Sci 122 Telecourse Program 30 Entropy & Chaos

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©1998 RCBrill. All rights reserved

Entropy & Chaos
Program 30
Lesson 4.8

Text References

Speilberg & Anderson 146-184

Booth & Bloom, none

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1. Introduction

2. Incomplete Conservation

3. History of Heat and Energy

4. Sadi Carnot

5. Rudolf Clausius

6. Entropy

7. Laws Of Thermodynamics

8. Entropy & Efficiency

9. Entropy & Order

10. Entropy & Probability

11. Entropy & Kinetic Theory

12. Entropy Respite

13. Chaos

14. Summary & Conclusions


1.1. In what important way is the concept of conservation of energy incomplete?
1.2. State the laws of thermodynamics.
1.3. What is entropy?
1.4 What is the relationship between information and entropy?
1.5. Discuss the relationship between entropy, order, probability, and the arrow of time?
1.6. What prevents the molecules of air in a room from collecting in one place leaving a vacuum elsewhere?
1.7. Give a brief explanation for why a heat engine can never operate at 100 percent efficiency.
1.8. You decide to close the windows and open the refrigerator door on a hot day and let it cool the house. Is this a good thermodynamic strategy? Why or why not?
1.9. What is chaos?
1.10. How do equilibrium systems differ from non equilibrium systems?
1.11. What is the butterfly effect?


1.1. Describe the incompleteness of the concept of energy conservation
1.2. Review the history of the mechanical theory of heat
1.3. Detail the contributions of Carnot and Clausius to the science of thermodynamics
1.4. Define entropy
1.5. State the laws of thermodynamics
1.6. Relate the concept of entropy to efficiency, order, probability, and kinetic theory
1.7. Describe the basics of the science of chaos

1. Introduction

"Chaos rules the universe. Scientists call it 'ENTROPY'" "Everything is breaking down tending towards GREATER AND GREATER DISORDER." "It's great to be on the winning side."

1.1. As any dog or cat owner knows, it is easier to destroy something that to build it. Dogs are particularly good at one but entirely ineffective at the other. Cats are about the same, although they go about their entropic bahavor in different ways.
1.2. So what's this got to do with the nature of physical science? Easy. The answer is. Everything
1.3. Cars rust, pendulums run down, people get old. Things like this don't happen backwards. Like the flow of time, they happen only in one sequence.
1.4. Thermodynamics, Entropy and Chaos

1.4.1. Are chaos and disorder the same thing?
1.4.2. Is there such as thing as structured disorder?
1.4.3. Can a system be in a constrained yet unpredictable state?
1.4.4. These questions and others follow from a flaw in the law of conservation of energy. We will examine that flaw in some detail in section 2 below. Meanwhile let's define "thermodynamics."

1.5. Thermodynamics

1.5.1. Thermo: Heat
1.5.2. Dynamics: Motion & Forces
1.5.3. Why Thermodynamics? Conservation of Energy is Incomplete Cannot explain irreversibility of some energy transformations Leads to other things

1.6. Entropy is a measure of disorder
1.7. Chaos is a state of organized disorder

2. Incomplete Conservation

2.1. Video Clip of Diver

2.1.1. Work is converted to energy Work is done to climb to the platform Diver has potential energy while on the platform Potential energy is converted to kinetic energy during the dive Diver does work on water as she loses energy upon impact Energy is conserved when the diver hits the water Kinetic energy of diver is converted to motion of the water energy of water increases, energy of diver decreases no violation of conservation principle is noted

2.1.2. Reverse video Clip of Diver

2.1.3. What prevents the reverse from happening? wave energy is used to raise diver's potential energy energy of water decreases diver is hurled out of water to land back on platform no violation of conservation of energy is noted

2.2. Something is missing from conservation principle!

2.2.1. certain processes do not occur although they are permitted by conservation of energy
2.2.2. any process which "looks funny" when played in reverse

2.3. Other examples

2.3.1. a jar of beans spills onto the floor
2.3.2. gas diffuses to fill available volume
2.3.3. heat flows from hot to cold

3. History of Heat and Energy (Review)

3.1. concepts of work and energy not yet clarified by Newton's death in 1729
3.2. Worked out by the late 1700s
3.3. Facts about heat and temperature known by early 1800s

