Information and Thermodynamic Entropy John D. Norton Department of History and Philosophy of Science Center for Philosophy of Science University of Pittsburgh 1 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

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

This Talk Background Maxwell’s demon and the molecular challenge to the second law of thermodynamics. Exorcism by principle Szilard’s Principle, Landauer’s principle 3 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.

Fluctuations and Maxwell’s demon 4

The original conception J. C. Maxwell in a letter to P. G. Tait, 11 th 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.” Divided chamber with a kinetic gas. Demon operates door intelligently “[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.” 5

Maxwell’s demon lives in 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 Could these momentary, miniature violations of the second law be accumulated to large-scale violations? Guoy (1888), Svedberg (1907) designed mini- machines with that purpose. 6 “One can almost see Maxwell’s demon at work.” Poincaré, 1905

Szilard’s One-Molecule Engine 7

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

The One-Molecule Engine Initial state A partition is inserted to trap the molecule on one side. The gas undergoes a reversible, isothermal expansion to its original state. 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:

The One-Molecule Engine Initial state A partition is inserted to trap the molecule on one side. The gas undergoes a reversible, isothermal expansion to its original state. 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:

Exorcism by principle 11

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

Failed proofs of Landauer’s Principle 13

Direct Proofs that model the erasure processes in the memory device directly. 14 Wrong sort of entropy. No connection to heat. 3. Information-theoretic Entropy “p ln p” Associate entropy with our uncertainty over which memory cell is occupied. A n inefficiently designed erasure procedure creates entropy. No demonstration that all must. 1. Thermalization 2. Phase Volume Compression aka “many to one argument” Erasure need not compress phase volume but only rearrange it. or 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

4. Indirect Proof: General Strategy 15 Process known to reduce entropy Arbitrary erasure process coupled to Assume second law of thermodynamics holds on average. Entropy must increase on average. Entropy reduces.

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

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

Dissipationless Erasure 18 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.

The Importance of Fluctuations 19

Marian Smoluchowski, Exorcism of Maxwell’s demon by fluctuations. 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. 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.

Fluctuations disprupt Reversible Expansion and Compression 21

The Intended Process 22 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 h eq = kT/Mg

The massive piston… 23 ….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

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

Fluctuations disrupt Measurement and Detection 25

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): (November, 1987). 26 …is fatally disrupted by fluctuations that leave the keel rocking wildly. FAILS

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

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

A General Model of Detection 29 First step: the detector is coupled with the target system. The process is isothermal, thermodynamically reversible: It proceeds infinitely slowly. The driver is in equilibrium with the detector. The process intended: The coupling is an isothermal, reversible compression of the detector phase space.

A General “No-Go” Result 30

Fluctuation Disrupt All Reversible, Isothermal Processes at Molecular Scales 31 Intended process = 1 = 2 Actual process = 1 = 2

Einstein-Tolman Analysis of Fluctuations 32 Probability density that system is in stage Total system of gas-piston or target-detector-driver is canonically distributed. p(x,  ) = (1/Z) exp(-E(x,  )/kT) Z( ) = ∫ exp(-E(x,  )/kT) dxd  Different stages p( ) proportional to Z( ) Free energy of stage F( ) = - kT ln Z( ) Probability density for fluctuation to stage  : p(λ) proportional to exp(-F(  )/kT) p( 2 ) p( 1 ) = exp ( - ) F(  2 )-F(  1 ) kT Different subvolumes of the phase space.

Equilibrium implies uniform probability over 33 Condition for equilibrium ∂F/∂ = 0 F( ) = constant Probability distribution over p( ) = constant p( 1 ) = p( 2 ) Time evolution over phase space Expected p( 2 ) p( 1 ) = exp ( - ) F(  2 )-F(  1 ) kT since Actual

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

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

What it takes to overcome fluctuations 36 Enforcing a small probability gradient… p( 2 ) p( 1 ) = exp ( - ) > exp(3) = 20 F(  2 )-F(  1 ) kT …requires a disequilibrium… F(  1 ) > F(  2 ) + 3kT …which creates entropy. S(  2 )-S(  1 ) – (E(  2 )-E(  1 ))/T = 3k Exceeds the entropy k ln2 = 0.69k tracked by Landauer’s Principle! No problem for macroscopic reversible processes. F(  1 ) - F(  2 ) = 25kT p(  2 )/p(  1 ) = 7.2 x = mean thermal energy of ten Oxygen molecules

More Woes 37

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

In Sum… We are selectively ignoring fluctuations. 39 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.

