Warm Up Computers Generate Lots of Heat

1. No-Go Resultfor the Thermodynamics of ComputationJohn D. NortonDepartment of History and Philosophy of ScienceCenter for Philosophy of ScienceUniversity of Pittsburgh

2. Warm UpComputers Generate Lots of Heat

3. Cray 2 computer (1985) From Cray sales brochure.

4. Cooling chips fan heat pipe vanes

5. What is the Minimum Heat Generation?

6. To minimize heat generation… molecular scale devices Use the smallest systems possible =

7. Landauer’s Principle 0 Logically irreversible operations like erasure pass heat/entropy to the environment. Erasure 1 or RESTORE TO ONE 1 Pass kT ln 2 heat k ln 2 entropy Erase one bit Logically reversible operations can be carried out non-dissipatively. Rolf Landauer (1961) "Irreversibility and heat generation in the computing process," IBM Journal of Research and Development, 5, pp. 183–191. (p.265)

8. The Standard Erasure Procedure Model of binary memory. One molecule gas in a divided chamber. Heat kT ln 2 Entropy k ln 2 passes to environment. Plausibility of Landauer’s Principle 2 Information theoretic entropy of ln 2 is associated with distribution P(L) = P(H) = 1/2 3 Erasure compresses the space by a factor of 2. Corresponds to entropy change of k ln 2. Credible erasure procedures generate kT ln 2 of heat. 1 Sketchy… but someone has worked out the details…..??

9. Bennett’s 2003 statement of Landauer’s Principle “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….” Must pass entropy to environment Logically irreversible operation Logically reversible operation Can be thermodynamically reversible “…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.” Bennett, Charles H. (2003). “Notes on Landauer’s Principle, Reversible Computation, and Maxwell’s Demon,” Studies in History and Philosophy of Modern Physics, 34, pp. 501-10.

10. Bennett’s 2003 statement of Landauer’s Principle Why not… Must pass entropy to environment Logically irreversible operation Thermodynamically irreversible process …create entropy in a thermodynamically irrreversible process? Logically reversible operation Can be thermodynamically reversible Thermodynamically reversible process

11. Erasing Random versus Nonrandom/Known Data NO entropy created “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 the operation as a whole is thermodynamically reversible. … [I]n computations, logically irreversible operations are usually applied to nonrandom data deterministically generated by the computation. When erasure is applied to such data, the entropy increase of the environment is not compensated by an entropy decrease of the data, and the operation is thermodynamically irreversible.” Random data erase Known data Entropy created ? ? ? ? Erasure procedure on known data… Erasure is just rearrangement. No entropy created. …DOES use what is known. Bennett, Charles H. (1988). “Notes on the History of Reversible Computation,” IBM Journal of Research and Development, 32 (No. 1), pp. 16-23 No difference in erasing known and random data. …does NOT use what is known.

12. To Come

13. This Presentation No-Go Result The standard inventory of processes in the thermodynamics of computation neglects fluctuations. All process must create entropy to overcome them and the quantities created swamp those tracked by Landauer’s principle. Demonstrations of Landauer’s Principle fail. 1. Thermalization 2. Compression of phase space 3. Information entropy (4. New indirect proof)

14. Failedproofs of Landauer’s Principle

15. 1.

16. 1.Thermalization Reversible isothermal compression passes heat kT ln 2 to heat bath. Initial data L or R Irreversible expansion “thermalization” Data reset to L Entropy k ln 2 created in heat bath !!! !!! Entropy created in this ill-advised, dissipative step. Proof shows only that an inefficiently designed erasure procedure creates entropy. No demonstration that all must. Mustn’t we thermalize so the procedure works with arbitrary data? No demonstration that thermalization is the only way to make procedure robust.

17. Dissipationless Erasure or First method. 1. Dissipationlessly detect memory state. 2. If R, shift to L. Second method. 1. Dissipationlessly detect memory state. 2. If R, remove and reinsert partition and go to 1.Else, halt.

18. 2.

19. 2.Phase Volume Compressionaka “many to one argument” …if entropy is to connect to heat via Clausius’ dS = dqrev/T thermodynamic entropy = k ln (accessible phase volume) “random” data occupies twice the phase volume of reset data Erasure halves phase volume. Erasure reduces entropy of memory by k ln 2. Entropy k ln 2 must be created in surroundings to conserve phase volume.

