EEL 5930 sec. 5, Spring ‘05 Physical Limits of Computing - PowerPoint PPT Presentation

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EEL 5930 sec. 5, Spring ‘05Physical Limits of Computing

Slides for a course taught byMichael P. Frankin the Department of Electrical & Computer Engineering

Review of Basic Physics Background

(Module 2)

• Unit prefixes
• Basic quantities
• Units of measurement
• Planck units
• Physical constants

M. Frank, "Physical Limits of Computing"

• See http://www.bipm.fr/enus/3_SI/si-prefixes.htmlfor the official international standard unit prefixes.
• When measuring physical things, these prefixes always stand for powers of 103 (1,000).
• But, when measuring digital things (bits & bytes) they often stand for powers of 210 (1,024).
• See also alternate kibi, mebi, etc. system at http://physics.nist.gov/cuu/Units/binary.html
• Don’t get confused!

M. Frank, "Physical Limits of Computing"

M. Frank, "Physical Limits of Computing"

M. Frank, "Physical Limits of Computing"

• We’ll skip magnetism & related quantities this semester.

M. Frank, "Physical Limits of Computing"

• These are important physical quantities also
• But, are different from other physical quantities
• They are based on combinatorics and statistics
• But, we’ll wait to explain them till we have a whole lecture on this topic later.
• Interestingly, there have been attempts to describe all physical quantities & entities in terms of information (e.g., Frieden, Fredkin).

M. Frank, "Physical Limits of Computing"

• See http://www.cise.ufl.edu/~mpf/physlim/units.txt for definitions of the above-mentioned units, and more. (Source: Emacs calc software.)
• Many mathematics applications have built-in support for physical units, unit prefixes, unit conversions, and physical constants.
• Emacs calc package (by Dave Gillespie)
• Mathematica
• Matlab - ?
• Maple - ?
• You can also do conversions using Google or using other web-based calculators.

M. Frank, "Physical Limits of Computing"

• Speed of light c = 299,792,458 m/s
• Planck’s constant h = 6.6260755×1034 J s
• Reduced Planck’s constant  = h / 2
• Remember this with the analogy: (h : 360°) :: ( : 1 radian)
• In fact, later we’ll see it’s valid to view h, as being these angles.
• Newton’s gravitational constant: G = 6.67259×1011 Nm2 / kg
• Boltzmann’s constant:k = kB = log e = 1.3806513×1023 J / K
• Others: permittivity of free space, Stefan-Boltzmann constant, etc. to be introduced later, as we go along.

M. Frank, "Physical Limits of Computing"

• Basic Newtonian mechanics
• Newton’s laws, motion, energy, etc.
• Basic electrostatics
• Ohm’s law, Kirchoff’s laws, etc.
• Also helpful, but not prerequisite (we’ll introduce them as we go along):
• Basic statistical mechanics & thermodynamics
• Basic quantum mechanics
• Basic relativity theory

M. Frank, "Physical Limits of Computing"

Generalized Classical Mechanics

• Classical mechanics can be expressed most generally and concisely in the Lagrangian and Hamiltonian formulations.
• Based on simple functions of the system state:
• The Lagrangian: Kinetic minus potential energy.
• The Hamiltonian: Kinetic plus potential energy.
• The dynamical laws can be derived from either of these energy functions.
• This framework generalizes to be the basis for quantum mechanics, quantum field theories, etc.

M. Frank, "Physical Limits of Computing"

Note the over-dot!

Where:

• L(q, v) is the system’s Lagrangian function.
• qi :≡ Generalized position coordinate w. index i.
• vi :≡ Generalized velocity coordinate i,
• or (as appropriate)
• t :≡ Time coordinate
• In a given frame of reference.

or just

M. Frank, "Physical Limits of Computing"

• Let q = (qi) (with i {1,2,3}) be the ordinary x, y, z coordinates of a point particle with mass m.
• Let L = ½mvi2 − V(q). (Kinetic minus potential.)
• Then, ∂L/∂qi = − ∂V/∂qi = Fi
• The force component in direction i.
• Meanwhile, ∂L/∂vi = ∂(½mvi2)/∂vi = mvi = pi
• The momentum component in direction i.
• And,
• Mass times acceleration in direction i.
• So we get Fi = mai or F = ma (Newton’s 2nd law)

M. Frank, "Physical Limits of Computing"

A.k.a.Hamilton’sprinciple

• The action of an energy quantity means the integral of that quantity over time.
• The trajectory specified by the Euler-Lagrange equation is one that locally extremizes the action of the Lagrangian:
• Among trajectories s(t)between specified pointss(t0) and s(t1).
• Infinitesimal deviations from this trajectory leave the action unchanged, to 1st order.

