Transcript modeller - Studentportalen

Models, theory, reality
Lars-Göran Johansson
Uppsala University
What is a model?

A model resembles reality in certain respects, not in others.

Could one say that a model claims anything about reality?

How to analyze the relation between model and reality?

Two examples:

Causal network behind heart attacks.

Commuter train network in Stockholm.
Pendeltågsnätet i Storstockholm
About the commuter train map

The map is a model of the commuter train network because:

The topological properties of the commuter train network are correctly
represented.

The metrical properties are not correctly represented.

The model can be used if one knows which of its properties represents
reality and which not.

In this case all knows how to use the map, albeit almost no one have heard
about the distinction between metrical anc topological properties!
The causal model for heart attack

The model represents causal links between different factors

The model does not tell us anything about the relative strength of different
causal factors

The model is correct if it all arrows represent a causal link, and no links are
missing

Nothing in the model excludes the possibility that it is incomplete
Theoretical models

Theoretical models are mostly mathematical objects

What kind of similarity can there be between a mathematical object and a
part of reality?

Answer: structure

An important species of structural similarity is isomorphy.
Theoretical model: pendulum

The motion of a pendulum can be described by the equation

d2θ/dt2 +g/l sinθ =0

Hard to solve! For small angels θ we may put sinθ=θ!

Then

d2θ/dt2 +g/l θ =0
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Solutions:

θ(t) = A sin ωt+ B cos ωt, där ω2= g/l
Conditions for the correctness of this
model

Small angels

Constant length of the pendulum (i.e. the thread is virtually non-elastic.)

Continuous motion

If these conditions are fulfilled there is a reasonable similarity of structural
similarity between the real motion and the function!
Isomorphy

Isomorphism is a special case of homomorphism

A homomorphism is a mapping from a group to another (or the
same) group
Groups





A set  consisting of a number of elements is a group if:
There is a binary operation o on the elements in the set such
that if a, bM, then aob  
There is an inverse a-1 to every element a  
There is a unit element e  such that for each a  
aoe=eoa=a
The operation o is associative: for all a,b, c   (aob)oc=
ao(boc)
homomorphisms

Given: two groups M and N with respectively the operations o1
och o2 defined

A mapping h:M→N is an homomorphism iff for each pair
a,b∈M, it holds
h(ao1b)=h(a) o2h(b)

A homomorphism h:M→N is an isomorphism iff the mapping h
is 1-1 and if there are no elements in N which are not the map of
an element in M
group –representation of rotations

Represent an anti-clockvise rotation by the function R(f), where f is the
rotation angle.

Two rotations can be added: R(f)+R(g)= R(f+g)
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There is a unit element R(0)

To every rotation there is an inverse, the clockvise rotation, -R(f)=R(-f)

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The associative law holds: (R(f)+R(g))+R(h)= R(f)+(R(g)+R(h))
Hence, the set of all rotations in a plane form a group!
Mathematical model for rotations

The set of rotations make up a group!

The group of rotations is isomorphic with the set of real numbers!

The structures of the set of rotations and that of the set of real numbers are
similar.

For some purposes we can use the real numbers as a model for rotations, if
we don’t care about the periodic character of rotations, i.e., R(2p+f)=R(f)

If we care about periodicity we can add this condition on the set; if so, the
compact interval [0,2p] on R is a model for rotations.
Summary

Models are concrete or abstract objects

Models can be used for making assertions about reality.

The model itself does not claim anything; But we can use it for making
claims about reality (Analogy: An equation doesn’t claim anything in itself;
but we can use it for making assertions)

In many contexts we can use the words ‘model’ and ‘theory’
interchangeably; but not always!
An observation

We can not directly compare a model with reality!

We cannot as it were, just look at a real phenomenon and compare it with a
model.

We need some kind of representation of reality as a starting point for model
construction, for example, a data set of measured values, a set of
descriptions, a set of pictures.
Theories
The word ‘theory’ is used in several different ways

One use is to make the distinction observation- theory;

Another use is to make a distinction between certain facts and mere
hypotheses; a set of hypotheses may be called a theory.

