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Approximationand Idealization:
Why theDifference Matters
John D. NortonDepartment of History and
Philosophy of ScienceCenter for Philosophy of Science
University of Pittsburgh
1
Pitt-Tsinghua Summer School for Philosophy of ScienceInstitute of Science, Technology and Society, Tsinghua UniversityCenter for Philosophy of Science, University of PittsburghAt Tsinghua University, Beijing June 27- July 1, 2011
This Talk
Stipulate that:
“Approximations” are inexact descriptions of a target system.
“Idealizations” are novel systems whose properties provide inexact descriptions of a target system.
2
1Infinite limit systems can have quite unexpected behaviors and fail to provide idealizations.
Extended example:Thermodynamic and other limits of statistical mechanics.
Dominance argument:Infinite idealizations should be replaced by limiting property approximations.
2
CharacterizingApproximation and
Idealization
3
Approximation
Stipulate …
Idealization
Types of analyses
4
Target system(boiling stew at roughly 100oC )
“The temperature is 100oC.”
Inexact description(Language)
Another Systemwhose properties are an inexact description of the target system.
…and an idealization is more like a model, the more it has properties disanalogous to the target system.
A Well-Behaved Idealization
5
Target:Body in free fall
dv/dt = g – kv
v(t) = (g/k)(1 – exp(-kt))
= gt - gkt2/2 + gk2t3/6 - …
v = gtInexact description for the the first moments of fall (t is small).
Approximation
Body in free fall
in a vacuum
v = gtExact description
Idealization forfirst moments of fall.
Approximation only
6
Bacteria grow with generations roughly following an exponential formula.
Approximatewithn(t) = n(0) exp(kt)
System of infinitely many bacteria
fails to be an idealization.
fit improves at n grows large.
Take limit as n
infiniten(t)
=infiniten(0) exp(kt)
??
??
∞
Using infinite Limits
to formidealizations
7
“Limit property”
“Limit system”
Limit Property and Limit System Agree
8
Infinite cylinder has area/volume = 2.
system1, system2, system3, … , limit systemagrees
withproperty1, property2, property3, … , limit property
Infinite cylinder is an idealization for
large capsules.
areavolume
= 4 + 2a4 + a
4 + 24 +
4 + 44 + 2
, 4 + 64 + 3
, , … , 2
Limit property
There is no limit system.
There is no Limit System
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There is no such thing as an “infinitely big sphere.”
system1, system2, system3, … , ???
property1, property2, property3, … , limit property
Limit property is an
approximationfor large spheres.
There is no idealization.
?, … , 0
areavolume
= 4r2
4r3 = 3/r
3/1 , 3/2 , 3/3 ,
“Limit system”
Limit Property and Limit System Disagree
10
Infinite cylinder has
area/volume = 2.
system1, system2, system3, … , limit systemDISagrees
withproperty1, property2, property3, … , limit property
Infinite cylinder is NOT an idealization for large ellipsoids.
“Limit property”
areavolume
= 2a4a
, , … , 3Area formula holds only for large a.
formula for a=1
,formula for a=2
,formula for a=3
formula for a=4
Limits in Statistical
Physics
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Recovering thermodynamicsfrom statistical physics
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Very many small components interacting.
Thermodynamic system of continuous substances.
Treated statistically
often behaves almost exactly like…
Analyses routinely take “limit as the number of
components go to infinity.”
The question of this talk:how is this limit used?
?∞
Two ways to take the infinite limit
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Idealization
The “limit system” of infinitely many
components analyzed.
Its properties provide inexact descriptions of the target system.
Infinite systems may have properties very different from finite systems.
∞
Systems with infinitely many components are never considered.
∞Approximation
Consider properties as a function of number n
of components.“Properties(n)”
“Limit properties”Limn∞Properties(n)provide inexact descriptions of the properties of target system.
Thermodynamic limit as an
idealization
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Two forms of the thermodynamic limit
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Number of components
Volume
n ∞
V ∞
such that n/V isconstant
Strong. Consider a system of infinitely many components.
“The physical systems to which the thermodynamic formalism applies are idealized to be actually infinite, i.e. to fill Rν (where ν=3 in the usual world). This idealization is necessary because only infinite systems exhibit sharp phase transitions. Much of the thermodynamic formalism is concerned with the study of states of infinite systems.”
Ruelle, 2004
Idealization
Weak. Take limit only for properties.
Property(n)volume
well-defined limit density
Le Bellac, et al., 2004.
