THE PHYSICS OF STUFF: WHY MATTER IS MORE THAN THE SUM OF ITS PARTS

Transcription

1 THE UNIVERSITY OF CHICAGO, ENRICO FERMI INSTITUTE ARTHUR H. COMPTON LECTURES 71 st SERIES THE PHYSICS OF STUFF: WHY MATTER IS MORE THAN THE SUM OF ITS PARTS JUSTIN C. BURTON -- APRIL 3 RD JUNE 12 TH 2009

2 WELCOME! The purpose of the Compton lectures is to share with the public some of the exciting research and new concepts in physics.

3 WELCOME! The purpose of the Compton lectures is to share with the public some of the exciting research and new concepts in physics. This Spring s lectures will focus on the physics of stuff

4 WELCOME! The purpose of the Compton lectures is to share with the public some of the exciting research and new concepts in physics. This Spring s lectures will focus on the physics of stuff -matter we interact with on a daily basis

5 WELCOME! The purpose of the Compton lectures is to share with the public some of the exciting research and new concepts in physics. This Spring s lectures will focus on the physics of stuff -matter we interact with on a daily basis -complex and emergent properties

6 WELCOME! The purpose of the Compton lectures is to share with the public some of the exciting research and new concepts in physics. This Spring s lectures will focus on the physics of stuff -matter we interact with on a daily basis -complex and emergent properties Lectures will be held every Saturday at 11am until June 12th -there is no lecture on May 29 th (Memorial Day weekend)

7 WELCOME! The purpose of the Compton lectures is to share with the public some of the exciting research and new concepts in physics. This Spring s lectures will focus on the physics of stuff -matter we interact with on a daily basis -complex and emergent properties Lectures will be held every Saturday at 11am until June 12th -there is no lecture on May 29 th (Memorial Day weekend) Make sure to grab a lecture handout -all handouts and notes will be available on the website

8 OUTLINE 1) The matter we know: from the ordinary to the exotic 2) Solids: crystals and symmetry 3) Fluids and interfacial physics 4) Phase transitions: a universal theme 5) Super-stuff: quantum matter 6) Disorder and Glassiness 7) From the old to the new: soft matter I 8) From the old to the new: soft matter II 9) Let s put it to use: materials science past and present 10) Much more than the sum of its parts: living matter and evolution

9 OUTLINE 1) The matter we know: from the ordinary to the exotic 2) Solids: crystals and symmetry 3) Fluids and interfacial physics 4) Phase transitions: a universal theme 5) Super-stuff: quantum matter 6) Disorder and Glassiness 7) From the old to the new: soft matter I 8) From the old to the new: soft matter II 9) Let s put it to use: materials science past and present 10) Much more than the sum of its parts: living matter and evolution

10 SOME PRELIMINARY CONSIDERATIONS What is matter?

11 SOME PRELIMINARY CONSIDERATIONS What is matter? A substance or object that occupies space and has mass

12 SOME PRELIMINARY CONSIDERATIONS What is matter? A substance or object that occupies space and has mass

13 SOME PRELIMINARY CONSIDERATIONS What is matter? A substance or object that occupies space and has mass

14 SOME PRELIMINARY CONSIDERATIONS What is matter? A substance or object that occupies space and has mass

15 SOME PRELIMINARY CONSIDERATIONS What is matter? A substance or object that occupies space and has mass Can we be more specific (reductionist)?

16 SOME PRELIMINARY CONSIDERATIONS What is matter? A substance or object that occupies space and has mass Can we be more specific (reductionist)? are there common components??

17 SOME PRELIMINARY CONSIDERATIONS What is matter? A substance or object that occupies space and has mass Can we be more specific (reductionist)? yes! molecules are there common components??

18 SOME PRELIMINARY CONSIDERATIONS molecule p+

19 SOME PRELIMINARY CONSIDERATIONS molecule p+

20 SOME PRELIMINARY CONSIDERATIONS molecule p+

21 SOME PRELIMINARY CONSIDERATIONS molecule atom p+ p+

22 SOME PRELIMINARY CONSIDERATIONS molecule atom p+ p+ different atoms = elements

23 SOME PRELIMINARY CONSIDERATIONS molecule atom p+ p+ Periodic Table (Dmitri Mendeleev 1869) different atoms = elements

24 WHAT MAKES UP AN ATOM? e- p+ n p+

25 WHAT MAKES UP AN ATOM? Atom from átomos (Greek) uncuttable e- p+ n p+

26 WHAT MAKES UP AN ATOM? Atom from átomos (Greek) uncuttable Nuclear fission breaks atoms into smaller parts usually lighter atoms p+ e- p+ n

27 WHAT MAKES UP AN ATOM? Atom from átomos (Greek) uncuttable Nuclear fission breaks atoms into smaller parts usually lighter atoms p+ e- n p+ 141 Ba 92 Kr

28 WHAT MAKES UP AN ATOM? Atom from átomos (Greek) uncuttable Nuclear fission breaks atoms into smaller parts usually lighter atoms p+ e- n p+ First nuclear reaction took place here at UofC, designed by physicist Enrico Fermi 141 Ba 92 Kr