20. 2.Phase Volume Compressionaka “many to one argument” FAILS “random” data DOES NOT occupy twice the phase volume of reset data It occupies the same phase volume. Confusion with thermalized data

21. Hence confusion over random and known data. A Ruinous Sense of “Reversible” Random data and insertion of the partition removal of the partition thermalized data have the same entropy because they are connected by a reversible, adiabatic process??? DS = 0 random data thermalized data No. Under this sense of reversible, entropy ceases to be a state function. DS = k ln 2

22. 3.

23. 3. Information-theoretic Entropy “p ln p” Information entropy Sinf = - k Si Pi ln Pi “random” data PL = PR = 1/2 Sinf = k ln 2 reset data PL = 1; PR = 0 Sinf = 0 Hence erasure reduces the entropy of the memory by k ln 2, which must appear in surroundings. does NOT equal Thermodynamic entropy Information entropy But… in this case, Thermodynamic entropy is attached to a probability only in special cases. Not this one.

24. What it takes… Information entropy DOES equal Thermodynamic entropy “p ln p” Clausius dS = dQrev/T IF… A system is distributed canonically over its phase space p(x) = exp( -E(x)/kT) / Z Z normalizes For details of the proof and the importance of the accessibility condition, see Norton, “Eaters of the Lotus,” 2005. AND All regions of phase space of non-zero E(x) are accessible to the system over time. Accessibility condition FAILS for “random data” since only half of phase space is accessible.

25. 4.

26. Ladyman et al., “The connection between logical and thermodynamic irreversibility,” 2007. 4. Indirect Proof: General Strategy Process known to reduce entropy Entropy reduces. Assume second law of thermodynamics holds on average. coupled to Arbitrary erasure process Entropy must increase on average.

27. The Debate is Ongoing

28. The Debate is Ongoing My concerns: The inventory of processes assumed as possible is… …survives only with artificial restrictions on what is possible. (“controlled operations”) …inconsistent with the statistical form of the second law. …selectively ignores fluctuations present in all molecular scale processes. No-go result Ladyman, James; Presnell, Stuart; Short, Anthony J. and Groisman, Berry (2007). “The connection between logical and thermodynamic irreversibility,” Studies in the History and Philosophy of Modern Physics, 38, pp. 58–79. Ladyman, James; Presnell, Stuart and Short, Anthony J. (2008). ‘The Use of the Information-Theoretic Entropy in Thermodynamics’, Studies in History and Philosophy of Modern Physics, 39, pp. 315-324. Ladyman, James and Robertson, Katie (forthcoming). “Landauer Defended: Reply to Norton, Studies in History and Philosophy of Modern Physics. Norton, John D. (2011). “Waiting for Landauer.” Studies in History and Philosophy of Modern Physics, 42, pp. 184–198. Norton, John D. (2013). “Author's Reply to Landauer Defended” Studies in History and Philosophy of Modern Physics. Available online, May 24, 2013

29. “…the same bitcannot be both the control and the target of a controlled operation…” The Most Beautiful Machine2003 Trunk, prosthesis, compressor, pneumatic cylinder 13,4 x 35,4 x 35,2 in. “…the observers are supposed to push the ON button. After a while the lid of the trunk opens, a hand comes out and turns off the machine. The trunk closes - that's it!” http://www.kugelbahn.ch/sesam_e.htm Every negative feedback control device acts on its own control bit. (Thermostat, regulator.)

30. Failedproofs ofConverse Landauer’s Principle

31. Brownian Computation is the Constructive Proof “…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.” Closer analysis… (Bennett, 2003) Brownian computation is thermodynamically irreversible, the thermodynamic analog of the uncontrolled expansion of a one molecule gas. Norton, John D. (Manuscript on website). “Brownian Computation is Thermodynamically Irreversible.”

32. No-Go ResultIllustrated

33. Fluctuations disruptReversible Expansion and Compression

34. The Intended Process Infinitely slow expansion converts heat to work in the raising of the mass. Mass M of piston continually adjusted so its weight remains in perfect balance with the mean gas pressure P= kT/V. Equilibrium height is heq = kT/Mg Heat kT ln 2 = 0.69kT passed in tiny increments from surrounding to gas.