A third use is to call a network of conceptual relations a theory.

The third use is often connected to talk about models; a theory is a
conceptual structure and a model is an application of this structure to a
class of phenomena.
Theories
Consider the three laws,

Newton’s second law: F=ma
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Newton’s third law: to every force there is an opposite force of equal strength.

The law of gravitation
These three laws doesn’t say anything particular about anything at all; they
merely tell us how a number of mechanical quantities are related.
Newton’s second law, for example may be regarded as an implicit definition of
the concept of force in terms of the concepts mass and acceleration.
Example: Bohrs model of the atom

An atom consists of a nucleus surrounded by electrons

Electrons are in stationary states with constant energy

Electrons can only change energy in discrete jumps.

Question: how do electrons move when being in a stationary state?
Example: Bohrs model of the atom

Electrons bound to a nucleus is often depicted as moving like a planet
orbiting the sun.

But if an electron is orbiting, it is accelerated.

According to electromagnetic theory, an accelerated charge radiates energy:
Example: Bohr’s model of the atom

If the electron radiates energy, it must rapidly spiral down into the nucleus.
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That is said to contradict Bohr’s theory!
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Wrong!
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Bohr’s model of the atom does not say that the electron move!
Example: Bohr’s model of the atom

If we accept that electromagnetic theory is true, we must say that electrons
are not accelerated when being bound in atoms!

But how can a be bound to a nucleus without moving around?

When this appears to be a contradiction, it is so because we assume, tacitly,
that the electron is a particle, i.e. a small object having a well defined
position in the coordinate system where the nucleus is at the centre at every
moment in time.
Example: Bohr’s model of the atom

In order to avoid the contradiction we must give up the assumption that the
electron is a particle.

What is it then? Is any visualisation possible?

Yes!

It may be seen as a standing wave.

In an s-state, the maximum intensity of this wave is at the centre of the
nucleus!
Bohr on complementarity

Bohr’s lessons:

We need two models for a complete description of atomic phenomena; but
they cannot be united into one single model.

Sometimes we may thing of objects as particles with definite positions

Sometimes we may think of objects as a kind of waves having no definite
positions

These models cannot be used simultaneously; but both are needed.
Bohr on complementarity
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The models does not directly contradict each other.

But the conditions for applying one of the models contradict the conditions
for applying the other.
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This is complementarity!
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The uncertainty relations should properly be called indeterminacy relations!
Uncertainty- indeterminacy

The impossibility of applying at the same time a particle model and a wave
model for ‘small’ objects is not a limitation of our knowledge.

It is not the case that electrons are particles but that we are unable to know
that; for if they were particles orbiting the nucleus, we would have a
contradiction with electromagnetism, independent of any observations.

We must conclude that ‘small’ objects sometimes are particles, sometimes
are a kind of waves; their nature is indeterminate!

Hence, the uncertainty relations should be called indeterminacy relations!
Uncertainty- indeterminacy

The indeterminacy relations does not tell us about statistical errors in
measurements.

When collecting data about atomic events, the product of standard
deviations of e.g., position and momentum measurements cannot be less
than h/2

But this is because of indeterminacy, not error.

When we talk about error in a measurement, we assume that
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Measured value=real value + error
Uncertainty- indeterminacy

But in atomic physics we cannot in general assume that the results of
measurements are more or less accurate descriptions of pre-existing facts.

Since measurements are interactions by which energy is exchanged, the
energy (and momentum) states of the measured objects are influenced by
that very interaction.

Measurements sometimes change the object measured upon!
Astronomical measurements
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Mostly, (always?) measurement upon astrophysical objects consists in
collecting quantum objects, viz., photons, or neutrinos

So all measurements of this kind is at bottom quantum interactions.

Is it in astrophysics necessary to consider the quantum character of these
measurements?