29 WHAT MAKES UP AN ATOM? Atom from átomos (Greek) uncuttable Nuclear fission breaks atoms into smaller parts usually lighter atoms p+ e- n p+ First nuclear reaction took place here at UofC, designed by physicist Enrico Fermi 141 Ba 92 Kr

30 WHAT MAKES UP AN ATOM? Atom from átomos (Greek) uncuttable Nuclear fission breaks atoms into smaller parts usually lighter atoms p+ e- n p+ First nuclear reaction took place here at UofC, designed by physicist Enrico Fermi 141 Ba 92 Kr Fermi worked in the metallurgy lab of Arthur Compton

31 WHAT MAKES UP AN ATOM? Atom from átomos (Greek) uncuttable Nuclear fission breaks atoms into smaller parts usually lighter atoms p+ e- n p+ First nuclear reaction took place here at UofC, designed by physicist Enrico Fermi 141 Ba 92 Kr Fermi worked in the metallurgy lab of Arthur Compton How far can this go? What determines an elementary particle?

32 ELEMENTARY PARTICLES Our standard model of particle physics is composed of elementary particles and force carriers

33 ELEMENTARY PARTICLES Our standard model of particle physics is composed of elementary particles and force carriers

34 ELEMENTARY PARTICLES Our standard model of particle physics is composed of elementary particles and force carriers These particles obey the modern laws of physics (quantum mechanics, general relativity )

35 ELEMENTARY PARTICLES Our standard model of particle physics is composed of elementary particles and force carriers These particles obey the modern laws of physics (quantum mechanics, general relativity ) All of the observable matter in The universe is composed of some combination of these particles

36 ELEMENTARY PARTICLES Our standard model of particle physics is composed of elementary particles and force carriers These particles obey the modern laws of physics (quantum mechanics, general relativity ) All of the observable matter in The universe is composed of some combination of these particles p+

37 ELEMENTARY PARTICLES Our standard model of particle physics is composed of elementary particles and force carriers These particles obey the modern laws of physics (quantum mechanics, general relativity ) All of the observable matter in The universe is composed of some combination of these particles p+

38 ELEMENTARY PARTICLES Our standard model of particle physics is composed of elementary particles and force carriers These particles obey the modern laws of physics (quantum mechanics, general relativity ) All of the observable matter in The universe is composed of some combination of these particles p+

39 ELEMENTARY PARTICLES Our standard model of particle physics is composed of elementary particles and force carriers These particles obey the modern laws of physics (quantum mechanics, general relativity ) All of the observable matter in The universe is composed of some combination of these particles p+ What do we mean observable?

40 THE STUFF WE CAN T SEE (BUT KNOW ITS THERE) Dark matter is the name given to mass in the universe that we cannot detect with optical instruments (i.e. does not interact with light)

41 THE STUFF WE CAN T SEE (BUT KNOW ITS THERE) Dark matter is the name given to mass in the universe that we cannot detect with optical instruments (i.e. does not interact with light) Dark matter does however interact gravitationally (it has mass), so we have indirect methods of detection (gravitational lensing)

42 THE STUFF WE CAN T SEE (BUT KNOW ITS THERE) Dark matter is the name given to mass in the universe that we cannot detect with optical instruments (i.e. does not interact with light) Dark matter does however interact gravitationally (it has mass), so we have indirect methods of detection (gravitational lensing) The distribution of dark matter can be determined from images, such as those from the Hubble Space Telescope

43 THE STUFF WE CAN T SEE (BUT KNOW ITS THERE) Dark matter is the name given to mass in the universe that we cannot detect with optical instruments (i.e. does not interact with light) Dark matter does however interact gravitationally (it has mass), so we have indirect methods of detection (gravitational lensing) The distribution of dark matter can be determined from images, such as those from the Hubble Space Telescope ~4% of the matter-energy in the universe is visible! (stuff we are made of)

44 THE STUFF WE CAN T SEE (BUT KNOW ITS THERE) Dark matter is the name given to mass in the universe that we cannot detect with optical instruments (i.e. does not interact with light) Dark matter does however interact gravitationally (it has mass), so we have indirect methods of detection (gravitational lensing) The distribution of dark matter can be determined from images, such as those from the Hubble Space Telescope ~4% of the matter-energy in the universe is visible! (stuff we are made of) ~22% of the matter-energy in the universe is dark matter!

45 THE STUFF WE CAN T SEE (BUT KNOW ITS THERE) Dark matter is the name given to mass in the universe that we cannot detect with optical instruments (i.e. does not interact with light) Dark matter does however interact gravitationally (it has mass), so we have indirect methods of detection (gravitational lensing) The distribution of dark matter can be determined from images, such as those from the Hubble Space Telescope ~4% of the matter-energy in the universe is visible! (stuff we are made of) ~22% of the matter-energy in the universe is dark matter! The nature of dark matter still eludes us and is an open and very active question in physics!

