The Unreasonable Effectiveness of

Mathematics

Honors Workshop

November 13, 2006

Steve Pennell

Department of Mathematical Sciences

University of Massachusetts Lowell

Topics:

I Roles of mathematics in science and

engineering

II Example - Population Modeling

III Chaotic systems - How have they changed

our thinking?

IV Carbon Dioxide Sequestration

References

Wigner, E. P., “The Unreasonable Effective-

ness of Mathematics in the Natural Sciences,”

1960, Communications on Pure and Applied

Mathematics, vol. 13, pp. 1 - 14.

Kalman, Dan, 1997, Elementary Mathemati-

cal Models: Order Aplenty and a Glimpse of

Chaos, The Mathematical Association of Amer-

ica.

Gleick, James, 1987, Chaos: Making a New

Science, Viking Penguin.

I. Roles of mathematics in science and

engineering:

1. Math is a concise & relatively unambiguous

language.

“The laws of Nature are written in the

language of mathematics.” - Galileo

“The rules that describe nature seem to be

mathematical.” - Richard Feynman

2. Math is an analytical tool.

Mathematical Modeling

Physical System

MathematicalModel

MathematicalSolution

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Translation

Analysis

Interpretation

Topics:

I Roles of mathematics in science and

engineering

II Example - Population Modeling

III Chaotic systems - How have they changed

our thinking?

IV Carbon Dioxide Sequestration

II. Example - Population Modeling

Consider a population of 1000 fish in HaggettsPond.

• Assume 500 are female.

• Assume each female produces 3 viable off-spring each year.

• Assume half the current population dies eachyear.

How many fish will there be next year?

1000+3 · 500− 500 = 2000

II. Example - Population Modeling

Consider a population of 1000 fish in HaggettsPond.

• Assume 500 are female.

• Assume each female produces 3 viable off-spring each year.

• Assume half the current population dies eachyear.

How many fish will there be next year?

1000+3× 500−500 = 2000

II. Example - Population Modeling

Consider a population of 1000 fish in HaggettsPond.

• Assume 500 are female.

• Assume each female produces 3 viable off-spring each year.

• Assume half the current population dies eachyear.

How many fish will there be next year?

1000+3× 500−500= 2000

II. Example - Population Modeling

Consider a population of 1000 fish in HaggettsPond.

• Assume 500 are female.

• Assume each female produces 3 viable off-spring each year.

• Assume half the current population dies eachyear.

How many fish will there be next year?

1000+3× 500−500 = 2000

Physical System

Let p0 denote the initial fish population.

Let pn denote the fish population after n years.

Assume

• No fish enter or leave except by births and

deaths.

• Number of fish born each year equals 1.5

times population.

• Number of fish that die each year equals

0.5 times population.

Mathematical Model

Next year’s population equals this year’s pop-

ulation plus the number of fish born minus the

number of fish that die.

pn+1︸ ︷︷ ︸Next year′s pop.

= pn︸︷︷︸Current pop.

+1.5pn︸ ︷︷ ︸Births

− 0.5pn︸ ︷︷ ︸Deaths

⇒ pn+1 = 2 × pn

Mathematical Model

pn+1 = 2 × pn

Analysis

p1 = 2 × p0

p2 = 2 × p1 = 2 × 2 × p0 = 22 × p0

p3 = 2p2 = 2 × 22p0 = 23 × p0...

⇒ pn = 2n × p0

Interpretation

Population increases without bound.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5x 10

4

n

pn

pn+1

=2 pn

Physical System

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Translation

Analysis

Interpretation

Modify assumption about number of births:

Original model: # born = 1.5pn

“Logistic” model:

# born = (1.5 − pn/2000) × pn

(Birth rate per individual declines as population

rises.)

Now our model is

pn+1 = pn + (1.5 − pn/2000) pn︸ ︷︷ ︸Births

− 0.5pn︸ ︷︷ ︸Deaths

⇒ pn+1 = (2 − pn/2000)× pn

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

500

1000

1500

2000

2500

3000

3500

4000

n

pn

pn+1

=(2 − pn/2000) p

n

Interpretation:

Population stabilizes at 2000.

(Birth rate equals death rate, so system is in

equilibrium.)