3.3.1. heat flows from warm to cold
3.3.2. specific heats, latent heats
3.3.3. gas laws

3.4. atomic theory began in the early 1800s with Dalton
3.5. kinetic theory was popularized in mid 1800s by Joule, Dalton's student.
3.6. Heat as a form of energy is modern concept

3.6.1. thought to be a form of matter (Caloric)
3.6.2. mechanical equivalent was a significant step
3.6.3. formulated by Joule and Mayer in early 1800s

3.7. Conservation of Energy

3.7.1. Published by Joule and Mayer in 1843
3.7.2. Energy can be transformed or used to do work
3.7.3. Amount of energy does not change

3.8. Symbolism

3.8.1. physicists use Q for heat, chemists use H.

4. Sadi Carnot (1796-1832)

4.1. a man ahead of his time, ideas ignored initially
4.2. mathematically modeled the ideal heat engine
4.3. using expansion and contraction of ideal gases
4.4. showed how to calculate maximum efficiency
4.5. showed that heat engines cannot operate at 100% efficiency
4.6. 100% efficiency implies cold reservoir at absolute zero

4.7. Maximum efficiency depends upon temperature difference

5. Rudolf Clausius (1822-88)

5.1. regarded as the founder of thermodynamics
5.2. using the work of Carnot was the first to state explicitly the second law
5.3. introduced the concept of entropy

6. Entropy

6.1. A way to quantify spontaneous processes
6.2. Defined as ratio of heat transferred to Kelvin temperature
6.3. Defined such that entropy change is always nonnegative for spontaneous processes
6.4. Example: Heat Transfer


All heat flows from hot to cold, no work is done. Heat out of TH -QH) equals heat into TC (QC)

6.4.1. Heat Transfer

6.4.2. entropy does not decrease during spontaneous heat flow (it increases in this case. The amount of change is 1 J/K)
6.4.3. heat lost by hot object equals heat gained by cold object
6.4.4. heat is transferred from hot to cold
6.4.5. these are laws 2, 1, 0

7. Laws Of Thermodynamics

7.1. formulated to explain lack of irreversibility
7.2. based on observed properties of heat
7.3. wide range of applicability
7.4. significance and applications not yet fully understood
7.5. can be stated in many different terms
7.6. Developed through high degree of mathematical sophistication

7.6.1. beyond casual observer
7.6.2. has power to describe and explain many situations
7.6.3. led to design and construction of energy transformation machines

7.7. Coherent theory developed near end of 19th century

7.7.1. required statistical mechanics, atomic theory and kinetic theory to be understood first
7.7.2. concepts are simple and understandable without sophisticated mathematics

7.8. Zeroth Law

7.8.1. Heat flows spontaneously only from hot to cold
7.8.2. Added as an afterthought to other laws

7.9. First Law

7.9.1. The Work Energy Theorem
7.9.2. energy in many forms can be transformed or transferred between objects work is a mechanical process by which energy is transformed or transferred note how we define work in terms of energy whereas before we defined energy in terms of work

7.9.3. total energy in universe is constant
7.9.4. increase in energy one place results in a decrease somewhere else
7.9.5. efficiency can't be greater than 100%
7.9.6. can't get something for nothing

7.10. Second Law

7.10.1. many different ways to say it
7.10.2. imposes maximum efficiency on heat transformations
7.10.3. some energy becomes unavailable during transformations
7.10.4. no transformation is completely reversible
7.10.5. some forms of heat or energy are more useful or available than others
7.10.6. energy is degraded in quality during transformations
7.10.7. heat cannot be completely transformed into work
7.10.8. can only break even at absolute zero

7.11. Third Law

7.11.1. entropy would be zero at absolute zero
7.11.2. gives another method to define absolute temperature
7.11.3. atoms would stop moving at absolute zero
7.11.4. Can't get to absolute zero can be deduced from zeroth law heat cannot flow unless there is a difference of temperature absolute zero is the absence of heat in the same way that you cannot remove all of the air from a jar without a perfect vacuum

7.11.5. Cooling to absolute zero would require a reservoir below absolute zero (not possible by definition) an infinitely large reservoir at absolute zero (not possible either)

