Adiabatic Processes: Limits and Efficiency

1. Physical Limits of ComputingDr. Mike Frank CIS 6930, Sec. #3753XSpring 2002 Lecture #21Principles of Adiabatic ProcessesWed., Feb. 27

2. Administrivia & Overview • Don’t forget to keep up with homework! • We are 7 out of 14 weeks into the course. • You should have earned ~50 points by now. • Course outline: • Part I&II, Background, Fundamental Limits - done • Part III, Future of Semiconductor Technology - done • Part IV, Potential Future Computing Technologies - done • Part V, Classical Reversible Computing • Fundamentals of Adiabatic Processes - TODAY • Part VI, Quantum Computing • Part VII, Cosmological Limits, Wrap-Up

3. Terminology Shift • The word “infropy” sounds a bit too goofy. • Unlikely to be accepted into widespread use. • Shift in terminology used in this course:Before today: After today: “Infropy R”  “Physical Information I” (“Information”, “Pinformation”?) “Information I”  “Known information K” (“Kinformation”?) “Entropy S”  “Entropy S”

4. Adiabatic Processes - overview • Adiabatic steps in the reversible Carnot cycle • Evolution of the meaning of “adiabatic” • Time-proportional reversibility (TPR) of quasi-adiabatic processes • Adiabatic theorem of quantum mechanics • Adiabatic transitions of a two-state system • Logic & memory in irreversible and adiabatic processes.

5. The Carnot Cycle • In 1822-24, Sadi Carnot analyzed the efficiency of an ideal heat engine all of whose steps were reversible, and furthermore proved that: • Any reversible engine (regardless of details) would have the same efficiency (THTL)/TH. • No engine could have greater efficiency than a reversible engine w/o producing work from nothing • Temperature itself could be defined on a thermodynamic scale based on heat recoverable by a reversible engine operating between TH and TL

6. Steps of Carnot Cycle P • Isothermal expansion at TH • Adiabatic (without flow ofheat) expansion THTL • Isothermal compression at TL • Adiabatic compression TLTH TH TL V

7. Carnot Cycle Terminology • Adiabatic (Latin): orig. “Without flow of heat” • I.e., no entropy enters or leaves the system • Isothermal: “At the same temperature” • Temperature of system remains constant as entropy enters or leaves. • Both kinds of steps, in the case of the Carnot cycle, are examples of isentropic processes • “at the same entropy” • I.e., no (known) information is transformed into entropy in either process • But “adiabatic” has mutated to mean isentropic.

8. Old and New “Adiabatic” • Consider a closed system where you just lose track of its detailed evolution: • It’s adiabatic (no heat flow), • But it’s not “adiabatic” (not isentropic) • Consider a box containing some heat,flying ballistically out of the system: • It’s not adiabatic, • because heat is “flowing” out of the system • But it’s “adiabatic” (no entropy is generated) • Hereafter, we bow to the 20th century’s corrupt usage: let adiabatic : isentropic “The System” Box o’ Heat

9. Quasi-Adiabatic Processes • No real process is completely adiabatic • Because some outside system may always have enough energy to interact with & disturb your system’s evolution - e.g., cosmic ray, asteroid •  Evolution of system state is never perfectly known • Unless you know the exact quantum state of the whole universe • Entropy of your system always increases. • Unless it is already at a maximum (at equilibrium) • Can’t really be at complete equilibrium with its surroundings • unless whole universe is at utterly stable “heat death” state. • Systems at equilibrium are sometimes called “static.” • Non-equilibrium, quasi-adiabatic processes are sometimes also called quasi-static • Changing, but near a local equilibrium otherwise

10. Degree of Reversibility • The degree of reversibility (a.k.a. reversibility, a.k.a. thermodynamic efficiency) of any quasi-adiabatic process is defined as the ratio of: • the total free energy at the start of the process •  by the total energy spent in the process • Or, equivalently: • the known, accessible information at the start •  by the amount that is converted to entropy • This same quantity is referred to as the (per-cycle) “quality factor” Q for any resonant element (e.g., LC oscillator) in EE.