What happens if we change the parameters?

Replace

pn+1 = pn + (1.5 − pn/2000) pn︸ ︷︷ ︸Births

− 0.5pn︸ ︷︷ ︸Deaths

with

pn+1 = pn + (2.6 − pn/2000)pn︸ ︷︷ ︸Births

− 0.5pn︸ ︷︷ ︸Deaths

= (3.1 − pn/2000)× pn

0 1 2 3 4 5 6 7 8 9 100

1000

2000

3000

4000

5000

6000

n

pn

pn+1

=(3.1 − pn/2000) p

n

Interpretation:

Population oscillates between two values.

Let’s try another parameter change:

Now use pn+1 = (4 − pn/2000)× pn

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

n

pn

pn+1

=(4 − pn/2000) p

n

Interpretation:No discernible pattern.

Even worse than the lack of pattern:

Systems that start “near” each other do not

stay near each other.

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

n

pn

pn+1

=(4 − pn/2000) p

n

p0=1000

p0=999

p0=1001

Chaos!

Even worse than the lack of pattern:

Systems that start “near” each other do not

stay near each other.

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

n

pn

pn+1

=(4 − pn/2000) p

n

p0=1000

p0=999

p0=1001

Chaos!

Topics:

I Roles of mathematics in science and

engineering

II Example - Population Modeling

III Chaotic systems - How have they changed

our thinking?

IV Carbon Dioxide Sequestration

III. Chaos

Systems believed to exhibit chaotic behavior:

• Orbit of Pluto

• Orbit of Saturn’s moon Hyperion

• Heart fibrillation

• Measles epidemics

• Turbulent fluid flow

• Weather?

Key characteristic of chaotic systems:Sensitive Dependence on Initial Conditions(a.k.a. The Butterfly Effect)

Slight changes in the initial state of a systemcan lead to large changes in the long-term be-havior of the system.

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

n

pn

pn+1

=(4 − pn/2000) p

n

p0=1000

p0=999

p0=1001

Question: If a system exhibits sensitive depen-

dence on initial conditions, what are the im-

plications for computing future states of the

system?

Topics:

I Roles of mathematics in science and

engineering

II Example - Population Modeling

III Chaotic systems - How have they changed

our thinking?

IV Carbon Dioxide Sequestration

CO2 Sequestration

• Why? To keep it out of the atmosphere,

alleviating global warming.

• Where? Deep ocean or underground.

• Collaborators: D. Golomb, D. Ryan, E.

Barry, P. Swett, M. Woods, Tiffany Kot

Goals of CO2 ocean sequestration

• Avoid seawater acidification

• Create sinking plume to prolong sequestra-

tion time

• Allow release at minimum possible depth

(about 500 m)

• Minimize expense

Solution: Liquid CO2 - in - water emulsion

stabilized by fine limestone particles

Mean diameter of pulverized limestone parti-

cles: 10 - 20 µm

Typical globule diameter 200 - 300 µm

Open Ocean Release

Independent variable: depth z

State variables:

• plume radius r

• plume velocity u

• plume density ρp

Equations:

• Conservation of mass

• Conservation of buoyancy

• Conservation of momentum

Rate in = Rate out

πρpur2∣∣∣z+ 2πρaEur∆z = πρpur2

∣∣∣z+∆z

⇒

ρpur2∣∣∣z+∆z

− ρpur2∣∣∣z

∆z= 2ρaEur ⇒

d

dz

[ρpur2

]= 2ρaEur

Steady-state Plume Model

d

dz

[r2u

]= 2Eru

d

dz

[r2uρp

]= 2Eruρa

d

dz

[r2u2ρp

]= r2g (ρp − ρa)

10−6

10−5

10−4

0

100

200

300

400

500

600

700

800

N2 (s−2)

Ver

tical

plu

me

leng

th (

m)

Plume Sinking Distance

E = 0.05, CO2 flux = 250 kg/s

E = 0.05, CO2 flux = 125 kg/s

E = 0.05, CO2 flux = 62.5 kg/s

E = 0.1, CO2 flux = 250 kg/s

E = 0.1, CO2 flux = 125 kg/s

E = 0.1, CO2 flux = 62.5 kg/s

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