46 STATES OF MATTER

47 STATES OF MATTER Gas

48 STATES OF MATTER Gas Liquid

49 STATES OF MATTER Gas Liquid Solid

50 STATES OF MATTER Gas Liquid Solid

51 STATES OF MATTER Gas phase transitions (week 4) Liquid Solid

52 STATES OF MATTER Gas phase transitions (week 4) Liquid Solid Plasma

53 STATES OF MATTER Gas phase transitions (week 4) Liquid Solid ionization Plasma

54 STATES OF MATTER Gas phase transitions (week 4) Liquid Solid ionization Plasma Quantum Condensates -superfluid (week 5) -superconductor -Bose-Einstein condensate

55 STATES OF MATTER Gas phase transitions (week 4) Liquid Solid ionization Plasma Quantum Condensates -superfluid (week 5) -superconductor -Bose-Einstein condensate

56 STATES OF MATTER Gas phase transitions (week 4) Liquid Solid ionization Plasma Quantum Condensates -superfluid (week 5) -superconductor -Bose-Einstein condensate Exotic Forms -dark matter -core of neutron star -quark-gluon plasma/fluid -transparent aluminum

57 STATES OF MATTER Gas phase transitions (week 4) Liquid Solid ionization Plasma Quantum Condensates -superfluid (week 5) -superconductor -Bose-Einstein condensate Exotic Forms -dark matter -core of neutron star -quark-gluon plasma/fluid -transparent aluminum

58 FROM THE BOTTOM UP?

59 FROM THE BOTTOM UP?

60 FROM THE BOTTOM UP? p+

61 FROM THE BOTTOM UP? p+ p+

62 FROM THE BOTTOM UP? p+ p+

63 FROM THE BOTTOM UP? p+ p+

64 FROM THE BOTTOM UP? p+ p+

65 FROM THE BOTTOM UP? p+ p+

66 FROM THE BOTTOM UP? p+ p+ Can the complex properties of emergent phenomena be derived solely from the properties of elementary particles?

67 FROM THE BOTTOM UP? p+ p+ Can the complex properties of emergent phenomena be derived solely from the properties of elementary particles? In principle, maybe

68 FROM THE BOTTOM UP? p+ p+ Can the complex properties of emergent phenomena be derived solely from the properties of elementary particles? In principle, maybe In practice, it seems nearly impossible

69 BREAKING DOWN THE SCIENCE There currently exist a scientific and philosophic debate about emergence and reductionism

70 BREAKING DOWN THE SCIENCE There currently exist a scientific and philosophic debate about emergence and reductionism the properties of large systems can be derived from properties of its constituents.

71 BREAKING DOWN THE SCIENCE There currently exist a scientific and philosophic debate about emergence and reductionism the properties of large systems can be derived from properties of its constituents. However, empirical evidence implies, for example, that Darwin s theory of evolution does not require the existence of elementary particle physics

72 BREAKING DOWN THE SCIENCE There currently exist a scientific and philosophic debate about emergence and reductionism the properties of large systems can be derived from properties of its constituents. However, empirical evidence implies, for example, that Darwin s theory of evolution does not require the existence of elementary particle physics In addition, it is likely impossible to derive Darwin s theory starting from particle physics.

73 BREAKING DOWN THE SCIENCE There currently exist a scientific and philosophic debate about emergence and reductionism Science X obeys the laws of science Y the properties of large systems can be derived from properties of its constituents. However, empirical evidence implies, for example, that Darwin s theory of evolution does not require the existence of elementary particle physics P.W. Anderson, Science, 1972 In addition, it is likely impossible to derive Darwin s theory starting from particle physics.

74 BREAKING DOWN THE SCIENCE There currently exist a scientific and philosophic debate about emergence and reductionism Science X obeys the laws of science Y the properties of large systems can be derived from properties of its constituents. However, empirical evidence implies, for example, that Darwin s theory of evolution does not require the existence of elementary particle physics In addition, it is likely impossible to derive Darwin s theory starting from particle physics. P.W. Anderson, Science, 1972 Although science X must obey the laws of science Y, science X requires new formalisms and theories to explain emergent phenomena

75 BREAKING DOWN THE SCIENCE There currently exist a scientific and philosophic debate about emergence and reductionism Science X obeys the laws of science Y the properties of large systems can be derived from properties of its constituents. However, empirical evidence implies, for example, that Darwin s theory of evolution does not require the existence of elementary particle physics In addition, it is likely impossible to derive Darwin s theory starting from particle physics. P.W. Anderson, Science, 1972 Although science X must obey the laws of science Y, science X requires new formalisms and theories to explain emergent phenomena Or X is more (and different) than the sum of its parts

76 HOW LARGE IS LARGE? 1 particle p+

77 HOW LARGE IS LARGE? 1 particle a few particles p+

78 HOW LARGE IS LARGE? 1 particle a few particles particles! p+

79 HOW LARGE IS LARGE? 1 particle a few particles particles! p+ How big is 10 23?

80 HOW LARGE IS LARGE? 1 particle a few particles particles! p+ How big is 10 23? There are about cells in the human body

81 HOW LARGE IS LARGE? 1 particle a few particles particles! p+ How big is 10 23? There are about cells in the human body cells X 10 9 humans human cells in the world!

82 HOW LARGE IS LARGE? 1 particle a few particles particles! p+ How big is 10 23? There are about cells in the human body cells X 10 9 humans human cells in the world! How can we possibly study such large systems?

83 THE PHYSICS OF MANY BODIES What do we need to describe a single particle?

84 THE PHYSICS OF MANY BODIES What do we need to describe a single particle? z x y

85 THE PHYSICS OF MANY BODIES What do we need to describe a single particle? v z x y 3 position coordinates (where?) and 3 velocity coordinates (how fast? and which direction?)