7.11.6. Modern techniques can approach very close on single atoms or small groups of atoms to 1 x 10-6 K near absolute zero matter displays strange properties superconductivity, Bose-Einstein condensates

8. Entropy & Efficiency

8.1. When work is extracted heat flow to cold reservoir is diminished
8.2. QH = QC + W
8.3. Efficiency maximum exists because entropy must not decrease
8.4. Example: Work Extracted at Maximum Efficiency

8.4.1. Narrative TH = 400K, TC = 300 K Efficiency = 0.25 Work = .25 x 1200 J = 300 J QH = 1200J, QC = 900 = 0 at maximum efficiency no additional work can be extracted because entropy change would be negative second law is limiting factor: entropy change must be nonnegative

9. Entropy & Order

9.1. Murphy's Law

If anything can go wrong it will.

9.1.1. Because there are more ways for something to go wrong than to go right

9.2. Entropy can be seen as a measure of disorder
9.3. lower order = lower entropy
9.4. spontaneous processes maximize entropy change
9.5. changes happen spontaneously to increase disorder

9.5.1. the room gets messy

9.6. disorganization requires energy to organize

9.6.1. it takes work to clean up your room

9.7. Spontaneous processes tend toward disorder

9.7.1. spilled jar of beans is more disorderly than when in the jar
9.7.2. gases diffuse to fill available space thereby increasing disorder
9.7.3. A room becomes cluttered unless an effort is made to keep it in order

9.8. There are more ways to be disordered than to be ordered

9.8.1. there are more places to be out of place than in the proper place
9.8.2. in many cases there is only one right place and an infinite number of wrong ones
9.8.3. it is extremely unlikely that an object will spontaneously find the proper place

10. Entropy & Probability

10.1. Schlindler's Law

Likely events happen more often than unlikely events.

10.2. most likely arrangement is most likely to occur
10.3. randomness is the most disorderly state
10.4. random motions do not have a preferred direction
10.5. randomness is difficult to define
10.6. heat and wave energy of swimming pool cannot organize to throw the diver backonto board
10.7. no physical law prevents it, it is just extremely unlikely
10.8. Boltzman defined the relationship

10.8.1. S = k log W
10.8.2. W = number of accessible states if there are only two accessible states it is an orderly system

10.8.3. k = Boltzman constant
10.8.4. same as constant in ideal gas equation, PV = kT note that k = PV/T has the same units as entropy (Joule/Kelvin)

11. Entropy & Kinetic Theory

11.1. Random motion of gaseous molecules is highly disordered and entropically favorable
11.2. Two containers of gas at different temperatures is more ordered than one container at the same temperature
11.3. mixing of gases of different temperatures is an increase in entropy
11.4. distinction between collections of molecules disappears with mixing
11.5. Why do gases fill available space?

11.5.1. no physical law prevents molecules of air from collecting in one corner of the room leaving a vacuum everywhere else
11.5.2. why doesn't it happen?

12. Entropy Respite

12.1. We have seen a different kind of connection between heat and gases in terms of randomness and probability, order and disorder
12.2. Galileo's legacy still guides scientific inquiry
12.3. science is concerned with how things are related, not why they behave the way they do
12.4. Is Thermodynamics Difficult to Understand?

12.4.1. "How" of relationships is not difficult Entropy is simply defined Mathematics of thermodynamics is simple ratios Relationships are quantitatively sound, qualitatively logical, and agree with observations

12.4.2. "Why" of relationships is difficult philosophically deep the mystery of our time similar to the inverse square central force question in Newton's time not at all clear why these things should be related in this way not at all clear what the relationships are

12.5. Is Thermodynamics Important?

12.5.1. has generated much discussion
12.5.2. has spawned several new areas of study
12.5.3. has provided new explanations for the behavior of the universe
12.5.4. has allowed linkages between diverse areas of study

13. Chaos

13.1. randomness and pseudo randomness

13.1.1. is there such a thing as pure randomness?
13.1.2. if so, how would you recognize it?
13.1.3. can we define patterns of pseudo randomness?
13.1.4. what does it take to recognize a pattern?

