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Tony Rothman

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One of the Deepest Paradoxes of Modern Physics: Microscopic Physics is Time Reversal Symmetric. Nature is Not!!

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The Fundamental, Microscopic Theories of Physics Don’t Care Which Way Time Goes Newtonian Mechanics Newtonian Mechanics Electromagnetism Electromagnetism Relativity Relativity Quantum Mechanics Quantum Mechanics

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Nevertheless, Nature Cares!!!

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One of the single most surprising things about the universe is that things change!! Eggs break, ice cubes melt, stars emit radiation, we record memories of the past. And it all happens in a consistent “direction” (from the past to the future) throughout the universe: The Arrow of Time.

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Why does time have a direction? It’s certainly NOT a feature of the microscopic laws of physics!! Those work equally well forwards or backwards in time; they are “invariant under time reversal” (t -t, p i -p i )

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A common answer to why there is an arrow of time is: Entropy & The 2 nd Law of Thermodynamics: This tells us that entropy tends to increase (in closed systems) as a function of time. The 2 nd Law is NOT invariant under time reversal!

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Roger Penrose: It’s not quite that simple! He lists 7 ways in which Nature distinguishes the past from the future: The 7 “arrows of time” Roger Penrose: (Cosmologist) It’s not quite that simple! He lists 7 ways in which Nature distinguishes the past from the future: The 7 “arrows of time”

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The Seven Arrows of Time (according to Penrose!) 1. Memory Arrow 2. Electromagnetic Arrow 3. Quantum Arrow 4. Black Hole Arrow 5. Cosmological Arrow 6. Kaon Arrow 7. Entropy Arrow Here, we’re interested mainly in #7.

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1. The “Memory” Arrow Memory only works Backwards! (We don't “remember” the future!)

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2. The Electromagnetic Arrow Electromagnetic Causes Always Precede Effects (The Principle of Causality)

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3. The Quantum Arrow Wave Functions “Collapse” But They Don’t “Uncollapse”!!!

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4. The Black Hole Arrow Black Holes Exist But White Holes Don’t (Apparently)

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5. The Cosmological Arrow 1. We live in an expanding universe, not a contracting one. a contracting one. 2. Would time change direction if the universe started contracting? universe started contracting?

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6. The Kaon Arrow Kaon decay is not time reversal symmetric! K o ππ Theoretically, in a time-reversed world, this particle would decay at a different rate than in our world!

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The CPT Theorem CPT = “Charge-Parity-Time” If the charge of all particles in universe were changed to the opposite charge (so that all particles change to their antiparticles); & at the same time, all were reflected in a mirror; & at the same time, time started to run backwards: This new world would be indistinguishable from the old world.

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The Princeton Experiment (1964): Found that: Kaon decay is not invariant under CP-reversal. Kaon decay is not invariant under CP-reversal. Kaon decay must also change under T-reversal Kaon decay must also change under T-reversal

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7. The Entropy Arrow The Arrow of Time that is the most relevant to this course! Total Entropy always increases (The 2 nd Law of Thermodynamics)

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Philosophical Question: Are these arrows of time independent? Debate Among Cosmologists: Hawking: The memory & entropy arrows are linked. (Energy is required to read one bit. Reading increases the entropy) Rothman to Penrose: One arrow is arbitrary, so there are only six independent ones. Sudarshan to Rothman: All particle decay is CPT invariant but such decays are irreversible: The particle- decay arrow. Rothman to Sudarshan: Is there a temperature dependence to these decays?

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Entropy As we’ve seen: Entropy can be thought of as a a quantity proportional to the waste heat generated in any realistic process. Entropy can be thought of as a a quantity proportional to the waste heat generated in any realistic process. Entropy can also be thought of as a measure of the “disorder” in a system Entropy can also be thought of as a measure of the “disorder” in a system

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The 2 nd Law of Thermodynamics: In any process, the entropy of a system always increases under realistic (“irreversible”) conditions This is the only law of nature that exhibits a direction of time!

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Phase Space: Sets of macroscopically indistinguishable microstates Entropy measures volumes in phase space.

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Boltzmann: Total Entropy increases because there are more high-entropy than low-entropy microstates. The 2 nd Law: “Starting in a low-entropy state, evolving to a higher entropy state is natural”.

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Penrose: The low entropy near the Big Bang is responsible for everything about the (7 th ) arrow of time. Life and death Biological evolution Memory The “flow” of time Without the arrow of time, we would be in thermal equilibrium -- everything would be static, nothing would ever change! Question: Why aren’t we in thermal equilibrium? Answers: Controversial issues in cosmology! Beyond the scope of this course!

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Sir Arthur Eddington (1931) Still another version of the 2 nd Law of Thermodynamics! “A universe containing mathematical physicists will at any assigned date be in the state of maximum disorganization which is not inconsistent with the existence of such creatures.”

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Consider a “Thermodynamic System”: A many particle s ystem (perhaps a gas) Obeys the 2 nd Law of Thermodynamics Consider a “Thermodynamic System”: A many particle s ystem (perhaps a gas) Obeys the 2 nd Law of Thermodynamics which says that real processes are irreversible & that total entropy can only increase However, the individual particles in the system obey either Newton’s Laws or Quantum Mechanical Laws Back to Entropy & the Arrow of Time!

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Microscopic reversible, macroscopic irreversible? Raindrop splash and displacement of soil particles. Source: USDA Natural Resources Conservation Service.

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How is irreversibility described quantitatively? Example: Mixing alcohol with water Time’s Arrow: Maximizes the “number of ways” to arrange the molecules. That is, it maximizes the entgropy.

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The Arrow of Time & The Free Expansion of a Gas See Figure We wouldn’t expect a gas undergoing free expansion to spontaneously revert to a situation with all molecules back in the left container.

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The Entropy Arrow of Time The universe is a closed system (it is everything we know of), so its entropy must be increasing since no processes in the universe are reversible. The 2 nd Law of Thermodynamics: Entropy must increase as processes go forward, so the Arrow of Time must point towards increased entropy. That is, Time always points from the past to the future where the entropy is maximized.

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So, there is a Major Paradox: 1. A many particle system is a thermodynamic system which obeys the 2 nd Law of Thermo. 2. But if the particles individually obey time reversible Newtonian (or Quantum) physics, how can arrow of time even arise????? So, there is a Major Paradox: 1. A many particle system is a thermodynamic system which obeys the 2 nd Law of Thermo. 2. But if the particles individually obey time reversible Newtonian (or Quantum) physics, how can arrow of time even arise?????

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Boltzmann: Attempted to derive an increase in entropy from Newton’s Laws of Motion. “Boltzmann’s H Theorem”

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Loschmidt Strikes Back

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Boltzmann Uses Probability Theory “An ordered deck of cards becomes disordered on shuffling”

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Gibbs strikes back

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Prigogine’s Solution to this paradox: 1. The 2 nd Law of Thermodynamics is a fundamental law governing the universe. It can’t be derived from other laws (such as Newton’s Laws) It can’t be derived from other laws (such as Newton’s Laws) 2. Chaotic systems intrinsically manifest irreversible behavior. 3. So, the 2 nd Law’s Entropy increase is due to chaos.

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Objections to Prigogine’s Solution: 1. The equations governing motion in chaotic systems are time-symmetric. 2. If a system is chaotic in the future, it was also chaotic into the past.

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Conclusion: To date, there has been no satisfactory explanation for direction of time!!

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