M. Frank, "Physical Limits of Computing"

Implicitsummationover i here.

• The Hamiltonian is defined as H :≡ vipi − L.
• Equals Ek + Ep if L = Ek − Ep and vipi = 2Ek = mvi2.
• We can then describe the dynamics of (q, p) states using the 1st-order Hamilton’s equations:
• These are equivalent to (but often easier to solve than) the 2nd-order Euler-Lagrange equation.
• Note that any Hamiltonian dynamics is what we might call bi-deterministic
• Meaning, deterministic in both the forwards and reverse time directions.

M. Frank, "Physical Limits of Computing"

• Here the space of indexes i of the generalized coordinates is continuous, thus uncountable.
• Usually it forms some topological space T, e.g., R3.
• We often use φ(x) notation in place of qi.
• In local field theories, the Lagrangian L(φ) is the integral of a Lagrange density function ℒ(x) where the point x ranges over the entire space T.
• This ℒ(x) depends only locally on the field φ, e.g.,

ℒ(x) = ℒ[φ(x), (∂φ/∂xi)(x), (x)]

• All successful physical theories can be explicitly written down as local field theories!
• Thus, there is no instantaneous action at a distance.

M. Frank, "Physical Limits of Computing"

Special Relativity and the Speed-of-Light Limit

• No form of information (including quantum information) can propagate through space at a velocity (relative to its local surroundings) that is greater than the speed of light, c ≈ 3×108 m/s.
• Some consequences:
• No closed system can propagate faster than c.
• Although you can define open systems that do, by definition
• No given “chunk” of matter, energy, or momentum can propagate faster than c.
• The influence of all of the fundamental forces (including gravity) propagates at (at most) c.
• The probability mass associated with a quantum particle flows in an entirely local fashion, at no faster than c.

M. Frank, "Physical Limits of Computing"

• The principle of locality was first anticipated by Newton
• He wished to get rid of the “action at a distance” aspects of his law of gravitation.
• The fact of the finiteness of the speed of light (SoL) was first observed experimentally by Roemer in 1676.
• The first decent speed estimate was obtained by Fizeau in 1849.
• Weber & Kohlrausch derived a constant velocity of c from empirical electromagnetic constants in 1856.
• Kirchoff pointed out the match with the speed of light in 1857.
• Maxwell showed that his EM theory implied the existence of waves that always propagate at c in 1873.
• Hertz later confirmed experimentally that EM waves indeed existed
• Michaelson & Morley (1887) observed that the empirical SoL was independent of the observer’s state of motion!
• Maxwell’s equations are apparently valid in all inertial reference frames!
• Fitzgerald (1889), Lorentz (1892,1899), Larmor (1898), Poincaré (1898,1904), & Einstein (1905) explored the implications of this...

M. Frank, "Physical Limits of Computing"

• How can the speed of something be a fundamental constant? Seemed broken...
• If I’m moving at velocity v towards you, and I shoot a laser at you, what speed does the light go, relative to me, and to you?Answer: both c!(Notv+c.)
• Newton’s laws were the same in all frames of reference moving at a constant velocity.
• Principle of Relativity (PoR): All laws of physics are invariant under changes in velocity
• Einstein’s insight: The PoR is consistent w. Maxwell’s theory!
• But we must change the definition of space+time.

M. Frank, "Physical Limits of Computing"

• Measured lengths and time intervals in a system vary depending on the system’s velocity relative to observers.
• Lengths are shortened in direction of motion.
• Moving clocks run slower.
• Sounds paradoxical, but isn’t!
• Mass of moving objects is amplified.
• Energy and mass are really the same quantity measured in different units: E=mc2.
• Nothing (including energy, matter, information, etc.) can go faster than light! (SoL limit.)

M. Frank, "Physical Limits of Computing"

• Energy of motion contributes to mass of object.
• Mass approaches  as velocity  c.
• Infinite energy would be needed to reach c.
• Lengths, times in a faster-than-light moving object would become imaginary numbers!
• What would that even mean?
• Faster than light in one reference frame  Backwards in time in another reference frame
• Sending information backwards in time violates causality, leads to logical contradictions!

M. Frank, "Physical Limits of Computing"

• Sometimes you see statements about “non-local” effects in quantum systems. Watch out!
• Even Einstein made this mistake.
• Described a quantum thought experiment that seemed to require “spooky action at a distance.”
• Later it was shown that this experiment did not actually violate the speed-of-light limit for information.
• These “non-local” effects are only illusions, emergent phenomena predicted by an entirely local underlying theory respecting the SoL limit..
• Widely-separated systems can still maintain quantum correlations, but that isn’t “true” non-locality.