13.2. information theory

13.2.1. to recognize patterns and communicate information requires order
13.2.2. signal vs. noise

13.3. complex systems

13.3.1. nonlinear systems have feedback
13.3.2. "The Butterfly Effect": An extremely small cause can have very large effects. A butterfly flapping its wings in Asia affect the weather worldwide. Lorenz's weather machine rounded divergence folklore 1

For want of a nail, the shoe was loss
For want of a shoe, the horse was lost;
For want of a horse, the rider was lost;
For want of a rider, the battle was lost;
For want of a battle, the kingdom was lost.

13.3.3. systems have many variations depending on initial conditions

13.4. chaos

13.4.1. non-equilibrium systems how does order arise if entropy must increase thermodynamics equation are valid for systems near equilibrium in general the further out of equilibrium the more likely there will be ordered structures pouring water from a bottle tornadoes and hurricanes

13.4.2. state variables in dynamic systems are constrained by interrelationships act as feedback mechanisms that keep certain systems restrained

13.4.3. attractors
13.4.4. values of functions can bifurcate and attain different states from tiny differences in values The Lorenzian Waterwheel simple device proves capable of surprisingly complicated behavior long term behavior depends on how hard the driving force is, ie. on the intensity of the energy gradient show on graph that Force is the slope of an energy vs. displacement graph water pours into top bucket at steady rate too slow --> top bucket doesn't fill enough to overcome friction so the wheel never turns initial direction is extremely sensitive to initial position of top bucket try balancing a credit card (not the budget) by standing it on end If flow rate is faster the weight of the top bucket sets the wheel in motion. It can settle into an equilibrium, steady rotation rate change in speed or change in water flow rate causes nonlinear effects how much the buckets fill depends on the flow rate and the rate of spin slow flow rate means that the buckets take a longer time to fill and buckets start up the other side before they have time to empty a rapid spinning rate allows too little time for the buckets to fill heavy buckets going uphill slow the wheel and eventually reverse it Lorenz discovered that the spin will reverse itself many time, never settles down to a steady rate, never repeating itself in any as yet predictable pattern. similar to butterfly effect The compound pendulum (Video Demo) complex behavior generated from two simple laws unpredictable yet predictable exact path vs. containment field of paths

13.4.5. The Mandelbrot Set and Fractals See illustrations in "Chaos: Making a New Science", by James Gleick the most complex mathematical structure known a recursive fractal reveals infinitely fine structure through series of magnifications length of perimeter depends on scale number of dimension is somewhere between two and three. take complex number and square it, add the original number,repeat ((((((A2+A)2+A)2+A)2+A)2+A)2+A)2 . . .
In the figures below, each picture represents an enlargement of the white frame in the pircute before it. Mandelbrot 1 Mandelbrot 2 Mandelbrot 3 Mandelbrot 4 Mandelbrot 5 Mandelbrot 6 Mandelbrot 7 Mandelbrot 8

This last picture is "pixelated" to save time. In reality the set can be "magnified" indefinitely with no loss of clarity. fractal folklore

Turtle upon turtle upon turtle.

13.5. catastrophe

13.5.1. under some conditions rapid and catastrophic changes occur
13.5.2. dams burst, beams collapse, floods flash, eg.

13.6. time's arrow

13.6.1. why does time flow from the past to the present?
13.6.2. the fact that some kinds of processes can only happen in one direction is related to the flow of time
13.6.3. if all events were equally reversible, like Lorenz's waterwheel, there would be no flow of time

13.7. quantum thermodynamics

13.7.1.Steven Hawking's analysis of energy leaking from black holes is a theory that not everyone agrees with, but asks some interesting questions.
13.7.2. combines quantum principles with thermodynamic principles

14. Summary & Conclusions

14.1. Carnot showed relationship between heat, work, efficiency, entropy and absolute zero
14.2. Entropy was originally defined by Clausius as the ratio of heat transfer to temperature
14.3. Boltzman showed a relationship between entropy, probability and the ideal gas constant
14.4. Laws of Thermodynamics Summary

14.4.1. Zeroth. There is a strong incentive to play the game
14.4.2. First: You can't win, you can at best break even
14.4.3. Second: You can only break even at absolute zero
14.4.4. Third: You can't get to absolute zero

14.5. Chaos and Complex Systems

14.5.1. a whole new area of study
14.5.2. wide range of applicability in diversity of areas
14.5.3. limits the precision of predictions for complex systems
14.5.4 Suggests that there is a limit to the "knowability" of certain systems.