11. The “Adiabatic Principle” • Claim: Any ideal quasi-adiabatic process performed over time t has a thermodynamic efficiency that is proportional to t, • in the limit as t0. • We call processes that realize this idealization time-proportionally reversible (TPR) processes. • Note that the total energy spent (Espent), and the total entropy generated (S), are both inversely proportional to t in any TPR process. • The slower the process, the more energy-efficient.

12. Proving the Adiabatic Principle (See RevComp memo #M14) • Assume free energy is in generalized kinetic energy of motion Ek of system through its configuration space.Ek = ½mv2 v2 = (/t)2 t2 for m,  const. • Assume that every tf time, on average (mean free time), a constant fraction f of Ek is thermalized (turned into heat) • Whole process thermalizes energy f(t/tf)Ek  tt2 = t1. Constant in front is ½ fm2/tf: 2, whereh =½fm/tf is the effective viscosity.

13. Example: Electrical Resistance • We know Pspent=I2R=(Q/t)2R, or Espent=Pt = Q2R/t.  Note scaling with 1/t • Charge transfer through a resistor obeys the adiabatic principle! • Why is this so, microscopically? • In most situations, conduction electrons have a large thermal velocity relative to drift velocity. • Scatter off of of lattice-atom cross-sections with a mean free time tf that is fairly independent of drift velocity • Each scattering event thermalizes the electron’s drift kinetic energy - a frac. f of current’s total Ek • Therefore assumptions in prev. proof apply!

14. Adiabatic Theorem • A result of basic quantum theory • proved in many quantum mechanics textbooks • Paraphrased: A system initially in its ground state (or more generally, its nth energy eigenstate) will, after subjecting it to a sufficiently slow change of applied forces, remain in the corresponding state, with high probability. • Amount of leakage out of desired state is proportional to speed of transition at low speeds • Quantum systems obey the adiabatic principle!

15. Two-state Systems • Consider any system having an adjustable, bistable potential energy surface in its configuration space. • The two stable states form a natural bit. • One state represents 0, the other 1. • Consider now the well having twoadjustable parameters: • Height of the potential energy barrierrelative to the well bottom • Relative height of the left and rightstates in the well (bias) 0 1

16. Possible Parameter Settings • We will distinguish six qualitatively different settings of the well parameters, as follows… BarrierHeight Direction of Bias Force

17. Possible Adiabatic Transitions • Catalog of all the possible transitions in these wells, adiabatic & not... (Ignoring superposition states.) 1 1 1 leak Logicvalue 0 0 leak 0 BarrierHeight N 1 0 Direction of Bias Force

18. Ordinary Irreversible Logics • Principle of operation: Lower a barrier, or not, based on input. Series/parallel combinations of barriers do logic. Major dissipation in at least one of the possible transitions. 1 0 Example: CMOS logics 0

19. Ordinary Irreversible Memory • Lower a barrier, dissipating stored information. Apply an input bias. Raise the barrier to latch the new informationinto place. Remove inputbias. 1 1 0 0 Example:DRAM N 1 0

20. Input-Bias Clocked-Barrier Logic • Cycle of operation: • Data input applies bias • Add forces to do logic • Clock signal raises barrier • Data input bias removed Can amplify/restore input signalin clocking step. Retractinput 1 1 Retractinput Clockbarrierup Can reset latch reversibly givencopy of contents. 0 0 Clock up Input“1” Input“0” Examples: AdiabaticQDCA, SCRL latch, Rod logic latch, PQ logic,Buckled logic N 1 0

21. Input-Barrier, Clocked-Bias Retractile * Must reset outputprior to input.* Combinational logiconly! • Cycle of operation: • Inputs raise or lower barriers • Do logic w. series/parallel barriers • Clock applies bias force which changes state, or not 0 0 0 Examples:Hall logic,SCRL gates,Rod logic interlocks Input barrier height N 1 0 Clocked force applied 

22. Input-Barrier, Clocked-Bias Latching • Cycle of operation: • Input conditionally lowers barrier • Do logic w. series/parallel barriers • Clock applies bias force; conditional bit flip • Input removed, raising the barrier &locking in the state-change • Clockbias canretract 1 0 0 0 Examples: Mike’s4-cycle adiabaticCMOS logic N 1 0