86 THE PHYSICS OF MANY BODIES What do we need to describe a single particle? v z x y 3 position coordinates (where?) and 3 velocity coordinates (how fast? and which direction?) This give 6 total degrees of freedom

87 THE PHYSICS OF MANY BODIES What do we need to describe a single particle? What about N particles? (N ~ ) v z x y 3 position coordinates (where?) and 3 velocity coordinates (how fast? and which direction?) This give 6 total degrees of freedom

88 THE PHYSICS OF MANY BODIES What do we need to describe a single particle? What about N particles? (N ~ ) v z 6 degrees of freedom per particle X N particles = 6N total degrees of freedom! x y 3 position coordinates (where?) and 3 velocity coordinates (how fast? and which direction?) This give 6 total degrees of freedom

89 THE PHYSICS OF MANY BODIES What do we need to describe a single particle? What about N particles? (N ~ ) v z 6 degrees of freedom per particle X N particles = 6N total degrees of freedom! x That s a lot to keep track of! y 3 position coordinates (where?) and 3 velocity coordinates (how fast? and which direction?) This give 6 total degrees of freedom

90 THE PHYSICS OF MANY BODIES What do we need to describe a single particle? What about N particles? (N ~ ) v z 6 degrees of freedom per particle X N particles = 6N total degrees of freedom! x That s a lot to keep track of! y This was the dilemma in the later parts of the 19 th century 3 position coordinates (where?) and 3 velocity coordinates (how fast? and which direction?) This give 6 total degrees of freedom

91 THE PHYSICS OF MANY BODIES What do we need to describe a single particle? What about N particles? (N ~ ) v z 6 degrees of freedom per particle X N particles = 6N total degrees of freedom! x That s a lot to keep track of! y This was the dilemma in the later parts of the 19 th century 3 position coordinates (where?) and 3 velocity coordinates (how fast? and which direction?) This give 6 total degrees of freedom Issac Newton

92 THE PHYSICS OF MANY BODIES What do we need to describe a single particle? What about N particles? (N ~ ) v z 6 degrees of freedom per particle X N particles = 6N total degrees of freedom! x That s a lot to keep track of! y This was the dilemma in the later parts of the 19 th century 3 position coordinates (where?) and 3 velocity coordinates (how fast? and which direction?) This give 6 total degrees of freedom Issac Newton bulk properties of matter

93 A HISTORICAL PRIMER By the late 19 th century the principles of thermodynamics had been established : energy conservation, work, heat, entropy

94 A HISTORICAL PRIMER By the late 19 th century the principles of thermodynamics had been established : energy conservation, work, heat, entropy Sadi Carnot Robert Mayer William Thomson Hermann Helmholtz Rudolf Clausius James Joule

95 A HISTORICAL PRIMER By the late 19 th century the principles of thermodynamics had been established : energy conservation, work, heat, entropy Sadi Carnot Robert Mayer William Thomson Hermann Helmholtz Rudolf Clausius James Joule This provided a framework in order to define basic macroscopic ideas such as heat flow, thermal equilibrium, and efficiency of an engine

96 A HISTORICAL PRIMER By the late 19 th century the principles of thermodynamics had been established : energy conservation, work, heat, entropy Sadi Carnot Robert Mayer William Thomson Hermann Helmholtz Rudolf Clausius James Joule This provided a framework in order to define basic macroscopic ideas such as heat flow, thermal equilibrium, and efficiency of an engine How do we connect thermodynamics to the underlying material, namely a large collection of atoms?

97 A HISTORICAL PRIMER By the late 19 th century the principles of thermodynamics had been established : energy conservation, work, heat, entropy Sadi Carnot Robert Mayer William Thomson Hermann Helmholtz Rudolf Clausius James Joule This provided a framework in order to define basic macroscopic ideas such as heat flow, thermal equilibrium, and efficiency of an engine How do we connect thermodynamics to the underlying material, namely a large collection of atoms? atomistics

98 A HISTORICAL PRIMER By the late 19 th century the principles of thermodynamics had been established : energy conservation, work, heat, entropy Sadi Carnot Robert Mayer William Thomson Hermann Helmholtz Rudolf Clausius James Joule This provided a framework in order to define basic macroscopic ideas such as heat flow, thermal equilibrium, and efficiency of an engine How do we connect thermodynamics to the underlying material, namely a large collection of atoms? atomistics?

99 A HISTORICAL PRIMER By the late 19 th century the principles of thermodynamics had been established : energy conservation, work, heat, entropy Sadi Carnot Robert Mayer William Thomson Hermann Helmholtz Rudolf Clausius James Joule This provided a framework in order to define basic macroscopic ideas such as heat flow, thermal equilibrium, and efficiency of an engine How do we connect thermodynamics to the underlying material, namely a large collection of atoms? atomistics? thermodynamics

100 A HISTORICAL PRIMER By the late 19 th century the principles of thermodynamics had been established : energy conservation, work, heat, entropy Sadi Carnot Robert Mayer William Thomson Hermann Helmholtz Rudolf Clausius James Joule This provided a framework in order to define basic macroscopic ideas such as heat flow, thermal equilibrium, and efficiency of an engine How do we connect thermodynamics to the underlying material, namely a large collection of atoms? atomistics statistical mechanics thermodynamics

