Approximation
and Idealization
John D. Norton
Department of History and Philosophy of Science
Center for Philosophy of Science
University of Pittsburgh
4th Tuebingen Summer School
in History and Philosophy of Science, July 2015
1
This Lecture
1
Stipulate that:
“Approximations” are
inexact descriptions of a
target system.
“Idealizations” are novel
systems whose properties provide
inexact descriptions of a target
system.
3
2
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.
Fruitless debates over reduction and
reduction
emergence in phase transitions derive from
unnoticed differences in the notion of level. emergence
theory
scale
2
Characterizing
Approximation and
Idealization
3
The Proposal
Target
system
(boiling stew at
roughly 100oC )
“The
temperature
is 100oC.”
Inexact
description
(Proposition)
Stipulate …
Approximation
Idealization
Another System
whose 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.
4
A Well-Behaved Idealization
Target:
Body in
free fall
dv/dt = g – kv
v(t) = (g/k)(1 – exp(-kt))
= gt - gkt2/2 + gk2t3/6 - …
Body in
free fall
in a
vacuum
v = gt
Exact description
v = gt
Idealization for
Inexact description for the the first
moments of fall (t is small).
first moments of fall.
Approximation
5
Approximation only
Bacteria grow with
generations roughly following
an exponential formula.
Approximate
with
n(t) = n(0) exp(kt)
fit improves at
n grows large.
Take limit as n
infinite
n(t)
??
=
??

∞
infinite
n(0) exp(kt)
System of infinitely many bacteria
fails to be an idealization.
6
Using infinite
Limits
to form
idealizations
7
Two ways to take the infinite limit
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.
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.
Systems with
infinitely many
components are
never
considered.
8
Limit Property and Limit System Agree
Infinite cylinder has
area/volume = 2.
,
4p
4p/3 + 2p
area =
volume
4p + 2pa
4p/3 + pa
4p + 2p
4p/3 + p
4p +
Infinite cylinder is
an idealization for
large capsules.
,
6p
4p/3 + 3p
4p +
,… ,
2
“Limit
system”
“Limit
property”
system1, system2, system3, … , limit system
agrees
with
property1, property2, property3, … , limit property
9
There is no Limit System
?
3/1 ,
3/2 ,
3/3 ,
,… ,
There is no such
thing as an
“infinitely big
sphere.”
0
area = 4pr2
= 3/r
volume
4pr3/3
Limit property is an
approximation
for large spheres.
There is no idealization.
system1, system2, system3, … ,
There is no
limit system.
Limit
property
???
property1, property2, property3, … , limit property
10
Limit Property and Limit System Disagree
Infinite cylinder has
area/volume = 2.
formula
for a=1
,
formula
for a=2
,
formula
for a=3
area = p2a
volume
4pa/3
Infinite cylinder is
NOT an idealization
for large ellipsoids.
,
formula
for a=4
, … ,
3p/4
“Limit
system”
“Limit
property”
Area formula holds
only for large a.
system1, system2, system3, … , limit system
DISagrees
with
property1, property2, property3, … , limit property
11
When Idealization Succeeds and Fails
2
Property
1
2
2
1
2
…
System
N=1
…
N=2
N=3
N=∞
limit
property
limit
system
Limit property
and
limit system
Limit property
and
limit system
agree
Limit property
exists,
but NO limit
DISagree.
system.
Limit system is
idealization of large
N systems.
Limit property is an approximation for
large N systems. No idealization.
12
Limits in
Statistical
Physics
13
Thermodynamic
limit as an
idealization
14
Two forms of the thermodynamic limit
Number of
components
n
Volume
V
such that
n/V is
constant
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
15
Infinite one-dimensional crystal
then
then
Problem for
strong form.
Spontaneously
excites when
disturbance
propagates in
“from infinity.”
Determinism,
energy
conservation
fail.
then
then
This
indeterminism is
generic in
infinite systems.
16
Strong Form: Must Prove Determinism
Simplest one
dimensional
system of
interacting
particles.
Clause bars
monsters not arising
in finite case.
17
Inessential complications…??
“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.,…”
But now extra conditions have to
be added by hand to reproduce the
essential functions of the boundary
conditions.
Lanford, 1975, p.17
18
Continuum limit
as an
approximation
19
Useful for
spatially
inhomogeneous
systems.
Continuum limit
Number of
components
Volume
such that
n
V fixed
nd3 = constant
Portion of space occupied
by matter is constant.
d = component size
No limit state.
Stages do not approach
continuous matter distribution.
See “half tone printing” next.
Boltzmann’s k  0
Avogadro’s N 
Fluctuations obliterated
Continuum limit provides
approximation
Limit of properties is an
inexact description of
properties of systems with
large n.
Idealization fails.