Conclusions 40

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

Conclusions 42 Is a Maxwell demon possible? The best analysis is the Smoluchowski fluctuation exorcism of 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. It selectively considers and ignores fluctuation phenomena according to the result sought. Its inventory of processes is assembled inconsistently.

43

44 Finis

45 Appendix

A dilemma for information theoretic exorcisms 46

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

1. 48

1. Thermalization 49 Initial data L or R 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. Entropy created in this ill- advised, dissipative step. !!! Irreversible expansion “thermalization” Reversible isothermal compression passes heat kT ln 2 to heat bath. Data reset to L Entropy k ln 2 created in heat bath

2. 50

2. Phase Volume Compression aka “many to one argument” 51 Boltzmann statistical mechanics thermodynamic entropy k ln (accessible phase volume) = “random” data reset data occupies twice the phase volume of 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.

2. Phase Volume Compression aka “many to one argument” 52 “random” data reset data DOES NOT occupy twice the phase volume of thermalized data Confusion with It occupies the same phase volume. FAILS

A Ruinous Sense of “Reversible” 53 Random data and thermalized data have the same entropy because they are connected by a reversible, adiabatic process??? insertion of the partition removal of the partition No. Under this sense of reversible, entropy ceases to be a state function.  S = 0  S = k ln 2 random data thermalized data

3. 54

3. Information-theoretic Entropy “p ln p” 55 “random” data reset data Information entropy P i ln P i S inf = - k  i P L = P R = 1/2 S inf = k ln 2 P L = 1; P R = 0 S inf = 0 Hence erasure reduces the entropy of the memory by k ln 2, which must appear in surroundings. But… in this case, Information entropy Thermodynamic entropy does NOT equal Thermodynamic entropy is attached to a probability only in special cases. Not this one.

What it takes… 56 IF… Information entropy Thermodynamic entropy DOES equal “p ln p”Clausius dS = dQ rev /T A system is distributed canonically over its phase space p(x) = exp( -E(x)/kT) / Z Z normalizes All regions of phase space of non-zero E(x) are accessible to the system over time. AND For details of the proof and the importance of the accessibility condition, see Norton, “Eaters of the Lotus,” Accessibility condition FAILS for “random data” since only half of phase space is accessible.

4. 57

4. An Indirect Proof 58 gas memory One- Molecule Reduces entropy of heat bath by k ln 2. isothermal reversible expansion insert partition or shift cell to match dissipationlessly detect gas state or Dissipationlessly detect memory state. If R, shift to L. Net effect is a reduction of entropy of heat bath. Second law violated even in statistical form. (Earman and Norton, 1999, “no-erasure” demon.) Final step is a dissipationless erasure built out of processes routinely admitted in this literature. Fails

“…the same bit cannot be both the control and the target of a controlled operation…” 59 Every negative feedback control device acts on its own control bit. (Thermostat, regulator.) The Most Beautiful Machine 2003 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!”

Marian Smoluchowski, 1912 The second law holds on average only over time. Machines that try to accumulate fluctuations are disrupted fatally by them. 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 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. 60 Exorcism of Maxwell’s demon by fluctations.

The standard inventory of processes 61

We may… 62 Exploit the fluctuations of single molecule in a chamber at will. Insert and remove a partition Perform reversible, isothermal expansions and contractions Inventory read from steps in Ladyman et al. proofs.

We may… 63 Detect the location of the molecule without dissipation. ?? Shift between equal entropy states without dissipation. ? Trigger new processes according to the location detected. Gas Memory R L