About That Landauer Limit…

You may have heard of the Landauer principle or the Landauer limit. This defines a lower bound on switching power dissipation in any form of digital circuit. Rolf Landauer first presented this principle in 1961, while working at IBM. It’s not limited by how the circuit is built – you can use FinFETS or spintronics or even dilithium crystals. The bound is determined by the laws of thermodynamics and is believed to set a fundamental lower limit to switching power.

A condensed version of the principle goes roughly like this. You can’t do useful work (computation) without dissipating heat. For a digital system, this originates from a decrease in the entropy of the system as computation proceeds. When data passes through a 2-input NAND gate, information is lost (because you started with 2 bits but get out only 1 bit). You have effectively erased 1 bit, so information entropy (Shannon) decreases by ln(2). Landauer assumed a correspondence between physical and information entropies leading to a physical entropy loss of k[SUB]B[/SUB]ln(2), where k[SUB]B[/SUB] is Boltzmann’s constant. This decrease must be offset by an increase in entropy in the surrounding environment, at least some of which will manifest as dissipated heat. The second law of thermodynamics says this must be at least TdS (T is temperature and dS is the change in entropy). And that’s Landauer’s principle: each loss of information of one bit, resulting from convergence in logic or other erasure of bits (through a reset for example), results in at least k[SUB]B[/SUB]Tln(2) of heat dissipated.

At room temperature this is about 3×10[SUP]-21[/SUP] joules/bit, which sounds negligible. But consider a 1 billion gate circuit running at 100MHz. Maybe 10% of the circuit is active at any time so, assuming bit loss is the same order as this number, you have 10[SUP]8[/SUP]x10[SUP]8[/SUP]x3x10[SUP]-21[/SUP] or 30uW. Still not bad. Now run 10 iterations of Moore’s law (assuming it continues). That’s a 10[SUP]3[/SUP] increase in gate-count and a 10[SUP]3[/SUP]increase in speed. Now you’ve got 30W. Not so ignorable, especially if this is an absolute minimum and a real design would be worse. Going to more advanced technologies doesn’t help; the quantum computing folks feel they are already in range of the Landauer limit.

However, the physics debate is not yet over. There have been vigorous efforts to demonstrate that the theory is flawed, not in basic thermodynamic principles and the expectation that there will be some lower limit, but instead in the the methods of proof and the specific limit that Landauer sets. These argue that Landauer and defenses of Landauer are based on consideration only of special cases and ignore effects that should be included. Whether right or not, it looks like there are soft spots in the theory which maybe should be renamed “Landauer’s Conjecture”, rather than “Landauer’s Principle”.

Yet again, there have been recent experiments which confirm that heat dissipated in a specialized bit erasure aligns precisely with Landauer’s prediction. So while the theory may need to be shored up, the limit may still be a practical reality.

And yet again, Landauer himself argued that reversible logic could break this bound since entropy need not increase in reversible computations. In a reversible gate, you can recover the state of the inputs given the state of the outputs, which requires that each gate has as many outputs as inputs, so logic would be quite a bit bulkier. However, it has been pointed out that among other disadvantages, the increased interconnect required with this design style would result in increased interconnect power dissipation which would overwhelm any switching savings.

And for the final yet again, comments by Cavin and others (comments section here and conclusions here) point out that in addition to the basic information entropy limits used by Landauer, there are dissipative mechanisms in the control of any kind of switch which themselves will have some unavoidable minimum and should therefore be included in a minimum energy calculation. And of course these could only increase the minimum threshold.

Wrapping up, the theory behind the Landauer limit is not entirely solid, but experiment backs up the number and objections mostly seems to be along the lines of the theory being based on ideal switches with zero energy cost in switching. Real dissipation can only be worse and Landauer is very probably an absolute if unattainable lower bound to power. So get ready for the big showdown (in 15 years or so): Moore’s law versus the second law of thermodynamics.

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