Approximation
Infinite one-dimensional crystal
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Spontaneously excites when disturbance propagates in “from infinity.”
then
then
then
then
Determinism, energy conservation fail.
This indeterminism is generic in infinite systems.
Problem for strong form.
Strong Form: Must Prove Determinism
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Simplest one dimensional system of interacting particles.
Clause bars monsters not arising in finite case.
Continuum limit as an
approximation
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Continuum limit
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Number of components
Volume
n ∞
V fixed
such that
Portion of space occupied by matter is constant.d = component size
nd3 = constant Boltzmann’s k 0Avogadro’s N ∞Fluctuations obliterated
No limit state.Stages do not approach continuous matter distribution.See “half tone printing” next.
Continuum limit provides
approximationLimit of properties is an inexact description of properties of systems with large n.
Idealization fails.
Useful for spatially inhomogeneous systems.
Half-tone printing analogy
20
State at pointx = 1/3y = 2/5
Oscillates indefinitely: black, black, white, white, black, black, white, white, …
At all stages of division
point in space is
blackoccupied or
whiteunoccupied
limit state ofgray =everywhere uniformly 50% occupied
Boltzmann-Grad limit as an
approximation
21
Boltzmann-Grad Limit
22
Useful for deriving the Boltzmann equation (H-theorem).
Number of components
Volume
n ∞
V fixed
such that
d = component size
nd2 = constant Portion of space occupied by matter 0
Limit stateof infinitely many point masses of zero mass. Can no longer resolve collisions uniquely.
System evolution in time has become
indeterministic.Limit properties provide approximation.Idealization fails.
Lose these for point masses.4
take limit…Equations 1 energy conservation3 momentum conservation2 direction of perpendicular surface6
Resolving collisions
23
Variables
2 x 3 velocity components for
outgoing masses 6
Renormalization Group Methods
24
Renormalization Group Methods
25
forexperts
Best analysis ofcritical exponents.
Zero-field specific heatCH ~ |t|
…Correlation length ~ |t|
…for reduced temperaturet=(T-Tc)/Tc
Transformations are degenerate if we apply them to systems of infinitely many componentsN = ∞.
!!
Renormalization group transformation generated by suppressing degrees of freedom:
Ncomponents
N’=bdNclusters of components
such that total partition function is preserved (unitarity):
Hence generate transformations of thermodynamic quantities
Total free energy F’ = -kT ln Z = F
Free energy per component
Z’(N’) = Z (N)
f’ = F’/N’ = F/bdN = f/bd
Properties of critical exponents recovered by analyzing RNG flow in region of space asymptotic to fixed point =region of finite system Hamiltonians.
The Flow
26
space of reduced
Hamiltonians
forexperts
Lines corresponding to systems of infinitely many components (critical points) are added to close topologically regions of the diagram occupied by finite systems.
Analysis employs
approximation andnot (infinite) idealization.
Elimination ofInfinite
Idealizations
27
Finite Systems Control
28
“The existence of a phase transition requires an infinite system. No phase transitions occur in systems with a finite number of degrees of freedom.”
Kadanoff, 2000
Necessity of infinite systems
Finite systems control infinite.vs“We emphasize that we are not considering the theory of infinite systems for its own sake… i.e. we regard infinite systems as approximations to large finite systems rather than the reverse.”
Lanford, 1975
Infinite system needed only for a mathematical discontinuity in thermodynamic quantities, which is not observed…and if it were, it wouldrefute the atomic theory!
Properties of finite systems control the analysis.
Dominance argument
29
Use of infinite idealization
requires:
Properties of infinite limit
system
must match
Limit properties of finite systems.
IFwe already know the properties of the finite systems,THENwe do not need the infinite limit system.
IFwe DO NOT already know the properties of the finite systems,THENwe cannot responsibly use limit system.
Either way, we should eliminate the infinite
idealization.
Else we mis-characterize the finite systems.
Conclusion
30
This Talk
Stipulate that:
“Approximations” are inexact descriptions of a target system.
“Idealizations” are novel systems whose properties provide inexact descriptions of a target system.
31
1Infinite limit systems can have quite unexpected behaviors and fail to provide idealizations.
Extended example:Thermodynamic and other limits of statistical mechanics.
Dominance argument:Infinite idealizations should be replaced by limiting property approximations.
2
32http://www.pitt.edu/~jdnorton/lectures/Tsinghua/Tsinghua.html
The End
33