1. Misplaced Idealizations Entropy, Information and Maxwell's Demon John D. Norton Department of History and Philosophy of Science Center for Philosophy of Science University of Pittsburgh 1 4th Tuebingen Summer School in History and Philosophy of Science, July 2015

2. This Lecture The thermodynamics of computation presumes it is possible to… Chain molecular-scale computational steps that are thermodynamically reversible or nearly so. Bad Idealization Detection of memory device states. Moving data from one location to another. … Compression and expansion of components spaces. steps

3. This Lecture No-Go result Thermal fluctuations (noise) prevent completion of any individual, molecular scale step. Thermodynamic entropy must be created to complete each step. Minimum entropy creation not set by the logical specification of the computation, but by the number of steps chained.

4. Maxwell’s Demon 4

5. The Maxwell Era 1867-1905 5

6. Theory of Heat, 1871, first ed. 6 Also Letter to Tait, 1867; Rayleigh 1871

7. Theory of Heat 7 Better scan from 1872, 2 nd ed.

8. Maxwell’s Proposal 8 “He will thus, without expenditure of work, raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics.” air initially at uniform temperature

9. Maxwell’s Moral: The Demon Wins 9 “This is only one of the instances in which conclusions which we have drawn from our experience of bodies consisting of an immense number of molecules may be found not to be applicable to the more delicate observations and experiments which we may suppose made by one who can perceive and handle the individual molecules which we deal with only in large masses. In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus.” Theory of Heat. No compulsion to exorcise the demon to protect the Second Law. The demon illustrates that Second Law would fail if we could manipulate individual molecules. …. Nanotechnology has not yet overturned the Second Law.

10. The Fluctuation Era 1905-1929 10

11. Einstein’s Brownian Motion Paper 11 "On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat.” Annalen der Physik, 17(1905), pp. 549-560. (May 1905; received 11 May 1905)

12. 12 “…no longer strictly valid…” “If it is really possible to observe the motion discussed here …” “… then classical thermodynamics can no longer be viewed as strictly valid even for microscopically distinguishable spaces....” “… … and an exact determination of the real size of atoms becomes possible.”

13. Maxwell’s demon lives in the details of Brownian motion and other fluctuations Could these momentary, miniature violations of the second law be accumulated to large-scale violations? A real Maxwell’s demon? Guoy (1888), Svedberg (1907) designed mini- machines with that purpose. 13 “[…] we see under our eyes now motion transformed into heat by friction, now heat changed inversely into motion, and that without loss since the movement lasts forever. This is the contrary of the principle of Carnot. If this be so, to see the world return backward, we no longer have need of the infinitely keen eye of Maxwell's demon; our microscope suffices.” Poincaré, 1904

14. Casing heats Colloid cools Svedberg’s Proposal 14 Svedberg, The. “Über die Bedeutung der Eigenbewegung der Teilchen in kolloidalen Lösungen für die Beurteilung der Gültigkeitsgrenzen des zweiten Haupsatzes der Thermodynamik”.Annalen der Physik, 59 (1907) pp. 451–458. Charged colloid particles radiate their thermal energy. Tuned lead casing absorbs the radiation. …plus many more layers, details designed to prevent return of heat.

15. Marian Smoluchowski, 1912 15 Exorcism of Maxwell’s demon by fluctuations. Trapdoor hinged so that fast molecules moving from left to right swing it open and pass, but not vice versa. The second law holds on average only over time. Machines that try to accumulate fluctuations are disrupted fatally by them. BUT The trapdoor must be very light so a molecule can swing it open. AND The trapdoor has its own thermal energy of kT/2 per degree of freedom. SO The trapdoor will flap about wildly and let molecules pass in both directions.

16. Marian Smoluchowski, 1912 16 Other examples of defeated demons. The second law holds on average only over time. Machines that try to accumulate fluctuations are disrupted fatally by them. Later popularized by Feynman

17. The Information Era 1929- ???? 17

18. 18 “On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings.” Zeitschrift für Physik, 53 (1929), pp. 840-856. Szilard 1929

19. 1 The One-Molecule Engine Initial state 2 A partition is inserted to trap the molecule on one side. 3 The gas undergoes a reversible, isothermal expansion to its original state. 4 Work kT ln 2 gained in raising the weight. It comes from the heat kT ln 2, drawn from the heat bath. Szilard 1929 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. Net effect of the completed cycle:

20. Szilard’s Principle 20 Acquisition of one bit of information by the demon creates k ln 2 of thermodynamic entropy. Szilard 1929 Von Neumann 1932 Brillouin 1951+… Landauer’s Principle versus Landauer 1961 Bennett 1987+… Erasure of one bit of information by the demon creates k ln 2 of thermodynamic entropy. Real entropy cost only taken when the naturalized demon erases the memory of the position of the molecule. Szilard’s principle is false. FALSE

21. Process is thermodynamically reversible if data is “random”; not if “known” data. Landauer’s Principle 21 “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….” 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. Logically irreversible operation (e.g. erasure) Must pass entropy to environment “…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.” Logically reversible operation Can be thermodynamically reversible k ln 2 per bit erased

22. The Standard Erasure Procedure 22 Model of binary memory. One molecule gas in a divided chamber. Heat kT ln 2 Entropy k ln 2 passes to environment.

23. No-Go Result 23

24. No-Go Result 24 NO molecular-scale process that completes is thermodynamically reversible. Thermodynamic entropy must be created to complete each step. SS SS SS SS SS SS SS SS

25. No-Go Result Illustrated 25

26. Fluctuations disrupt Reversible Expansion and Compression 26

27. The Intended Process 27 Very slow expansion converts heat to work in the raising of the mass. Mass M of piston continually adjusted so its weight remains in near perfect balance with the mean gas pressure P= kT/V. Equilibrium height is h eq = kT/Mg Heat kT ln 2 = 0.69kT passed in tiny increments from surroundings to gas.

