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Physical Limits on Information Processing

Physical Limits on Information Processing. We have Murphy’s law and Moore’s law! Laws of physics imposes limits! These limiting factors due to: Thermodynamics Relativity Quantum physics Gravity Quantum gravity effects Others?

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Physical Limits on Information Processing

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  1. Physical Limits on Information Processing

  2. We have Murphy’s law and Moore’s law! • Laws of physics imposes limits! • These limiting factors due to: • Thermodynamics • Relativity • Quantum physics • Gravity • Quantum gravity effects • Others? • These impose limits, independent of hardware technology, on processing rate, operating and transmission speeds, memory storage capacity, etc.

  3. Thermodynamic Limit • Information costs energy: not free! • Landauer’s principle • To store, insert or erase one bit of information requires energy: kTln2 • If P Watts is power available, maximum processing rate: rmax = P/kTln2 bits/sec. • For 1W power, at room temp. • rmax = 1020 bits/s (100exabits/s) • In general multiply by efficiency factore

  4. Processing produces heat! • Human brain performs 10Quadrillion ops/sec (1016/s) in a volume of 1 litre! • This produces heat that has to be taken away • At body temp., heat produced is 104 ergs/s • Brain uses 10W, upper limit on the processing rate ~ 1021/s! • Far more efficient than state of the art computing systems, energy wise • Heat generated over volume & lost over the surface • Ratio proportional to size • So brain cannot be too large, it would melt • Literally HOT HEADED !

  5. As living organisms we also emit, mostly in IR, at temp. of 310K, l = 10m. • Per unit mass, we are 104 times more luminous than the sun! >5X104 times than ULIRG’s and > 105 times LIRG’s • (L/M)sun= 2 ergs/g/s • (L/M)human= 2X104 ergs/g/s • Metabolic rate for 60kg man is 2500Kcal/day • 2500 X 103 X 4.2J = 1014 ergs • (L/M) = (1014 ergs)/(6 X 104g X 8.4 X 104s) = 2X104 ergs/g/s

  6. Metabolic rate of wide range of organisms, from whales to wombat, scales as M3/4 (Kleiber’s law) • So L/M scales as M-1/4 • For smallest mammals L/M = 7 X 104 ergs/g/s • Strangely, for maximal Eddington luminosity • (L/M)Ed = 4pGmpc/sT = 7 X 104 ergs/g/s! • Maximum (L/M) tends to be same for stars and animals!

  7. A new class of galaxies discovered (Sanders, Ann. Rev. A. A. (1996)) that radiate the bulk of their luminosity in FIR (~ 5 to 1000m), greater than 1011-1012 Lsun • Classified as LIRG’s or ULIRG’s • Starbursts in them nearly always triggered by galaxy-galaxy ints. • More numerous in the past • Higher density, more gas etc.

  8. Sensory Organs • Ear threshold of intensity ~10-16 W/cm2 • 100 decibel is about 1010 times the threshold • Eye can see a candle 10km away (can detect an individual quanta of light) • We can perceive a wide range of intensity (20 orders of magnitude) • Moonlight is 109 times the threshold

  9. General law of nature: (Weber- Fechner law) • Effect on sensory organ a log(intensity) • All detectors are log: Richter Beaufort scale • Olfactory: something different • Smell- diffusion flow

  10. Special relativistic limits • Special relativity puts limits on the signal speed to less than or equal to the speed of light • Limit on communication speed and transmission of information

  11. Quantum limits • Upper limit on rate of information processing, i.e., maximal rate of computation with a mean supply of power P is: • P ≥ ħf2 • Or f = (P/ħ)1/2bits/s • If E is available energy and t the processing time • P = E/t • For P = 1 kW, f = 1018 bits/s (1 Exabyte) • Eg. laser intensity is 1016 W/cm2, maximum information processing density is (1016/ħ)1/2 which is of the order of 1022 bits/cm2

  12. From CMBR • Background energy density: aT4Ac Watts • A is the area, T = 2.7 Kelvin, c light speed (limit) • I.P rate ~ (aT4Ac/ħ) ~ 1012 bits/cm2/s • Upper limit: 1 tera Flop/cm2 • Suppose device is using CMBR from earliest epoch, temperature is time dependant • T = (3c2/32pGa)1/4.(1/t(s)1/2) = 1010/t1/2 Kelvin