101 A HISTORICAL PRIMER Statistical mechanics uses the mathematical tools of probability and statistics to describe thermodynamic phenomena

102 A HISTORICAL PRIMER Statistical mechanics uses the mathematical tools of probability and statistics to describe thermodynamic phenomena If the number of particles N is very large, then the dynamics of the system can be subject to a statistical interpretation

103 A HISTORICAL PRIMER Statistical mechanics uses the mathematical tools of probability and statistics to describe thermodynamic phenomena If the number of particles N is very large, then the dynamics of the system can be subject to a statistical interpretation 1860 s James Clerk Maxwell derives the velocity distribution of particles in an ideal gas (coins the term statistical mechanics ) Maxwell

104 A HISTORICAL PRIMER Statistical mechanics uses the mathematical tools of probability and statistics to describe thermodynamic phenomena If the number of particles N is very large, then the dynamics of the system can be subject to a statistical interpretation 1860 s James Clerk Maxwell derives the velocity distribution of particles in an ideal gas (coins the term statistical mechanics ) 1870 s Ludwig Boltzmann gives entropy a probabilistic interpretation, develops kinetic theory of gases Maxwell Boltzman

105 A HISTORICAL PRIMER Statistical mechanics uses the mathematical tools of probability and statistics to describe thermodynamic phenomena If the number of particles N is very large, then the dynamics of the system can be subject to a statistical interpretation 1860 s James Clerk Maxwell derives the velocity distribution of particles in an ideal gas (coins the term statistical mechanics ) 1870 s Ludwig Boltzmann gives entropy a probabilistic interpretation, develops kinetic theory of gases 1890 s Josiah Willard Gibbs perfects the theory of statistical mechanics, writes the bible of stat. mech. - developed idea of phase space and phase transitions Maxwell Boltzman - 1 st American theorist Gibbs

106 WHAT IS ENTROPY? Entropy is typically defined as a measure of how organized or disorganized a system is

107 WHAT IS ENTROPY? Entropy is typically defined as a measure of how organized or disorganized a system is equilibrium: max entropy non-equilibrium: less entropy

108 WHAT IS ENTROPY? Entropy is typically defined as a measure of how organized or disorganized a system is equilibrium: max entropy non-equilibrium: less entropy Second Law of Thermodynamics: The entropy of an isolated system which is not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.

109 WHAT IS ENTROPY? Entropy is typically defined as a measure of how organized or disorganized a system is equilibrium: max entropy non-equilibrium: less entropy Second Law of Thermodynamics: The entropy of an isolated system which is not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium. Any method involving the notion of entropy, the very existence of which depends on the second law of thermodynamics, will doubtless seem to many far-fetched, and may repel beginners as obscure and difficult of comprehension. - J. Willard Gibbs

110 WHAT IS ENTROPY? Entropy is typically defined as a measure of how organized or disorganized a system is equilibrium: max entropy non-equilibrium: less entropy Second Law of Thermodynamics: The entropy of an isolated system which is not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium. Any method involving the notion of entropy, the very existence of which depends on the second law of thermodynamics, will doubtless seem to many far-fetched, and may repel beginners as obscure and difficult of comprehension. - J. Willard Gibbs The law that entropy always increases holds, I think, the supreme position among the laws of Nature. if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. - Sir Arthur Stanley Eddington

111 STATISTICAL MECHANICS - ENTROPY Probabilistic Interpretation:

112 STATISTICAL MECHANICS - ENTROPY Probabilistic Interpretation: The entropy of a system depends on the number of possible states that are accessible.

113 STATISTICAL MECHANICS - ENTROPY Probabilistic Interpretation: The entropy of a system depends on the number of possible states that are accessible. What do we mean by a system state?

114 STATISTICAL MECHANICS - ENTROPY Probabilistic Interpretation: The entropy of a system depends on the number of possible states that are accessible. What do we mean by a system state? Let s say we flip a coin 1 time. We can either get heads or tails. -there are 2 possible states

115 STATISTICAL MECHANICS - ENTROPY Probabilistic Interpretation: The entropy of a system depends on the number of possible states that are accessible. What do we mean by a system state? Let s say we flip a coin 1 time. We can either get heads or tails. -there are 2 possible states What about flipping it 2 times? There are 4 possible states: -hh, ht, th, tt

116 STATISTICAL MECHANICS - ENTROPY Probabilistic Interpretation: The entropy of a system depends on the number of possible states that are accessible. What do we mean by a system state? Let s say we flip a coin 1 time. We can either get heads or tails. -there are 2 possible states What about flipping it 2 times? There are 4 possible states: -hh, ht, th, tt What about flipping it 3 times? There are 8 possible states: -hhh, hht, hth, htt, thh, tht, tth, ttt

117 STATISTICAL MECHANICS - ENTROPY Probabilistic Interpretation: The entropy of a system depends on the number of possible states that are accessible. What do we mean by a system state? Let s say we flip a coin 1 time. We can either get heads or tails. -there are 2 possible states What about flipping it 2 times? There are 4 possible states: -hh, ht, th, tt What about flipping it 3 times? There are 8 possible states: -hhh, hht, hth, htt, thh, tht, tth, ttt What about flipping it 4 times? There are 16 possible states: -hhhh, hhht, hhth, hhtt, hthh, htht, htth, httt thhh, thht, thth, thtt, tthh, ttht, ttth, tttt

