Approximationand Idealization

John D. NortonDepartment of History and Philosophy of Science

Center for Philosophy of ScienceUniversity of Pittsburgh

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4th Tuebingen Summer Schoolin History and Philosophy of Science, July 2015

This Lecture

Stipulate that:

“Approximations” are inexact descriptions of a target system.

“Idealizations” are novel systems whose properties provide inexact descriptions of a target system.

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1 Infinite limit systems can have quite unexpected behaviors and fail to provide idealizations.

Extended example:Thermodynamic and other limits of statistical mechanics.

Infinite idealizations are often only limiting property approximations.

2

Fruitless debates over reduction and emergence in phase transitions derive from unnoticed differences in the notion of level.

3 reduction theoryemergence scale

CharacterizingApproximation and

Idealization

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Approximation

Stipulate …

Idealization

The Proposal

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Target system(boiling stew at roughly 100oC )

“The temperature is 100oC.”

Inexact description(Proposition)

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

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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

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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

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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.

“Limit property”

“Limit system”

Limit Property and Limit System Agree

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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

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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

When Idealization Succeeds and Fails

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N=1 N=2 N=3

… limit system

System1 1

N=∞

… limit property

Property 2 2 2 2

Limit property exists,

but NO limit system.

Limit propertyand

limit system

DISagree.

Limit property is an approximation for large N systems. No idealization.

Limit propertyand

limit system

agree

Limit system is idealization of large N systems.

Limits in Statistical

Physics

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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.

But now extra conditions have to be added by hand to reproduce the essential functions of the boundary conditions.

Inessential complications…??

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“We emphasize that we are not considering the theory of infinite systems for its own sake so much as for the fact that this is the only precise way of removing inessential complications due to boundary effects, etc.,…”

Lanford, 1975, p.17

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

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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

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Boltzmann-Grad Limit

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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

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Variables

2 x 3 velocity components for

outgoing masses 6

Renormalization Group Methods

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Renormalization Group Methods

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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

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space of reduced

Hamiltonians

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.

Finite Systems Control

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“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.

Reduction?Emergence?

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Phase transitions are…

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… a success of the reduction of thermodynamics by statistical mechanics.

… a clear example ofnon-reductiveemergence.

Norton, Butterfield

Who is right?

BOTH!..and no one is more right.

Different Senses of “Levels”

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Molecular-statistical Description.Phase space of canonical positions and momenta.Hamiltonian, canonical distribution, Partition function.Canonical entropy, free energy….

Few component molecular-statistical level

Many component molecular-statistical level

p

q

Thermodynamic level.State space pressure, volume, temperature, …Internal energy, free energy, entropy, …

Where Reduction Succeeds

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Level of many component, molecular-statistical theory

Critical exponents in vicinity of critical points.

Renormalization group flow on space of reduced Hamiltonians.

Level of thermodynamic theorydeduce

(Augmented) Nagel-style reduction:

Higher level theory

deduce surrogate forLower

level theory

Where Emergence Happens

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Few component molecular-statistical level

A few components• by themselves do not manifest phase transitions• in the mean field of the restdo not manifest the observed phase transition behavior quantitatively.

Many component molecular-statistical level

Quantitatively correct results from considering many components and their fluctuations from mean quantities.

“More is Different…”

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"The constructionist hypothesis [ability to start from fundamental laws and reconstruct the universe] breaks down when confronted with the twin difficulties of scale and complexity. The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each level of complexity, entirely new properties appear...”

P. W. Anderson, Science, 1972.

NH

HH N

H

HH

invert

few atoms--symmetry

do not invert

many atoms—broken symmetry

Leo Kadanoff"More is the Same…." Journal of Statistical Physics, 137 (December 2009)

Phase transitions are “a prime example of Anderson’s thesis.”

A conjecture…

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Level = theory Level = processes at same scale

Cannot mix results from different theories in one level. (deductive closure)

Draw whichever results needed from any applicable theory.

Few-many distinction divides condensed matter physics from atomic and particle physics.

Few-many distinction is within one theory. Mean field theory is an approximation, not a level.

Philosophers tend to

divide by theory.

Physicists tend to

divide by scale.

Reduction/emergence between self-contained theories of thermodynamic and statistical mechanics.

Reduction/emergence between systems of few components and many components.

Condensed matter physics deals with systems of many components.Solids, liquids, condensates, …

Theory = deductive closure of a few apt propositions.

… but not philosophers suspicious of theories as units.

Conclusion

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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.

37

1 Infinite limit systems can have quite unexpected behaviors and fail to provide idealizations.

Extended example:Thermodynamic and other limits of statistical mechanics.

Infinite idealizations are often only limiting property approximations.

2

Fruitless debates over reduction and emergence in phase transitions derive from unnoticed differences in the notion of level.

3 reduction theoryemergence scale

The End

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Appendices

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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?

?∞