28. The massive piston… 28 ….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 = h eq Standard deviation = kT/Mg = h eq

29. Mean energy of gas 3kT/2 Standard deviation (3/2) 1/2 kT = 1.225kT What Happens. 29 Fluctuations obliterate the very slow expansion intended A better analysis (elsewhere) does not need external adjustment of weight during expansion. It replaces the gravitational field with piston energy = 2kT ln (height) Heat kT ln 2 = 0.69kT passed in tiny increments from surrounding to gas.

30. Fluctuations disrupt Measurement and Detection 30

31. Measurement is compression of detector phase space 31 First step: the detector is coupled with the target system. The process is isothermal, thermodynamically reversible: It proceeds very slowly. The driver is in equilibrium with the detector. The process intended: The coupling is an isothermal, reversible compression of the detector phase space.

32. Fluctuations Obliterate Reversible Detection 32 What happens: What we expected:

33. Bennett’s Machine for Dissipationless Measurement… 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). 33 …is fatally disrupted by fluctuations that leave the keel rocking wildly. FAILS

34. No-Go Result Preparatory notions 34

35. Thermodynamically Reversible Processes 35 For… Two systems interacting isothermally in thermal contact with constant temperature surroundings at T: 1 2 env T dU = dq –X dx internal energy change heat trans- ferred generalized force generalized displacement X = -∂F/∂ for process parameter Thermo- dynamically reversible process Set of irreversible processes that approach a perfect balance of all thermodynamic forces in the (unrealized) limit. Condition approached arbitrarily closely in the limit: Total entropy of universe is constant. Total generalized forces vanish. X 1 +X 2 =0 Total free energy F=U-TS is constant. F 1 +F 2 =constant

36. Self-contained thermodynamically reversible processes 36 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.

37. Computing Fluctuations 37 Canonically distributed system in heat bath at T. give equilibrium, macroscopic description of non-equilibrium state F = -kT ln Z(V) F = -kT ln P + constant P ∝ exp(-F/kT) probability system at point with energy E ∝ probability P that system is in non- equilibrium state with phase volume V ∝ Z(V)

38. No-Go Result It, at last. 38

39. Combine 1. and 2. 39 initial final middle 1. Process is thermo- dynamically reversible F init = F mid = F fin stages P init ∝ exp(-F init /kT) P mid ∝ exp(-F mid /kT) P fin ∝ exp(-F fin /kT) 2. Fluctuations carry the system from one stage to another any isothermal, reversible process P init = P middle = P fin No-Go result

40. Fluctuation Disrupt All Reversible, Isothermal Processes at Molecular Scales 40 Intended process = 1 = 2 Actual process = 1 = 2

41. Beating Fluctuations 41

42. What it takes to overcome fluctuations 42 initial final Downward gradient in free energy recapture in most likely state release from here..but system can also be found in undesired intermediate states. Process moves from high free energy state to low free energy state.  F sys Net creation of thermodynamic entropy.  S tot = -  F sys /T

43. odds of final state P init = probability that fluctuation throws the system back to the initial state. What it takes to overcome fluctuations 43 initial final free energy recapture in most likely state release from here Least dissipative case High free energy mountain makes it unlikely that system is in intermediate stage.

44. Doing the sums… 44 Macroscopic Scale Odds of completion O fin = 20 P fin = 0.95  S tot = k ln 20 = 3k compare Landauer’s principle k ln2 = 0.69 k Odds of completion O fin = 7.2x10 10  S tot = k ln (7.2x10 10 ) = 25k 25kT is the mean thermal energy of ten nitrogen molecules. Molecular Scale

45. Bead on a Wire 45

46. 46 Each position is an equilibrium position Slow motion of bead over wire is a thermodynamically reversible process. (Tilt wire minutely.) Macroscopically… For 5g bead and T=25C v rms = 9.071 x 10 -10 m/s Molecular scale… For 100 amu mass (n-heptane molecule) and T=25C v rms = 157 m/s Effect of thermal fluctuation s

47. Overcome fluctuations by tilting wire 47 Macroscopically… For 5g bead  = 5.8x10 -18 radians For P fin = 0.999 T=25C stages 1/10 th length Depress by ~10 -7 Bohr radius H atom per meter. n-heptane is volatile! Molecular scale… For 100 amu mass (n-heptane molecule), turning the wire vertically has negligible effect!

48. Least dissipative case 48

49. More complicated cases 49

50. 50 Electric field moves a charge through a channel. Two state dipole measures sign of target charge. Computed in “All Shook Up…”

51. Conclusion 51

52. Thermal Fluctuations Cannot be Idealized Away No-Go result Thermal fluctuations (noise) prevent completion of any individual, molecular scale step. Thermodynamic entropy must be created to complete each step. Minimum entropy creation not set by the logical specification of the computation, but by the number of steps chained.

53. 53 The End

54. 54 Appendices

55. A Measurement Scheme Using Ferromagnets 55 Charles H. Bennett, “The Thermodynamics of Computation—A Review,” In. J. Theor. Phys. 21, (1982), pp. 905-40,

56. A Measurement Scheme Using Ferromagnets 56 Charles H. Bennett, “The Thermodynamics of Computation—A Review,” In. J. Theor. Phys. 21, (1982), pp. 905-40,

57. Thermodynamically reversible processes are NOT… 57 …merely very slow processes. capacitor discharges very slowly through resistor balloon deflates slowly through a pinhole …merely processes that can go easily in either way. one molecule gas released

58. Computing Fluctuations 58 Isolated, microcanonically distributed system 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)