118 STATISTICAL MECHANICS - ENTROPY coin flips number of states N 2 N

119 STATISTICAL MECHANICS - ENTROPY coin flips number of states N 2 N the number of states grows exponentially in this system

120 STATISTICAL MECHANICS - ENTROPY coin flips number of states N 2 N the number of states grows exponentially in this system A very powerful result of statistical mechanics is that the entropy is proportional to the exponent in the number of states

121 STATISTICAL MECHANICS - ENTROPY coin flips number of states N 2 N the number of states grows exponentially in this system A very powerful result of statistical mechanics is that the entropy is proportional to the exponent in the number of states If we have 2 N states, then the entropy S N

122 STATISTICAL MECHANICS - ENTROPY coin flips number of states N 2 N the number of states grows exponentially in this system A very powerful result of statistical mechanics is that the entropy is proportional to the exponent in the number of states If we have 2 N states, then the entropy S N entropy number of states

123 STATISTICAL MECHANICS - ENTROPY coin flips number of states N 2 N the number of states grows exponentially in this system A very powerful result of statistical mechanics is that the entropy is proportional to the exponent in the number of states If we have 2 N states, then the entropy S N entropy number of states huge numbers!

124 STATISTICAL MECHANICS - ENTROPY coin flips number of states N 2 N the number of states grows exponentially in this system A very powerful result of statistical mechanics is that the entropy is proportional to the exponent in the number of states If we have 2 N states, then the entropy S N entropy Entropy has a logarithmic dependence on the number of states, the same as the Richter scale for measuring earthquake strength. number of states huge numbers!

125 EQUILIBRIUM AND ENTROPY We can agree that if we flip a coin N times, we will on average get heads N/2 times (since the probability of heads is ½)

126 EQUILIBRIUM AND ENTROPY We can agree that if we flip a coin N times, we will on average get heads N/2 times (since the probability of heads is ½) For N coin flips, we will call the N/2 state equilibrium

127 EQUILIBRIUM AND ENTROPY We can agree that if we flip a coin N times, we will on average get heads N/2 times (since the probability of heads is ½) For N coin flips, we will call the N/2 state equilibrium What does the distribution of states look like as we increase N?

128 EQUILIBRIUM AND ENTROPY We can agree that if we flip a coin N times, we will on average get heads N/2 times (since the probability of heads is ½) For N coin flips, we will call the N/2 state equilibrium What does the distribution of states look like as we increase N? Distribution of states probability density N=10 0 N/2 N number of heads

129 EQUILIBRIUM AND ENTROPY We can agree that if we flip a coin N times, we will on average get heads N/2 times (since the probability of heads is ½) For N coin flips, we will call the N/2 state equilibrium What does the distribution of states look like as we increase N? Distribution of states probability density N=32 N=10 0 N/2 N number of heads

130 EQUILIBRIUM AND ENTROPY We can agree that if we flip a coin N times, we will on average get heads N/2 times (since the probability of heads is ½) For N coin flips, we will call the N/2 state equilibrium What does the distribution of states look like as we increase N? Distribution of states probability density N=100 0 N/2 N number of heads

131 EQUILIBRIUM AND ENTROPY We can agree that if we flip a coin N times, we will on average get heads N/2 times (since the probability of heads is ½) For N coin flips, we will call the N/2 state equilibrium What does the distribution of states look like as we increase N? Distribution of states probability density N=317 0 N/2 N number of heads

132 EQUILIBRIUM AND ENTROPY We can agree that if we flip a coin N times, we will on average get heads N/2 times (since the probability of heads is ½) For N coin flips, we will call the N/2 state equilibrium What does the distribution of states look like as we increase N? Distribution of states probability density N= N/2 N number of heads

133 EQUILIBRIUM AND ENTROPY We can agree that if we flip a coin N times, we will on average get heads N/2 times (since the probability of heads is ½) For N coin flips, we will call the N/2 state equilibrium What does the distribution of states look like as we increase N? Distribution of states probability density N= N/2 N number of heads As N approaches (macroscopic size), it becomes outrageously unlikely to find the system in anything but the equilibrium state!

134 TEMPERATURE Until this point I have not mentioned temperature

135 TEMPERATURE Until this point I have not mentioned temperature Temperature adds stochastic or random fluctuations to a system

136 TEMPERATURE Until this point I have not mentioned temperature Temperature adds stochastic or random fluctuations to a system These fluctuations allow a system to jump between different energy states

137 TEMPERATURE Until this point I have not mentioned temperature Temperature adds stochastic or random fluctuations to a system These fluctuations allow a system to jump between different energy states E high E low

138 TEMPERATURE Until this point I have not mentioned temperature Temperature adds stochastic or random fluctuations to a system These fluctuations allow a system to jump between different energy states E high E E low

139 TEMPERATURE Until this point I have not mentioned temperature Temperature adds stochastic or random fluctuations to a system These fluctuations allow a system to jump between different energy states E high E E low

140 TEMPERATURE Until this point I have not mentioned temperature Temperature adds stochastic or random fluctuations to a system These fluctuations allow a system to jump between different energy states E high E E low

141 TEMPERATURE Until this point I have not mentioned temperature Temperature adds stochastic or random fluctuations to a system These fluctuations allow a system to jump between different energy states E high E low E At a given temperature, what is the probability of finding the system in certain energy state?

