Pitt-Tsinghua Summer School for Philosophy of Science Institute of Science, Technology and Society, Tsinghua University

1. Information and Thermodynamic EntropyJohn D. NortonDepartment of History and Philosophy of ScienceCenter for Philosophy of ScienceUniversity of Pittsburgh Pitt-Tsinghua Summer School for Philosophy of Science Institute of Science, Technology and Society, Tsinghua University Center for Philosophy of Science, University of Pittsburgh At Tsinghua University, Beijing June 27- July 1, 2011

2. Philosophy and Physics = Information ideas and concepts Entropy heat, work, thermodynamics And why not? Mass = Energy Particles = Waves Geometry = Gravity …. Time = Money

3. This Talk Foreground Failed proofs of Landauer’s Principle Thermalization, Compression of phase space Information entropy, Indirect proof The standard inventory of processes in the thermodynamics of computation neglects fluctuations. Background Maxwell’s demon and the molecular challenge to the second law of thermodynamics. Exorcismby principle Szilard’s Principle, Landauer’s principle

4. Fluctuations and Maxwell’s demon

5. The original conception Divided chamber with a kinetic gas. Demon operates door intelligently J. C. Maxwell in a letter to P. G. Tait, 11th December 1867 “…the hot system has got hotter and the cold system colder and yet no work has been done, only the intelligence of a very observant and neat-fingered being has been employed.” “[T]he 2nd law of thermodynamics has the same degree of truth as the statement that if you throw a tumblerful of water into the sea you cannot get the same tumblerful of water out again.”

6. Maxwell’s demon livesin the details of Brownian motion and other fluctuations “…we see under out eyes now motion transformed into heat by friction, now heat changed inversely into motion, and that without loss since the movement lasts forever. That is the contrary of the principle of Carnot.” Poincaré, 1907 “One can almost see Maxwell’s demon at work.” Poincaré, 1905 Could these momentary, miniatureviolations of the second law be accumulated to large-scale violations? Guoy (1888), Svedberg (1907) designed mini-machines with that purpose.

7. Szilard’sOne-Molecule Engine

8. Simplest case of fluctuations Many molecules A few molecules One molecule Can a demon exploit these fluctuations?

9. The One-Molecule Engine Szilard 1929 A partition is inserted to trap the molecule on one side. Initial state Work kT ln 2 gained in raising the weight. It comes from the heat kT ln 2, drawn from the heat bath. The gas undergoes a reversible, isothermal expansion to its original state. Net effect of the completed cycle: Heat kT ln 2 is drawn from the heat bath and fully converted to work. The total entropy of the universe decreases by k ln 2. The Second Law of Thermodynamics is violated.

10. The One-Molecule Engine Szilard 1929 A partition is inserted to trap the molecule on one side. Initial state Work kT ln 2 gained in raising the weight. It comes from the heat kT ln 2, drawn from the heat bath. The gas undergoes a reversible, isothermal expansion to its original state. Net effect of the completed cycle: Heat kT ln 2 is drawn from the heat bath and fully converted to work. The total entropy of the universe decreases by k ln 2. The Second Law of Thermodynamics is violated.

11. Exorcism by principle

12. Szilard’s Principle Landauer’s Principle versus Von Neumann 1932 Brillouin 1951+… Landauer 1961 Bennett 1987+… Acquisitionof one bit of information creates k ln 2 of thermodynamic entropy. Erasure of one bit of information creates k ln 2 of thermodynamic entropy. Proof: By “working backwards.” By suggestive thought experiments. (e.g. Brillouin’s torch) Szilard’s principle is false. Real entropy cost only taken when naturalized demon erases the memory of the position of the molecule Proof: …???...

13. Failedproofs of Landauer’s Principle

14. Direct Proofs that model the erasure processes in the memory device directly. or 1.Thermalization An inefficiently designed erasure procedure creates entropy. No demonstration that all must. 2.Phase Volume Compressionaka “many to one argument” Erasure need not compress phase volume but only rearrange it. 3. Information-theoretic Entropy “p ln p” Associate entropy with our uncertainty over which memory cell is occupied. Wrong sort of entropy. No connection to heat. See: "Eaters of the Lotus: Landauer's Principle and the Return of Maxwell's Demon." Studies in History and Philosophy of Modern Physics, 36 (2005), pp. 375-411.

15. 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.

16. Ladyman et al., “The connection between logical and thermodynamic irreversibility,” 2007. 4. An Indirect Proof One-Molecule gas or isothermal reversible expansion insert partition dissipationlessly detect gas state Reduces entropy of heat bath by k ln 2. One-Molecule memory or shift cell to match perform any erasure Assume second law of thermodynamics holds on average. Original proof given only in terms of quantities of heat passed among components. Erasure must create entropy k ln 2 on average.

17. 4. An Indirect Proof Fails Inventory of admissible processes allows: Processes that erase dissipationlessly (without passing heat to surroundings) in violation of Landauer’s principle. Processes that violate the second law of thermodynamics, even in its statistical form. See: “Waiting for Landauer,” Studies in History and Philosophy of Modern Physics, forthcoming.

18. 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.

