations in series, such that each operation involves nV of conﬁguration space. The operation is logically reversible but Found Phys (2013) 43:1384–1410 1409 its computation will create exactly as much thermodynamic entropy as the erasure of the N cell memory device above. What matters is not whether a logically reversible operation is computed, but whether the two computations are driven by the same expansion of phase space volume.
6.4 Landauer’s Principle as a Temporal Effect?
Bennett’s analysis differs from that just given, as it must. It cannot include the thermodynamic entropy (14), for his analysis neglects the entropic forces that create it.
Instead, Bennett’s concern is that exploration of the additional accessible phase will slow down the computation unacceptably. He writes (Bennett, , p. 922)9 In a Brownian computer, a small amount of logical irreversibility can be tolerated..., but a large amount will greatly retard the computation or cause it to fail completely, unless a ﬁnite driving force (approximately kT ln 2 per bit of information thrown away) is applied to combat the computer’s tendency to drift backward into extraneous branches of the computation. Thus driven, the Brownian computer is no longer thermodynamically reversible, since its dissipation per step no longer approaches zero in the limit of zero speed.
That is, we must create more thermodynamic entropy to drive the computation forward to its end state and keep it out of the extraneous branches. Bennett (, pp. 531–
532) gives the quantitative expression:
This in turn means (roughly speaking) that the dissipation per step must exceed kT ln m, where m is the mean number of immediate predecessors (1) averaged over states near the intended path, or (2) averaged over all accessible states, whichever is greater. For a typical irreversible computer, which throws away one bit per logical operation, m is approximately two, and thus kT ln 2 is, as Landauer has argued , an approximate lower bound on the energy dissipation of such machines.
Bennett leaves unclear whether the “energy dissipation” indicated is derived from a computation not provided or is presumed on the prior authority of Landauer’s principle. I will not pursue the question. Since this dissipation arises in addition to the entropy creation described in Sect. 6.1 above, it is at best only a part of the full account.
More generally, unless the branching structure introduces inﬁnite phase volume, the extra dissipation is unnecessary and can provide no vindication of Landauer’s principle. For Bennett’s concern over the speed of computation is misplaced. It is standard in thermodynamics to allow processes unlimited but ﬁnite time for completion, so that they can approach thermodynamic reversibility arbitrarily closely. If one’s interest is what is possible in principle by a thermodynamically reversible process, one should not create additional thermodynamic entropy merely to speed up the process. That will only confound the analysis.
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