Information and Thermodynamic Entropy or, Waiting for ...
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 4th Tuebingen Summer School in History and Philosophy of Science, July 2015 1 This Lecture The thermodynamics of computation presumes it is possible to
Chain molecular-scale computational steps that are thermodynamically reversible or nearly so. steps Compression and expansion of components spaces. Detection of memory device states. Moving data from one location to another. Bad Idealization
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. Pfin Stot = k ln = k lnO fin Pinit Minimum entropy creation not set by the logical specification of the computation, but by the number of steps chained.
Maxwells Demon 4 The Maxwell Era 1867-1905 5 Theory of Heat, 1871, first ed. Also Letter to Tait, 1867; Rayleigh 1871 6
Theory of Heat Better scan from 1872, 2nd ed. 7 Maxwells Proposal He will thus, without expenditure of work, air initially at uniform temperature raise the temperature of B and lower that of A, in contradiction to
the second law of thermodynamics. 8 Maxwells Moral: The Demon Wins 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. The demon illustrates that Second Law would fail if we could manipulate individual molecules. No compulsion to
exorcise the demon to protect the Second Law. . Nanotechnology has not yet overturned the Second Law. 9 The Fluctuation Era 1905-1929 10 Einsteins Brownian Motion Paper "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)
11 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. 12 Maxwells demon lives
in the details of Brownian motion and other fluctuations  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. Poincar, 1904 Could these momentary, miniature violations of the second law be accumulated to large-scale violations? A real Maxwells demon?
Guoy (1888), Svedberg (1907) designed minimachines with that purpose. 13 Svedbergs Proposal Colloid cools Charged colloid particles radiate their thermal energy. Casing heats Tuned lead casing
absorbs the radiation. plus many more layers, details designed to prevent return of heat. Svedberg, The. ber die Bedeutung der Eigenbewegung der Teilchen in kolloidalen Lsungen fr die Beurteilung der Gltigkeitsgrenzen des zweiten Haupsatzes der Thermodynamik.Annalen der Physik, 59 (1907) pp. 451458. 14 Exorcism of Maxwells demon by fluctuations. Marian Smoluchowski, 1912 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. 15 Marian Smoluchowski, 1912 Other examples of defeated demons. Later popularized by Feynman The second law holds on average only over time.
Machines that try to accumulate fluctuations are disrupted fatally by them. 16 The Information Era 1929- ???? 17 Szilard 1929 On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings. Zeitschrift fr Physik, 53 (1929), pp. 840-856.
18 The One-Molecule Engine Szilard 1929 1 4 2 3 A partition is Initial inserted to trap the molecule on one side.
state Work kT ln 2 The gas undergoes a reversible, isothermal expansion to its original state. gained in raising the weight. It comes from the heat kT ln 2, drawn from the heat bath. 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. Szilards Principle Szilard 1929 Von Neumann 1932 Brillouin 1951+ Acquisition of one bit of information by the demon creates k ln
2 of thermodynamic entropy. versus Landauers Principle Landauer 1961 Bennett 1987+ Szilards principle is false. 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.
20 Landauers Principle Landauers 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-informationbearing degrees of freedom of the information-processing apparatus or its 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 irreversible operation Must pass entropy to environment k ln 2 per bit erased (e.g. erasure)
Process is thermodynamically reversible if data is random; not if known data. Logically reversible operation Can be thermodynamically reversible Bennett, Charles H. (2003). Notes on Landauers Principle, Reversible Computation, and Maxwells Demon, Studies in History and Philosophy of Modern Physics, 34, pp. 501-10.
