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Time Arrow,Statistics

and the Universe IIIPhysics Summer School

18 July 2002K. Y. Michael Wong

Outline:* Alternative forms of the second law* Maxwell’s demon* Available energy in refrigerators and heat engines* Time arrow and cosmology

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Microstate = the detailed specification of the state of a collection of particles, particle-by-particle

Macrostate = the overall specification of the state of a collection of particles, irrespective of the detailed information of individual particles

Degeneracy = the number of microstates belonging to a macrostate

Basic assumption = all microstates are equally probable, hence the macrostate with the highest degeneracy or the largest volume in phase space is the most probable one.

Entropy: S = constant × log W (W=degeneracy or volume) 2nd law: S ≥ 0. Temperature of ideal gas: T E/N.

Review

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Heat will never of itself flow from a cold object to a hot object.

Second law of thermodynamics(2nd alternative form)

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Heat engines

Heat engines change heat energy into mechanical energy.It consists of 3 steps:1. extract heat energy from a hot reservoir2. change part of the heat energy into mechanical energy3. dump the remaining heat energy into a cold reservoirReverse process: the refrigerator, heat and energy flow in reverse direction.

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The refrigerator

The refrigerator is equivalent to a heat engine operated in the reverse direction.

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It is not possible to completely change heat energy into mechanical energy.

Second law of thermodynamics(3rd alternative form)

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Proof:

1. Suppose we can find a heat engine that can completely change heat energy from a hot reservoir into mechanical energy.2. Then we can use the mechanical energy of this heat engine to drive a refrigerator that transports heat energy from a cold reservoir back to the hot reservoir.3. The net effect of this engine-fridge system is thus to transport heat energy from the cold reservoir to the hot reservoir, which violates the 2nd alternative form of the second law of thermodynamics.

+ =

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Maxwell’s demon

If the demon opens the door when he sees fast molecules,but closes the door when he sees slow molecules,then the right side will get hot, and the left side will get cold, without any other changes in the environment.

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Question 6Is it possible for Maxwell’s demon to separate hot gas on one side and cold gas on the other without having to spend any energy? (Why?)A. Yes B. No

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ExplanationConsider Maxwell’s demon as a heat engine.1. When the demon opens the door for the hot molecule to

pass, it acts as a refrigerator.2. Heat flows from the cold side to the hot side. According to

the second law, this must be accompanied by mechanical work supplied by the demon.

3. For example, if the demon opens the door as shown in the figure, or a turnpike (similar to those at the entrance of MTR stations) for fast molecules, it will experience a strong opposing force from the other side, if it is really hotter. The demon must do work.

4. On the other hand, if it uses a sliding door, more molecules are likely to pass through in the reverse direction if the other side is really hotter. The demon does not do work, but it cannot complete the task of separation.

Answer: No.

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Ratchet and Pawl

In his lectures, Feynman considered other devices with apparent violations of the second law:1.When molecules in the right chamber bombard the vanes, the wheel oscillates and jiggles.2.However, the ratchet and pawl in the left chamber only allow it to turn in the direction that lifts up the bug.3.Hence, this forms a device which can convert heat to work without any other changes in the environment?

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Efficiency of heat engines

Thermodynamic arguments by Carnot show that the ideal efficiency of a heat engine is

efficiencyenergy we get

energy we pay forwork doneheat input

hot

coldhot

TT-Tefficiency

Ideal efficiency of heat engines

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Steam turbine

e.g. hot reservoir at 127oC (400 K), cold reservoir at 27oC (300 K).

efficiencyT - T

T400 - 300

400= 25%hot cold

hot

100% efficiency is not possible.Efficiency can be improved by raising the temperature of the hot reservoir, or lowering that of the cold reservoir.Heat energy dumped into the cold reservoir is not available.

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Which is possible?1) On a warm day a pool of water rejects heat to the air and freezes.2) A heat engine absorbs 10 J of heat and does 10 J of work, with no other changes.3) A heat engine absorbs 10 J of heat, does 3 J of work and dumps 7 J of heat to the surroundings.4) A heat engine absorbs 10 J of heat, does 3 J of work and returns 7 J of heat to the heat source. 5) A bacteria grows without any influence on its environment.

Question 7

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More efficient gasoline engines can be built if1) engines are made to rotate faster.2) a better antifreeze is developed.3) materials are developed that can withstand lower temperatures.4) materials are developed that can withstand higher temperatures.5) none of these -- they are already perfectly efficient.

Question 8

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Time arrow and cosmology

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The strongest support to the big bang theory came in 1964, when Penzias and Wilson accidentally discovered a noise in their radio telescope.

The radiation does not come from any focused source. It is persistent and uniform (same in all directions).

It has a wavelength of 7cm (microwave region).

It fits the blackbody radiation spectrum of 2.7K.

This is called the cosmic background radiation (CBR), which is the remaining photons predicted by the big bang theory.

Cosmic background radiation

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The following probes were launched: 1989: COBE satellite 1998: MAXIMA (Millimeter

Anisotropy eXperiment IMaging Array, balloon over Texas)

1998: Boomerang (balloon over Antarctica)

2001: MAP (Microwave Anisotropy Probe)

MAP satellite

Recent Developments

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This COBE observation shows that CBR is uniform, with extremely small fluctuations.

The temperature difference is extremely small, 1/50,000 K.

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The Fate of the Universe? The current density of the universe is and there are three

possible situations when it is compared with a critical density c:

> c (closed universe)The expansion will stop eventually.The universe will then start collapsing and end in a Big Crunch.

< c (open universe)It will expand forever. Galaxies become further and further apart. The universe will end in a Big Freeze.

= c (critical universe)The expansion will never stop but the speed will be ever-decreasing.

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The Future of the Universe?

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The Strange Big Crunch?Gold suggested that in a collapsing universe: Stars will absorb radiation from space! The incoming energy will convert helium

into hydrogen! Ice on Earth will radiate heat and grow

larger! Living things will grow from old age to

young!To summarize, time is running backwards!

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Will You Notice?However, Gold suggests that if time runs backwards, human intelligence will also run backwards!

We would still see heat flows from hot objects to cold ones etc!

You could be living in a contracting universe and are totally unaware of it!

Stephen Hawking changed his mind at least twice on the answer to this puzzle!

See “A Brief History of Time” about his present views.

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Assurance?There is now growing evidence that the Big Crunch will not happen.

It is likely that the Universe will expand forever.

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Other challengesHow to reconcile the time arrow with quantum mechanics?

How does the second law apply to black holes?

Does past, present and future exist ‘all the time’, and all that travels is our consciousness about ‘now’?

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ConclusionThe second law of thermodynamics is useful in describing many physical phenomena.

However, the problem of time arrow continues to pose many puzzles to physicists.

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Final Word

University education is not about providing standard answers.

Keep on thinking, listening, asking, trying.

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ears

eyes

mouth

hands

heart