35. The massive piston… ….is very light since it must be supported by collisions with a single molecule. It has mean thermal energy kT/2 and will fluctuate in position. Probability density for the piston at height h p(h) = (Mg/kT) exp ( -Mgh/kT) Mean height = kT/Mg = heq Standard deviation = kT/Mg = heq

36. What Happens. A better analysis does not need external adjustment of weight during expansion. It replaces the gravitational field with Fluctuations obliterate the infinitely slow expansion intended piston energy = 2kT ln (height) Heat kT ln 2 = 0.69kT passed in tiny increments from surrounding to gas. Mean energy of gas 3kT/2 Standard deviation (3/2)1/2kT = 1.225kT

37. Fluctuations disruptMeasurement and Detection

38. Bennett’s Machine for Dissipationless Measurement… FAILS Measurement apparatus, designed by the author to fit the Szilard engine, determines which half of the cylinder the molecule is trapped in without doing appreciable work. A slightly modified Szilard engine sits near the top of the apparatus (1) within a boat-shaped frame; a second pair of pistons has replaced part of the cylinder wall. Below the frame is a key, whose position on a locking pin indicates the state of the machine's memory. At the start of the measurement the memory is in a neutral state, and the partition has been lowered so that the molecule is trapped in one side of the apparatus. To begin the measurement (2) the key is moved up so that it disengages from the locking pin and engages a "keel" at the bottom of the frame. Then the frame is pressed down (3). The piston in the half of the cylinder containing no molecule is able to desend completely, but the piston in the other half cannot, because of the pressure of the molecule. As a result the frame tilts and the keel pushes the key to one side. The key, in its new position. is moved down to engage the locking pin (4), and the frame is allowed to move back up (5). undoing any work that was done in compressing the molecule when the frame was pressed down. The key's position indicates which half of the cylinder the molecule is in, but the work required for the operation can be made negligible To reverse the operation one would do the steps in reverse order. Charles H. Bennett, “Demons, Engines and the Second Law,” Scientific American 257(5):108-116 (November, 1987). …is fatally disrupted by fluctuations that leave the keel rocking wildly.

39. A Measurement Scheme Using Ferromagnets Charles H. Bennett, “The Thermodynamics of Computation—A Review,” In. J. Theor. Phys. 21, (1982), pp. 905-40,

40. A Measurement Scheme Using Ferromagnets Charles H. Bennett, “The Thermodynamics of Computation—A Review,” In. J. Theor. Phys. 21, (1982), pp. 905-40,

41. A General Model of Detection First step: the detector is coupled with the target system. The process intended: The process is isothermal, thermodynamically reversible: • It proceeds infinitely slowly. • The driver is in equilibrium with the detector. The coupling is an isothermal, reversible compression of the detector phase space.

42. Fluctuations Obliterate Reversible Detection What we expected: What happens:

43. No-Go Result

44. Preparatory notions Isothermal, thermodynamically reversible process Self-contained isothermal, thermodynamically reversible process. Computing fluctuations: How to do it.

45. Thermodynamically Reversible Processes For… Two systems interacting isothermally in thermal contact with constant temperature surrounding at T T 1 2 env Condition for thermodynamic reversibility All thermodynamic forces are in perfect balance (or minutely removed from it). internal energy change heat trans-ferred generalized force generalized displacement Process is a sequence of equilibrium states. dU = dq –X dx X = -∂F/∂l for process parameter l Total entropy of universe is constant. Total free energy F=U-TS is constant. F1+F2=constant Total generalized forces vanish. X1+X2=0

46. Thermodynamically reversible processes are NOT… balloon deflates slowly through a pinhole capacitor discharges very slowly through resistor …merely very slow processes. …merely processes that can go easily in either way. one molecule gas released

47. Self-containedthermodynamically reversible processes No interventions from non-thermal or far-from-equilibrium systems. External hand removes shot one at a time to allow piston to rise slowly. Slow compression by slowly moving, very massive body. Mass is far from thermal equilibrium of a one-dimensional Maxwell velocity distribution.

48. Isolated, microcanonically distributed system Computing Fluctuations probability P that system is in non-equilibrium state with phase volume V ∝ phase volume V give equilibrium, macroscopic description of non-equilibrium state S = k ln V S = k ln P+ constant P∝ exp(S/k)

49. Computing Fluctuations Canonically distributed system in heat bath at T. ∝ probability system at point with energy E probability P that system is in non-equilibrium state with phase volume V ∝ Z(V) give equilibrium, macroscopic description of non-equilibrium state F = -kT ln Z(V) F = -kT ln P+ constant P ∝ exp(-F/kT)

50. No-Go ResultIt, at last.