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Tony Rothman. One of the Deepest Paradoxes of Modern Physics:. Microscopic Physics is Time Reversal Symmetric. Nature is Not!!. The Fundamental, Microscopic Theories of Physics Don’t Care Which Way Time Goes. Newtonian Mechanics Electromagnetism Relativity Quantum Mechanics.

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Tony rothman

Tony Rothman

One of the deepest paradoxes of modern physics

One of the Deepest Paradoxes of Modern Physics:

Microscopic Physics is Time Reversal Symmetric.

Nature is Not!!

The fundamental microscopic theories of physics don t care which way time goes

The Fundamental, Microscopic Theories of Physics Don’t Care Which Way Time Goes

  • Newtonian Mechanics

  • Electromagnetism

  • Relativity

  • Quantum Mechanics

Nevertheless n ature cares

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 NOTa feature of the microscopic laws of physics!!

Those work equally well forwards or backwards in time;

they are “invariant under time reversal”

(t  -t, pi  -pi)

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A common answer to why there is an arrow of time is:

Entropy & The 2nd Law of Thermodynamics:

This tells us that entropy tends to increase (in closed systems)

as a function of time.

The 2nd Law is NOT invariant under time reversal!

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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”

The seven arrows of time according to penrose

The Seven Arrows of Time(according to Penrose!)

Memory Arrow

Electromagnetic Arrow

Quantum Arrow

Black Hole Arrow

Cosmological Arrow

Kaon Arrow

Entropy Arrow

Here, we’re interested mainly in #7.

1 the memory arrow

Memory only works Backwards!

(We don't “remember” the future!)

1. The “Memory” Arrow

2 the electromagnetic arrow

2. The Electromagnetic Arrow

Electromagnetic Causes Always Precede Effects(The Principle of Causality)

3 the quantum arrow

3. The Quantum Arrow

Wave Functions “Collapse”

But They Don’t “Uncollapse”!!!

4 the black hole arrow

4. The Black Hole Arrow

Black Holes Exist But White Holes Don’t (Apparently)

5 the cosmological arrow

5. The Cosmological Arrow

1. We live in an expanding universe, not

a contracting one.

2. Would time change direction if the

universe started contracting?

6 the kaon arrow

6. The Kaon Arrow

Kaon decay is not time reversal symmetric!

Ko ππ

Theoretically, in a time-reversed world, this particle would decay at a different rate than in our world!

The cpt theorem

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.

The princeton experiment 1964

The Princeton Experiment (1964):

Found that:

Kaon decay is not invariant under CP-reversal.

Kaon decay must also change under T-reversal

7 the entropy arrow the arrow of time that is the most relevant to this course

7. The Entropy ArrowThe Arrow of Time that is the most relevant to this course!

Total Entropy always increases

(The 2nd Law of Thermodynamics)

Philosophical question are these arrows of time independent debate among cosmologists

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?



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 also be thought of as a measure of the “disorder” in a system

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The 2nd 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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Entropy measures volumes in phase space.

Phase Space:

Sets of macroscopically



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Total Entropy increases because there are more

high-entropy than low-entropy microstates.

The 2nd Law:

“Starting in a low-entropy state, evolving to a higher

entropy state is natural”.

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The low entropy near the Big Bang is responsible

for everything about the (7th) arrow of time.

Life and death

Biological evolution


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

2nd 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.”

However the individual particles in the system obey either newton s laws or quantum mechanical laws

Back to Entropy & the Arrow of Time!

Consider a “Thermodynamic System”:A many particle system (perhaps a gas)Obeys the 2nd Law of Thermodynamicswhich 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

Microscopic reversible macroscopic irreversible

Microscopic reversible, macroscopic irreversible?

Raindrop splash and displacement of soil particles.

Source: USDA Natural Resources Conservation Service.

How is irreversibility described quantitatively

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.

The arrow of time the free expansion of a gas

The Arrow of Time&The Free Expansionof 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.

The entropy arrow of time

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 2nd Law of Thermodynamics: Entropy must increase as processes go forward, so the Arrow of Timemust 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 aMajor Paradox:1.A many particle system is a thermodynamic system which obeys the 2nd 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 Lawsof Motion. “Boltzmann’s H Theorem”

Loschmidt strikes back

Loschmidt Strikes Back

Boltzmann uses probability theory

Boltzmann Uses Probability Theory

“An ordered deck of cards becomes disordered on shuffling”

Gibbs strikes back

Gibbs strikes back

Prigogine s solution to this paradox

Prigogine’s Solutionto this paradox:

1. The 2nd Law of Thermodynamics is a fundamental law governing the universe.

It can’t be derived from other laws(such as Newton’s Laws)

2. Chaotic systems intrinsically manifest irreversible behavior.

3. So, the 2nd Law’s Entropy increase is due to chaos.

Objections to prigogine s solution

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.



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

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