20
Half-tone printing analogy
At all stages
of division
point in
space is
black
occupied
or
white
unoccupied
limit state of
gray =
everywhere
uniformly 50%
occupied
State at point
x = 1/3
y = 2/5
Oscillates indefinitely: black, black, white, white, black, black, white, white, …
21
Boltzmann-Grad
limit as an
approximation
22
Useful for deriving
the Boltzmann
equation (Htheorem).
Boltzmann-Grad Limit
Number of
components
Volume
such that
n
V fixed
nd2 = constant
d = component size
Limit state
of infinitely many point
masses of zero mass. Can no
longer resolve collisions
uniquely.
Portion of space
occupied by matter
0
System evolution in
time has become
indeterministic.
Limit properties provide
approximation.
Idealization fails.
23
Resolving collisions
2 x 3 velocity
components for
outgoing masses
Variables
6
Equations 1 energy conservation
4
6
3 momentum
conservation
2 direction of
perpendicular surface
take limit…
Lose these
for point
masses.
24
Renormalization
Group Methods
25
Renormalization Group Methods
Best analysis of
critical exponents.
Zero-field specific heat
CH ~ |t|-a
…
Correlation length
x ~ |t|-n
…
for reduced temperature
Renormalization group transformation
generated by suppressing degrees of
freedom:
N
components
N’=bdN
clusters of
components
such that total partition function is
preserved (unitarity):
t=(T-Tc)/Tc
Z’(N’) = Z (N)
Hence generate transformations of
thermodynamic quantities
Total free energy F’ = -kT ln Z = F
Free energy
f’ = F’/N’
per component
= F/bdN = f/bd
!! Transformations
are degenerate if
we apply them to
systems of
infinitely many
components
N = ∞.
26
The Flow
space of
reduced
Hamiltonians
Properties of
critical exponents
recovered by
analyzing RNG
flow in region of
space asymptotic
to fixed point
=
region of finite
system
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 and
not (infinite)
idealization.
27
Finite Systems Control
Necessity of infinite
systems
“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
vs
Finite systems control
infinite.
“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 would
refute the atomic theory!
Properties of finite systems
control the analysis.
28
Reduction?
Emergence?
29
Phase transitions are…
Norton, Butterfield
… a success of the reduction
of thermodynamics by
statistical mechanics.
… a clear example of
non-reductive
emergence.
Who is right?
BOTH!
..and no one is more right.
30
Different Senses of “Levels”
p
q
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
Thermodynamic level.
State space pressure, volume, temperature, …
Internal energy, free energy, entropy, …
31
Where Reduction Succeeds
Level of many component,
molecular-statistical theory
Renormalization group flow on
space of reduced Hamiltonians.
deduce
Level of
thermodynamic theory
Critical exponents in
vicinity of critical points.
(Augmented) Nagel-style reduction:
Lower
level
theory
deduce
surrogate for
Higher
level
theory
32
Where Emergence Happens
Few component
molecular-statistical level
Many component
molecular-statistical level
A few components
• by themselves do not
manifest phase transitions
• in the mean field of the rest
do not manifest the observed
phase transition behavior
quantitatively.
Quantitatively correct
results from considering
many components and
their fluctuations from
mean quantities.
33
“More is Different…”
P. W. Anderson, Science, 1972.
"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...”
few atoms--symmetry
N
H
H
H
invert
H
H
H
many atoms—broken symmetry
N
Leo Kadanoff
"More is the Same…." Journal of
Statistical Physics, 137 (December 2009)
do not
invert
Phase transitions are “a prime
example of Anderson’s thesis.”
34
A conjecture…
Philosophers tend to
Physicists tend to
divide by theory.
divide by scale.
Theory = deductive closure of a few
apt propositions.
Condensed matter physics deals with
systems of many components.
Solids, liquids, condensates, …
Level = theory
Level = processes at same scale
Reduction/emergence between selfcontained theories of thermodynamic
and statistical mechanics.
Reduction/emergence between
systems of few components and
many components.
Cannot mix results from different
theories in one level. (deductive closure)
Draw whichever results needed
from any applicable theory.
Few-many distinction is within
one theory. Mean field theory is an
Few-many distinction divides
condensed matter physics from
atomic and particle physics.
approximation, not a level.
35
Conclusion
36
This Talk
1
Stipulate that:
“Approximations” are
inexact descriptions of a
target system.
“Idealizations” are novel
systems whose properties provide
inexact descriptions of a target
system.
3
2
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.
Fruitless debates over reduction and
reduction
emergence in phase transitions derive from
unnoticed differences in the notion of level. emergence
theory
scale
37
The End
38
Appendices
39
Recovering thermodynamics
from statistical physics
Treated statistically
often behaves almost
exactly like…
Very many small
components
interacting.
Analyses routinely take
“limit as the number of
components go to infinity.”
Thermodynamic
system of continuous
substances.
The question of this talk:
how is this limit used?
?
40