19. The Importance of Fluctuations

20. Exorcism of Maxwell’s demon by fluctuations. Marian Smoluchowski, 1912 The best known of many examples. Trapdoor hinged so that fast molecules moving from left to right swing it open and pass, but not vice versa. BUT AND SO The trapdoor must be very light so a molecule can swing it open. The trapdoor has its own thermal energy of kT/2 per degree of freedom. The trapdoor will flap about wildly and let molecules pass in both directions. The second law holds on average only over time. Machines that try to accumulate fluctuations are disrupted fatally by them.

21. Fluctuations dispruptReversible Expansion and Compression

22. 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

23. 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

24. What Happens. Fluctuations obliterate the infinitely slow expansion intended This analysis is approximate. The exact analysis replaces the gravitational field with piston energy = 2kT ln (height)

25. Fluctuations disruptMeasurement and Detection

26. 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.

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

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

29. 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.

30. A General “No-Go” Result

31. Fluctuation Disrupt All Reversible, Isothermal Processes at Molecular Scales Intended process l l=l2 l=l1 Actual process l l=l2 l=l1

32. Einstein-Tolman Analysis of Fluctuations Total system of gas-piston or target-detector-driver is canonically distributed. p(x, p) = (1/Z) exp(-E(x,p)/kT) Different stages l Different subvolumes of the phase space. Probability density that system is in stage l p(l) proportional to Z(l) Z(l) = ∫l exp(-E(x,p)/kT) dxdp Free energy of stage l F(l) = - kT ln Z(l) p(λ) proportional to exp(-F(λ)/kT) Probability density for fluctuation to stage λ: = exp(-) p(l2) F(λ2)-F(λ1) kT p(l1)

33. Equilibrium implies uniform probability over l Condition for equilibrium = exp(-) ∂F/∂l = 0 F(l) = constant p(l2) F(λ2)-F(λ1) kT p(l1) Probability distribution over l p(l) = constant p(l1) = p(l2) since Time evolution over phase space Expected Actual

34. One-Molecule Gas/Piston System Overlap of subvolumes corresponding to stages h = 0.5H h=0.75H h=H h=1.25H Slice through phase space.

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

36. What it takes to overcome fluctuations = exp(-) > exp(3) = 20 p(l2) F(λ2)-F(λ1) Enforcing a small probability gradient… kT p(l1) …requires a disequilibrium… F(λ1) > F(λ2) + 3kT …which creates entropy. S(λ2)-S(λ1) – (E(λ2)-E(λ1))/T = 3k Exceeds the entropy k ln2 = 0.69k tracked by Landauer’s Principle! No problem for macroscopic reversible processes. F(λ1) - F(λ2) = 25kT p(λ2)/p(λ1) = 7.2 x 1010 = mean thermal energy of ten Oxygen molecules

37. More Woes

38. Dissipationless Insertion of Partition? No friction-based device is allowed to secure the partition. With a conservative Hamiltonian, the partition will bounce back. Arrest partition with a spring-loaded pin? The pin will bounce back. Feynman, ratchet and pawl.

39. In Sum… We are selectively ignoring fluctuations. Dissipationless detection disrupted by fluctuations. Reversible, isothermal expansion and contraction does not complete due thermal motions of piston. Inserted partition bounces off wall unless held by… what? Friction?? Spring loaded pin??... Need to demonstrate that each of these processes is admissible. None is primitive. Inventory assembled inconsistently. It concentrates on fluctuations when convenient; it ignores them when not.

40. Conclusions

41. Why should we believe that… …the reason for the supposed failure of a Maxwell demon is localizable into some single information theoretic process? (detection? Erasure?) …the second law obtains even statistically when we deal with tiny systems in which fluctuations dominate?

42. Conclusions Is a Maxwell demon possible? The best analysis is the Smoluchowski fluctuation exorcism of 1912. It is not a proof but a plausibility argument against the demon. Efforts to prove Landauer’s Principle have failed. …even those that presume a form of the second law. It is still speculation and now looks dubious. Thermodynamics of computation has incoherent foundations. The standard inventory of processes admits composite processes that violate the second law and erase without dissipation. Its inventory of processes is assembled inconsistently. It selectively considers and ignores fluctuation phenomena according to the result sought.

43. http://www.pitt.edu/~jdnorton/lectures/Tsinghua/Tsinghua.htmlhttp://www.pitt.edu/~jdnorton/lectures/Tsinghua/Tsinghua.html

44. Finis

45. Appendix

46. A dilemmafor information theoretic exorcisms

47. Do information theoretic ideas reveal why the demon must fail? Total system = gas + demon + all surrounding. Earman and Norton, 1998, 1999, “Exorcist XIV…” EITHER Canonically thermal = obeys your favorite version of the second law. Demon’s failure assured by our decision to consider only system that it cannot breach. the total system IS canonically thermal. (sound horn) Principles need independent justifications which are not delivered. (…and cannot? Zhang and Zhang pressure demon.) the total system is NOT canonically thermal. (profound horn) OR Profound “ …the real reason Maxwell’s demon cannot violate the second law …uncovered only recently… energy requirements of computers.” Bennett, 1987. Cannot have both! Sound Deduce the principles (Szilard’s, Landauer’s) from the second law by working backwards. and

48. 1.

49. 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.

50. 2.