one dimensional flow derivations
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Philosophy, Purdue University
Derivation and application
conditions of theone-dimensional heat equation
Christopher Pincock ([email protected])
May 20, 2005
C. Pincock: The heat equation, 1
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Outline
Background and terminology
Derivation conditions
Application conditions
Tentative conclusions
C. Pincock: The heat equation, 2
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Background and terminology
Besides the indispensability argument, what is left for thephilosopher to worry about when it comes to the applicationof mathematics in science?
Example: which mathematical truths concerning the realnumbers play a role in using real numbers to represent
temperature? “temperature and other scalar fields used in physics are
assumed to be continuous, and this guarantees that if point x
has temperature ψ(x ) and point z has temperature ψ(z ) andr is a real number between ψ(x ) and ψ(z ), then there will bea point y spatio-temporally between x and z such thatψ(y ) = r ” (Field 1980, 57).
C. Pincock: The heat equation, 3
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Background and terminology
Response: not all mathematical properties transfer to
temperatures. There is no least real number but there is a lowest
temperature.
Case study: For u (x , t ) representing the temperature of pointx at time t , we can derive the partial differential equation(Boyce and DiPrima 1986, 514-584):
α2u xx = u t (1)
where α2 = κ/ρs , κ is the thermal conductivity of thematerial, ρ its density and s the specific heat of the material.Throughout subscripts indicate partial differentiation withrespect to that variable, e.g. u t = ∂
∂ t u (x , t ).
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Background and terminology
Derivation conditions: under what conditions are scientists
warranted in adding this equation to their scientific theory of heat?
Application conditions: under what conditions are scientistswarranted in using this equation to describe a particularphysical system?
Conclusions:
1. There are two attitudes that we can take to (1), only one of which is warranted by the evidence that scientists typicallyhave available.
2. Even in this simple case, the derivation and applicationconditions diverge.
3. This account can give insight into some interpretativequestions about the role of mathematics in the sciences.
C. Pincock: The heat equation, 5
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Derivation conditions
To add (1) to their theories scientists appeal to two experimentallydetermined laws:
Newton’s law of cooling: the amount of heat per unit of timethat passes from warmer plate 2 to cooler plate 1 is
H =∆Q
∆t =
κA|T 2 − T 1|
d (2)
where T 2 and T 1 are the respective temperatures of theplates, A is their area, d their distance from each other and κis the thermal conductivity of the material.
A claim about the average change in temperature: ∆u
increases in proportion to Q ∆t , but is inversely proportionalto s ∆m = s ρA∆x , where ∆m is the mass of the element:
∆u =Q ∆t
s ρA∆x (3)
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Derivation conditions
Step 1
Use (2) to determine the net heat flow Q for our element:
Q = κA(u x (x 0 + ∆x , t )− u x (x 0, t )) (4)
Deriving (4) from (2) requires taking d in (2) to 0 once for eachboundary of the element.
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Derivation conditions
Step 2
Assume that there is some position x 0 + θ∆x (0 < θ < 1) wherethe actual change in temperature equals the average change in
temperature in the entire element:
Q ∆t = [u (x 0 + θ∆x , t + ∆t )− u (x 0 + θ∆x , t )][s ρA∆x ] (5)
C. Pincock: The heat equation, 8
D i i di i
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Derivation conditions
Step 3
Multiply both sides of (4) by ∆t and identify the right-hand sidewith the right-hand side of (5):
κA(u x (x 0 + ∆x , t )− u x (x 0, t ))∆t =
[u (x 0 + θ∆x , t + ∆t )− u (x 0 + θ∆x , t )][s ρA∆x ] (6)
Dividing both sides by ∆x ∆t , taking the limit of ∆x → 0, ∆t → 0
and rearranging gives (1).
C. Pincock: The heat equation, 9
D i i di i
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Derivation conditions
Two derivation conditions:
(D1) Various limits are well defined.
(D2) Given the average change in temperature ∆u in the barelement, there is some point in the bar x whose temperaturechange is identical to ∆u .
Two attitudes are possible here:
1. Physical attitude: Throughout the derivation we are talkingdirectly about physical systems and physical magnitudes.
2. Mathematical attitude: Throughout the derivation we are
talking directly about mathematical entities and theirproperties and only indirectly about physical systems.
C. Pincock: The heat equation, 10
D i ti diti
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Derivation conditions
A Choice
The physical attitude would ground Field’s claim abouttemperatures, but such an attitude is not warranted by theexperimental evidence available. No iron bar has elements
corresponding to the elements assumed in the derivation.
The mathematical attitude allows the derivation to go throughas there is no doubt about its mathematical correctness. Forit to succeed, though, we must explain how it involves indirect
claims about physical systems and what these claims are.