142 PROBABILITY OF STATES

143 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is:

144 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is:

145 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is: T = temperature k B = Boltzmann s constant = 1.38 x Joules/Kelvin

146 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is: T = temperature k B = Boltzmann s constant = 1.38 x Joules/Kelvin exponential decay probability energy

147 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is: T = temperature k B = Boltzmann s constant = 1.38 x Joules/Kelvin As the temperature decreases, it becomes less likely to find the system in the high energy state E high probability exponential decay energy

148 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is: T = temperature k B = Boltzmann s constant = 1.38 x Joules/Kelvin As the temperature decreases, it becomes less likely to find the system in the high energy state E high As the E increases, it becomes less likely to find the system in the high energy state E high probability exponential decay energy

149 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is: T = temperature k B = Boltzmann s constant = 1.38 x Joules/Kelvin Question: what is the pressure at the top of Mt. Everest?

150 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is: T = temperature k B = Boltzmann s constant = 1.38 x Joules/Kelvin Question: what is the pressure at the top of Mt. Everest? Answer: about 1/3 of atmospheric pressure

151 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is: T = temperature k B = Boltzmann s constant = 1.38 x Joules/Kelvin Question: what is the pressure at the top of Mt. Everest? Answer: about 1/3 of atmospheric pressure Potential energy of a nitrogen molecule in gravity = mass x height x acceleration of gravity Increasing gravitational potential energy

152 PROBABILITY OF STATES Boltzmann showed that the relative probability P of a system being in a state with a given energy E high is: T = temperature k B = Boltzmann s constant = 1.38 x Joules/Kelvin Question: what is the pressure at the top of Mt. Everest? Answer: about 1/3 of atmospheric pressure Potential energy of a nitrogen molecule in gravity = mass x height x acceleration of gravity Increasing gravitational potential energy

153 IDEAL GAS Ideal gas: the simplest large collection of particles

154 IDEAL GAS Ideal gas: the simplest large collection of particles Consider N particles confined in a box of volume V:

155 IDEAL GAS Ideal gas: the simplest large collection of particles Consider N particles confined in a box of volume V:

156 IDEAL GAS Ideal gas: the simplest large collection of particles Consider N particles confined in a box of volume V:

157 IDEAL GAS Ideal gas: the simplest large collection of particles Consider N particles confined in a box of volume V: Assumptions: particles are not interacting

158 IDEAL GAS Ideal gas: the simplest large collection of particles Consider N particles confined in a box of volume V: Assumptions: particles are not interacting system is at high temperatures, dominated by kinetic energy

159 IDEAL GAS Ideal gas: the simplest large collection of particles Consider N particles confined in a box of volume V: Assumptions: particles are not interacting system is at high temperatures, dominated by kinetic energy particles are indistinguishable

160 IDEAL GAS Ideal gas: the simplest large collection of particles Consider N particles confined in a box of volume V: Assumptions: particles are not interacting system is at high temperatures, dominated by kinetic energy particles are indistinguishable This model works amazingly well for many gases at room temperature, especially monatomic gases such as Argon

161 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume:

162 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N

163 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law:

164 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure

165 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume

166 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume = number of particles

167 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume = number of particles x temperature

168 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume = number of particles x temperature x constant

169 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume = number of particles x temperature x constant PV = NTk B

170 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume = number of particles x temperature x constant PV = NTk B The point of all this is to show that the thermodynamics and physics of a collection of particles boils down to a counting problem

171 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume = number of particles x temperature x constant PV = NTk B The point of all this is to show that the thermodynamics and physics of a collection of particles boils down to a counting problem Statistical mechanics provides a pathway between the distribution of energy states of a system and its macroscopic properties.

172 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume = number of particles x temperature x constant PV = NTk B The point of all this is to show that the thermodynamics and physics of a collection of particles boils down to a counting problem Statistical mechanics provides a pathway between the distribution of energy states of a system and its macroscopic properties. -heat capacity (how much heat is needed to change the temperature)

173 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume = number of particles x temperature x constant PV = NTk B The point of all this is to show that the thermodynamics and physics of a collection of particles boils down to a counting problem Statistical mechanics provides a pathway between the distribution of energy states of a system and its macroscopic properties. -heat capacity (how much heat is needed to change the temperature) -compressibility (speed of sound)

174 COUNTING STATES AND THERMODYNAMICS For this system of non-interacting particles (ideal gas), it can be shown that the number of states is proportional to the Nth power of the volume: # of states V N Turns out that if this relation is true, then the system will obey the ideal gas law: pressure x volume = number of particles x temperature x constant PV = NTk B The point of all this is to show that the thermodynamics and physics of a collection of particles boils down to a counting problem Statistical mechanics provides a pathway between the distribution of energy states of a system and its macroscopic properties. -heat capacity (how much heat is needed to change the temperature) -compressibility (speed of sound) -thermal conductivity (how well does it conduct heat?)