21 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. 22 No-Go Result 23 No-Go Result
NO molecular-scale process that completes is thermodynamically reversible. Thermodynamic entropy must be created to complete each step. Pfin Stot = k ln = k lnO fin Pinit S S S S S
S S S 24 No-Go Result Illustrated 25 Fluctuations disrupt
Reversible Expansion and Compression 26 The Intended Process 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 heq = kT/Mg
Heat kT ln 2 = 0.69kT passed in tiny increments from surroundings to gas. 27 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
28 What Happens. Fluctuations obliterate the very slow expansion intended Heat kT ln 2 = 0.69kT passed in tiny increments from surrounding to gas. A better analysis (elsewhere) does not need external adjustment of weight during expansion. It replaces the gravitational field
withpiston = 2kT ln (height) energy Mean energy of gas 3kT/2 Standard deviation (3/2)1/2kT = 1.225kT 29 Fluctuations disrupt Measurement and Detection 30 Measurement is compression of detector phase space
First step: the detector is coupled with the target system. The process intended: The process is isothermal, thermodynamically reversible: It proceeds very slowly. The driver is in equilibrium with the detector. The coupling is an isothermal, reversible compression of the detector phase space. 31
Fluctuations Obliterate Reversible Detection What we expected: What happens: 32 Bennetts 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 boatshaped 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. is fatally disrupted by fluctuations that leave the keel rocking wildly. Charles H. Bennett, Demons, Engines and the Second Law, Scientific American 257(5):108-116 (November, 1987). 33 No-Go Result Preparatory notions 34
Thermodynamically Reversible Processes For Two systems interacting isothermally in thermal contact with constant temperature surroundings at T: Thermodynamically reversible process Set of irreversible processes that approach a perfect balance of all thermodynamic forces in the (unrealized) limit. T 1
2 env internal energy change heat transferred generalized generalized force displacement dU = dq X dx Condition approached
arbitrarily closely in the limit: Total entropy of universe is constant. X = -F/ for process parameter Total generalized forces vanish. X1+X2=0 Total free energy F=U-TS is constant. F1+F2=constant 35
Self-contained thermodynamically 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. 36
Computing Fluctuations probability system at point with energy E probability P that system is in nonequilibrium state with V phase volume Canonically distributed system in heat bath at T. E
exp kT Z(V) E(x) Z(V ) = V exp dx kT give equilibrium, macroscopic description of non-equilibrium state F = -kT ln Z(V) F = -kT ln P + constant P exp(-F/kT) 37
No-Go Result It, at last. 38 Combine 1. and 2. any isothermal, reversible process final middle initial
stages 1. Process is thermodynamically reversible Finit = Fmid = Ffin 2. Fluctuations carry the system from one stage to another Pinit exp(-Finit/kT) Pmid exp(-Fmid/kT) Pfin exp(-Ffin/kT) Pinit = Pmiddle = Pfin
No-Go result 39 Fluctuation Disrupt All Reversible, Isothermal Processes at Molecular Scales Intended process =1 =2 =1
=2 Actual process 40 Beating Fluctuations 41 What it takes to overcome fluctuations Downward gradient in
free energy final initial 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. Fsys recapture in
most likely state Net creation of thermodynamic entropy. Stot = -Fsys/T 42 What it takes to overcome fluctuations Least dissipative case free energy final
initial release from here High free energy mountain makes it unlikely that system is in intermediate stage. Ffin Finit Pfin Stot = exp = exp k Pinit kT Pinit = probability that fluctuation throws
the system back to the initial state. recapture in most likely state Pfin Stot = k ln = k lnO fin Pinit odds of final state 43 Doing the sums Molecular Scale Odds of completion Ofin = 20 Pfin = 0.95 Stot = k ln 20 = 3k
compare Landauers principle k ln2 = 0.69 k Macroscopic Scale Odds of completion Ofin = 7.2x1010 Stot = k ln (7.2x1010) = 25k 25kT is the mean thermal energy of ten nitrogen molecules. 44 Bead on a Wire
45 Each position is an equilibrium position Macroscopically Effect of thermal fluctuation s Slow motion of bead over wire is a thermodynamically reversible process. (Tilt wire minutely.) Molecular scale
For 5g bead and T=25C For 100 amu mass (n-heptane molecule) and T=25C vrms = 9.071 x 10-10 m/s vrms = 157 m/s 46 Overcome fluctuations by tilting wire Macroscopically For Pfin = 0.999 T=25C stages 1/10 length th
For 5g bead = 5.8x10-18 radians Depress by ~10-7 Bohr radius H atom per meter. Molecular scale For 100 amu mass (n-heptane molecule), turning the wire vertically has negligible effect! n-heptane is volatile! 47 Least dissipative case 48
More complicated cases 49 Electric field moves a charge through a channel. Two state dipole measures sign of target charge. Computed in All Shook Up 50 Conclusion
51 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. Pfin Stot = k ln = k lnO fin Pinit Minimum entropy creation not set by the logical specification of the computation, but by the number of steps
chained. The End 53 Appendices 54 A Measurement Scheme Using Ferromagnets Charles H. Bennett, The Thermodynamics of Computation A Review, In. J. Theor. Phys. 21, (1982), pp. 905-40, 55
A Measurement Scheme Using Ferromagnets Charles H. Bennett, The Thermodynamics of Computation A Review, In. J. Theor. Phys. 21, (1982), pp. 905-40, 56 Thermodynamically reversible processes are NOT merely very slow processes. merely processes that can go easily in either way. capacitor discharges
very slowly through resistor balloon deflates slowly through a pinhole one molecule gas released 57 Computing Fluctuations probability P that system is in nonequilibrium state with phase volume
V Isolated, microcanonically distributed system phase volume V give equilibrium, macroscopic description of non-equilibrium state S = k ln V
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