Transcript Approximation and Idealization: Why the Difference Matters John D. Norton Department of History and Philosophy of Science Center for Philosophy of Science University of Pittsburgh Pitt-Tsinghua Summer School for.

Approximation
and Idealization:
Why the
Difference Matters
John D. Norton
Department of History and
Philosophy of Science
Center for Philosophy of Science
University of Pittsburgh
Pitt-Tsinghua Summer School for Philosophy of Science
Institute of Science, Technology and Society, Tsinghua University
Center for Philosophy of Science, University of Pittsburgh
At Tsinghua University, Beijing
June 27- July 1, 2011
1
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.
2
Infinite 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
Characterizing
Approximation and
Idealization
3
Types of analyses
Target
system
(boiling stew at
roughly 100oC )
“The
temperature
is 100oC.”
Inexact
description
(Language)
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
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
8
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
9
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
10
Limits in
Statistical
Physics
11
Recovering thermodynamics
from statistical physics
Treated statistically
often behaves almost
exactly like…
Very many small
components
interacting.
Thermodynamic
system of continuous
substances.
Analyses routinely take
“limit as the number of
components go to infinity.”

The question of this talk:
how is this limit used?
?
12
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.
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.

Infinite systems
may have
properties very
different from
finite systems.

Systems with
infinitely many
components are
never
considered.
13
Thermodynamic
limit as an
idealization
14
Two forms of the thermodynamic limit
Number of
components
Volume
such that
n 
V
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
Continuum limit
as an
approximation
18
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.
19
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, …
20
Boltzmann-Grad
limit as an
approximation
21
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.
22
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.
23
Renormalization
Group Methods
24
Renormalization Group Methods
Best analysis of
critical exponents.
Zero-field specific heat
CH ~ |t|-a
…
Correlation length
x ~ |t|-n
…
for reduced temperature
for
experts
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 = ∞.
25
The Flow
for
experts
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.
26
Elimination of
Infinite
Idealizations
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
Dominance argument
Use of infinite
idealization
requires:
Properties of
infinite limit
system
must
match
IF
we already know the
properties of the finite
systems,
THEN
we do not need the infinite
limit system.
Limit properties
of finite systems.
Else we mischaracterize the
finite systems.
IF
we DO NOT already know
the properties of the finite
systems,
THEN
we cannot responsibly use
limit system.
Either way, we should
eliminate the infinite
idealization.
29
Conclusion
30
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.
2
Infinite 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.
31
http://www.pitt.edu/~jdnorton/lectures/Tsinghua/Tsinghua.html
32
The End
33