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http://www.eng.fsu.edu/~mpf. EEL 5930 sec. 5, Spring ‘05 Physical Limits of Computing. Slides for a course taught by Michael P. Frank in the Department of Electrical & Computer Engineering. Review of Basic Physics Background. (Module 2). Basic physical quantities & units. Unit prefixes

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eel 5930 sec 5 spring 05 physical limits of computing

http://www.eng.fsu.edu/~mpf

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

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

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

M. Frank, "Physical Limits of Computing"

unit prefixes
Unit Prefixes
  • 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"

three fundamental quantities
Three “fundamental” quantities

M. Frank, "Physical Limits of Computing"

some derived quantities
Some derived quantities

M. Frank, "Physical Limits of Computing"

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

M. Frank, "Physical Limits of Computing"

information entropy temperature
Information, Entropy, Temperature
  • 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"

unit definitions conversions
Unit definitions & conversions
  • 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"

some fundamental physical constants
Some fundamental physical constants
  • 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"

physics that you should already know
Physics that you should already know
  • 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 mechanics
Generalized 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"

euler lagrange equation
Euler-Lagrange Equation

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"

euler lagrange example
Euler-Lagrange example
  • 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"

least action principle
Least-Action Principle

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"

hamilton s equations
Hamilton’s Equations

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"

field theories
Field Theories
  • 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"

the speed of light limit
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"

early history of the limit
Early History of the Limit
  • 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"

relativity non intuitive but true
Relativity: Non-intuitive, but True
  • 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"

some consequences of relativity
Some Consequences of Relativity
  • 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"

three ways to understand the c limit
Three Ways to Understand the c limit
  • 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"

the c limit in quantum physics
The c limit in quantum physics
  • 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"

the lorentz transformation
The “Lorentz” Transformation

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"

some consequences of the lorentz transform
Some Consequences of the Lorentz Transform
  • 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"

lorentz transform visualization
Lorentz Transform Visualization

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"

mixed frame version of lorentz transformation
Mixed-Frame Version of Lorentz Transformation
  • 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"

visualization of the mixed frame perspective
Visualization of the Mixed Frame Perspective

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"

relativistic kinetic energy
Relativistic Kinetic Energy
  • 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"

spacetime intervals
Spacetime Intervals
  • 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"

relativistic momentum
Relativistic Momentum
  • 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"