C. Pincock: The heat equation, 11
Derivation conditions
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Derivation conditions
A Proposal
The derivation is directly about a wholly mathematical modelM 1 and indirectly about a physical system P when (i) there isa mathematical model M 2 that matches P and (ii) there is anacceptable mathematical transformation from M 1 to M 2.
Matching & Resolution: A mathematical model’s degree of resolution will be perfect when the constituents and processesin the model match precisely the constituents and processes of the physical situation modeled (mapping account). A modelwill lack (have excess ) resolution if its constituents andprocesses are of a larger (smaller ) scale than the constituentsand processes of the physical situation modeled. E.g. anactual discrete iron bar vs. the continuous bar.
C. Pincock: The heat equation, 12
Derivation conditions
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Derivation conditions
A Proposal (cont.)
When is a transformation acceptable?
Batterman 2002: In asymptotic reasoning we construct‘minimal’ models which lack resolution but that neverthelessrepresent the physical situation. This occurs when the relevant
features of the model correspond to the relevant features of the situation. E.g. the buckling load for the strut in theminimal model vs. the buckling load for the actual strut.
A generalization: Batterman has isolated just one type of
acceptable transformation between a matching model and amodel that lacks resolution. Others exist, as do acceptabletransformations between matching models and models withexcess resolution.
C. Pincock: The heat equation, 13
Derivation conditions
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Derivation conditions
The Heat Equation
Our derivation involves a model with excess resolution. Webelieve that adding details to it does not change itstemperature dynamics captured by our experimental laws (2),
(3). To prove: For reasonable temperature and time intervals, the
distribution of temperature magnitudes for the twomathematical models agree.
Objection: This is “lazy optimism” (Wilson 2000).
Derivation Model o o / / Matching Model k s + 3 Physical Situation
C. Pincock: The heat equation, 14
Application conditions
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Application conditions
A different set of conditions must be satisfied for (1) to beused to describe a particular physical system. We review
perhaps the simplest case. For boundary conditions
u (0, t ) = 0,
u (l , t ) = 0, t > 0 (7)
with initial conditions
u (x , 0) = f (x ), 0 ≤ x ≤ l
f (x ) =∞
n=1
b n sin(nπx /l ) (8)
every solution to (1) is of the form
u (x , t ) =∞
n=1
b n exp(−n2π2α2t /l 2) sin(nπx /l ) (9)
C. Pincock: The heat equation, 15
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Application conditions
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Application conditions
Two application conditions
(A1) Separation of variables is appropriate, i.e. there arefunctions X and T such that u (x , t ) = X (x )T (t ).
(A2) Given a function of a real variable, there are additionalparallel functions of a complex variable.
Neither condition is reasonable given the physical attitude. Themathematical attitude is required to make scientific practice
intelligible.
C. Pincock: The heat equation, 18
Tentative conclusions
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First conclusion
There are two attitudes that we can take to (1), only one of whichis warranted by the evidence that scientists typically have available.
Taking the physical attitude requires adopting the axiomsneeded for a Field-style nominalistic version of our physicaltheories.
Adopting the mathematical attitude allows us to reject theseaxioms and suggests a role for mathematics in science: we use
mathematics when we are unable to construct a matchingmodel.
C. Pincock: The heat equation, 19
Tentative conclusions
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Second conclusionEven in this simple case, the derivation and application conditions
diverge. Even when (1) is part of our physical theory we need
additional application conditions to be satisfied to solve it. The same equation can have associated with it different,
conflicting application conditions. It seems possible that the
application and derivation conditions could also conflict,requiring three distinct models:
Derivation Model i i
) )
o o / / / o / o / o / o / o / o / o / o / o / o / o / o / o / o / o / o / o / o Application Model 5 5
u uMatching Model
K S
Physical Situation
C. Pincock: The heat equation, 20
Tentative conclusions
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Third conclusion
This account can give insight into some interpretative questionsabout the role of mathematics in the sciences.
If the use of mathematics is tied to a lack of understanding of
the matching model, then we can see why highly mathematicaltheories pose the greatest interpretative challenges.
Moreover, by investigating the appropriate transformationsbetween the matching, derivation and application models, we
can make progress on some of these debates.
C. Pincock: The heat equation, 21
References
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Batterman, R. W. (2002).The Devil in the Details: Asymptotic Reasoning in Explanation,
Reduction, and Emerge nce .Oxford University Press.
Boyce, W. E. and R. C. DiPrima (1986).Elementary Differential Equations and Boundary Value Problems
(Fourth ed.).
John Wiley & Sons.
Field, H. (1980).Science Without Numbers: A Defence of Nominalism.Princeton University Press.
Wilson, M. (2000).The unreasonable uncooperativeness of mathematics in the
natural sciences.The Monist 83 , 296–314.
C. Pincock: The heat equation, 22