175 NOW THE INTERESTING STUFF

176 NOW THE INTERESTING STUFF There is no liquid phase in the ideal gas model

177 NOW THE INTERESTING STUFF There is no liquid phase in the ideal gas model There is no freezing and crystallization in the ideal gas model

178 NOW THE INTERESTING STUFF There is no liquid phase in the ideal gas model There is no freezing and crystallization in the ideal gas model Real matter is composed of particles that interact in many different ways

179 NOW THE INTERESTING STUFF There is no liquid phase in the ideal gas model There is no freezing and crystallization in the ideal gas model Real matter is composed of particles that interact in many different ways These interactions are what give rise to the vast array of properties of ordinary and exotic states of matter

180 NOW THE INTERESTING STUFF There is no liquid phase in the ideal gas model There is no freezing and crystallization in the ideal gas model Real matter is composed of particles that interact in many different ways These interactions are what give rise to the vast array of properties of ordinary and exotic states of matter For the past 100 years physicists have been developing experiments and theoretical tools to study large systems and their emergent properties Photo by Wilson Bentley

181 NOW THE INTERESTING STUFF There is no liquid phase in the ideal gas model There is no freezing and crystallization in the ideal gas model Real matter is composed of particles that interact in many different ways These interactions are what give rise to the vast array of properties of ordinary and exotic states of matter For the past 100 years physicists have been developing experiments and theoretical tools to study large systems and their emergent properties These lectures are intended to provide a look into how we understand these complex phenomena Photo by Wilson Bentley

182 ARROW OF TIME? (SOMETHING TO THINK ABOUT)

183 ARROW OF TIME? (SOMETHING TO THINK ABOUT) Nearly all the laws of physics are time-reversible, meaning they have no preference for the direction of time

184 ARROW OF TIME? (SOMETHING TO THINK ABOUT) Nearly all the laws of physics are time-reversible, meaning they have no preference for the direction of time The fact that entropy will always tend to increase (2 nd law) means that there is a preferred arrow of time (increasing entropy)

185 ARROW OF TIME? (SOMETHING TO THINK ABOUT) Nearly all the laws of physics are time-reversible, meaning they have no preference for the direction of time The fact that entropy will always tend to increase (2 nd law) means that there is a preferred arrow of time (increasing entropy) Is this related to the psychological arrow of time?

186 ARROW OF TIME? (SOMETHING TO THINK ABOUT) Nearly all the laws of physics are time-reversible, meaning they have no preference for the direction of time The fact that entropy will always tend to increase (2 nd law) means that there is a preferred arrow of time (increasing entropy) Is this related to the psychological arrow of time? expanding universe?

187 ARROW OF TIME? (SOMETHING TO THINK ABOUT) Nearly all the laws of physics are time-reversible, meaning they have no preference for the direction of time The fact that entropy will always tend to increase (2 nd law) means that there is a preferred arrow of time (increasing entropy) Is this related to the psychological arrow of time? expanding universe? Maxwell s Demon: imagine a external influence able to open a door between two volumes of gas, and only let fast particles move to one side. One side is hot and one is cold!?

188 ARROW OF TIME? (SOMETHING TO THINK ABOUT) Nearly all the laws of physics are time-reversible, meaning they have no preference for the direction of time The fact that entropy will always tend to increase (2 nd law) means that there is a preferred arrow of time (increasing entropy) Is this related to the psychological arrow of time? expanding universe? Maxwell s Demon: imagine a external influence able to open a door between two volumes of gas, and only let fast particles move to one side. One side is hot and one is cold!? Does this violate the second law of thermodynamics?

189 ARROW OF TIME? (SOMETHING TO THINK ABOUT) Nearly all the laws of physics are time-reversible, meaning they have no preference for the direction of time The fact that entropy will always tend to increase (2 nd law) means that there is a preferred arrow of time (increasing entropy) Is this related to the psychological arrow of time? expanding universe? Maxwell s Demon: imagine a external influence able to open a door between two volumes of gas, and only let fast particles move to one side. One side is hot and one is cold!? Does this violate the second law of thermodynamics? NO! We must include the demon in the calculation of entropy!

190 ARROW OF TIME? (SOMETHING TO THINK ABOUT) Nearly all the laws of physics are time-reversible, meaning they have no preference for the direction of time The fact that entropy will always tend to increase (2 nd law) means that there is a preferred arrow of time (increasing entropy) Is this related to the psychological arrow of time? expanding universe? Maxwell s Demon: imagine a external influence able to open a door between two volumes of gas, and only let fast particles move to one side. One side is hot and one is cold!? Does this violate the second law of thermodynamics? NO! We must include the demon in the calculation of entropy! The demon must do work to store information (memories), the entropy of the whole system will increase

191 ARROW OF TIME? (SOMETHING TO THINK ABOUT) Nearly all the laws of physics are time-reversible, meaning they have no preference for the direction of time The fact that entropy will always tend to increase (2 nd law) means that there is a preferred arrow of time (increasing entropy) Is this related to the psychological arrow of time? expanding universe? Maxwell s Demon: imagine a external influence able to open a door between two volumes of gas, and only let fast particles move to one side. One side is hot and one is cold!? Does this violate the second law of thermodynamics? NO! We must include the demon in the calculation of entropy! The demon must do work to store information (memories), the entropy of the whole system will increase The connection between entropy and information is part of the basis for modern information theory (developed by Claude Shannon 1949)