M. Frank, "Physical Limits of Computing"

Actually it was written down earlier; e.g., one form by Voigt in 1887

• Lorentz, Poincaré: All the laws of physics remain unchanged, relative to the reference frame (x′,t′) of an object moving with constant velocity v = Δx/Δt in another reference frame (x,t), under the following substitutions:

Where:

Note: our γ here is the reciprocal of the quantity denoted γ by other authors.

M. Frank, "Physical Limits of Computing"

• Length contraction: (Fitzgerald 1889, Lorentz 1892)
• An object having length  in its rest frame appears, when measured in a relatively moving frame, to have the (shorter) length γ.
• For lengths that are parallel to the direction of motion.
• Time dilation: (Poincaré, 1898)
• If time interval τ is measured between two co-located events in a given frame, a (larger) time t = τ/γ will be measured between those same two events in a relatively moving frame.
• Mass expansion: (Einstein’s fix for Newton’s F=ma)
• If an object has mass m0>0 in its rest frame, then it is seen to have the larger mass m = m0/γ in a relatively moving frame.

M. Frank, "Physical Limits of Computing"

increasing t′

increasing t

increasing x′

increasing x

x=0

x′=0

Original x,t(“rest”) frame

Line colors:

Isochrones(space-like)

t′=0

Isospatials(time-like)

New x′,t′(“moving”) frame

Light-like

In this example:

v = Δx/Δt = 3/5γ = Δt′/Δt = 4/5vT = v/γ = Δx/Δt′ = 3/4

t = 0

The “tourist’s velocity.”

M. Frank, "Physical Limits of Computing"

• Usual version (with c=1):
• Letting (xA,tA)=(x, t′) and (xB,tB)=(x′, t), andsolving for (xA,tA), we get:
• Or, in matrix form:
• The Lorentz transform is thus revealed as a simple rotation of the mixed-frame coordinates!

(Where θ = arctan vT)

M. Frank, "Physical Limits of Computing"

t′

t

t′

StandardFrame #1

MixedFrame #1

t

x

x

In this example:

v = Δx/Δt = 3/5

vT = Δx/Δt′ = 3/4γ = Δt′ /Δt = 4/5

Note that (Δt)2 = (Δx)2 + (Δt′)2by the PythagoreanTheorem!

Rememberthe slogan:

“My space isperpendicularto your time.”

x

t′

t

x′

StandardFrame #2

MixedFrame #2

x′

x′

Note the obvious complete symmetryin the relation between the two mixed frames.

M. Frank, "Physical Limits of Computing"

• Total relativistic energy E of any object is E = mc2.
• For an object at rest with mass m0, Erest = m0c2.
• For a moving object, m = m0/γ
• Where m0 is the object’s mass in its rest frame.
• Energy of the moving object is thus Emoving = m0c2/γ.
• Kinetic energy Ekin :≡ Emoving − Erest= m0c2/γ − m0c2 = Erest(1/γ − 1)
• Substituting γ = (1−β2)1/2 and Taylor-expanding gives:

Higher-orderrelativistic corrections

Pre-relativistic kinetic energy ½ m0v2

M. Frank, "Physical Limits of Computing"

• Note that the lengths and times between two events are not invariant under Lorentz transformations.
• However, the following quantity is an invariant: The spacetime interval s, where:

s2 = (ct)2− xi2

• The value of s is also the proper timeτ:
• The elapsed time in rest frame of object traveling on a straight line between the two events. (Same as what we were calling t′ earlier.)
• The sign of s2 has a particular significance:

s2 > 0 - Events are timelike separated (s is real)May be causally connected.

s2 = 0 - Events are lightlike separated (s is 0) Only 0-rest-mass signals may connect them.

s2 < 0 - Events are spacelike separated (s is imaginary)Not causally connected at all.

M. Frank, "Physical Limits of Computing"

• The relativistic momentum p = mv
• Same as classical momentum, except that m= m0/γ.
• Relativistic energy-momentum-rest-mass relation:E2 = (pc)2 + (m0c2)2If we use units where c = 1, this simplifies to just:E2 = p2 + m02
• Note that if we solve this for m02, we get:

m02 = E2 − p2

• Thus, E2 − p2 is another relativistic invariant!
• Later we will show how it relates to the spacetime interval s2 = t2 − x2, and to a computational interpretation of relativistic physics.

M. Frank, "Physical Limits of Computing"