  13. Total information processed till present?! • Total I = ∫(acA/ħ)1/2(1020/t) dt = 1033 bits/cm2 • Multiplying by Hubble area: • Itotal = 1090 bits • It is the total entropy of CMBR • This is the number of photons in the CMBR • Entropy of radiation = number of photons a T3 • This is a conserved quantity • These are cosmological limits on processing

  14. Ultimate limit • Collapsing universe • Radiation gets blue shifted • T of background tends to infinity as t tends to 0 • Above integral diverges • So a dedicated computer (observer) can perform indefinitely large number of computation as he collapses into the big crunch! • A dark energy dominated universe is bad news for IT as the observer is dissociated from rest of the universe: Big Rip, etc…

  15. Phase space constraints • Quantum phase space: d3x d3p ≥ ħ3 • Not more than 2 bits per ħ3 • n/(mV)3 ≥ ħ-3, n ≤ m3V3/ħ • Degenerate gas can be used to store information • For electrons, max. allowed bits: 1032/cc • For holographic data storage (colour centres in crystals, etc.) 1023/cc HOLDOR

  16. Coherent optical information storage (group velocity of light, specially prepared media, brought almost to rest  COIS, in thin layer of volume V): n = (Nkc3)/48p2, kc skin depth, cut of wavelength • Analogy with black holes: C Sivaram (2005) • Ultimate phase space limit (t’Hooft) • Phase space of Universe/ħ3 = 10365 upper limit of storage • d3p= (Mc)3=10198; d3x= 2p2RH3 = 1085; ħ3= 10-81

  17. Astrophysical limits • Maximum luminosity of star: Eddington luminosity • The radiation pressure > gravity • This gives upper limit on processing rate: • (4pGmpc/sTħ)1/2.M1/2 ~ 1032 bits/s (M/Msun) • For total baryon mass of 1022Msun the upper limit is: 1043 bits/s • This strangely coincides with the upper limit allowed by general relativity

  18. General relativistic limits • GR gives upper limit to power generated as (Gunn luminosity): • Pmax = c5/G =1052 Watts (mass independent) • P= Mc2/(GM/c3): collapse to black hole over Schwarzschild time • This gives quantum gravity upper limit on processing as: • Imax = (Pmax/ħ)1/2 = (c5/Għ) 1/2 = 1043bits/s

  19. Hawking PBH • Luminosity due to Hawking radiation: • LH = (c5/G)(MPl/M)2; MPl= Planck mass (2X10-5 g) • It is maximum for M = MPl; LH = c5/G =1052 W • For solar mass BH, LH = 10-24 W • BH temp. = ħc/8pGMkB = 10-6 K for solar mass • For M = Msun, processing rate for Hawking flux is 106 bits/s (1 Mega Flop!) • Black hole has maximum information storage for given mass M • For the BH: I.S ≤ GM2/ħc = 1076 bits for M = Msun

  20. Given a region of size R, maximum information you can store is: • Imax ≤ R2/LPl2 = (c3/ħG)R2 • For Hubble radius of 1025 m • Imax = 10123 bits (1 quadragintillion bits) • Holographic upper bound

  21. Evaporating PBH: LBH = 1014 Watts • I ~ 1023 bits/s, total information 1040 bits • This system will run for Hubble age (1010 yrs) at 1023 bits/s • Particle spectrum, described in final stage by string degrees of freedom, no. of states grows exponentially with mass as: • m-3exp(m/TH) • Super string theory gives modified relation between input power P and processing rate f • P ≤ ħf2(1+ls2f2/c2); lsis the string scale • Max is given by (c5/Għ)(ls/lp)2

  22. Modified GUP for super string • For cosmic strings, gravitational radiation power is GT2 (T is the string tension), giving: • f ~ (GT2/ħ)1/2 • T ~ c2/G(Mcs/MPl)2; Mcs ~ 10-3 MPl • f ~ 1040 bits/s • Duration of computation ~ LT/GT2 = L/GT, L = 1kpc, t ~ 1017 seconds

  23. Showing complex organised structure in terms of their volume and processing power. Top right is the most complex. Libraries can store large info, but cannot process. Computers, living organisms can. Most complex: store large info and process rapidly

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