August 12, 2022

The science magazine of Imperial College

# The Arrow of Time (Part 2)

The Thermodynamic “Arrow of Time” in Physics (And to a Lesser Extent, in Snooker)
PART 1 | PART 2 | PART 3

Jiggling Particles

Let’s look again at the third film. The rolling ball possesses an initial kinetic energy, which gradually dissipates away through friction: the particles in the fabric that the ball travels over, and of the air it moves through, have their randomly-directed kinetic energies increased slightly – in effect, the particles “jiggle about” a bit more. This randomly-directed jiggling is what we mean by “heat”. The principle of energy conservation says that the total increase in heat content of the air and the table (and the ball) is exactly equal to the amount of kinetic energy the ball loses.

Since this transfer of energy is a classical process, it should be possible to run it “backwards”, and so return the ball to its original state of motion – in the green baize, each particle could, without violating the laws of physics, happen to move in the opposite direction, and likewise the slightly heated air molecules. Similarly any sound-waves created could converge back to the point of contact between ball and table – the net effect of all this would be to provide exactly enough of a push to set the ball rolling in the reverse direction.

So why doesn’t this happen in real-life?

Fun and Games

To understand why, let’s invent a game: imagine that I give you a large tray containing fifty small tiles. Each tile is uniquely numbered. The tray is just big enough to hold the flat array of tiles in place with no large gaps, and it has a lid that fits over the top. Furthermore the tiles are arranged in sequential order, so that from the top left corner they begin 1, 2, 3…. I call this particular arrangement the “S” state, where S stands for Special (or Sequential).

You now put the lid on and give it a shake. On removing the lid, you observe that the pattern is messed up and the tiles are in a different order. Put the lid on and shake it once more – again, we have a new arrangement. Now, here’s the game: try to get back to the original “S” state, simply by putting the lid on and giving the tray a succession of random shakes. If you get bored and give up, you lose (and I win, since someone has to).

Note the two essential features of the game: 1) it involves a large number of individual elements (the numbered tiles) and 2), one particular arrangement of the tiles is designated as “Special”. Also note that the choice of which arrangement is Special is essentially arbitrary – since the tiles are uniquely numbered, I could have picked any other ordering and called that the “S” state instead. Having done so however, every other possible configuration is immediately relegated to being a “non – S” state. As far as the game is concerned, these are all on an equal footing – none of them are the “S” state, and so all are equally useless in terms of winning.

Spaces and Systems

How can we visualize the way that the arrangement of the tiles evolves as you play the
game? There are about 3 x 1064 different ways to order the numbers 1 to 50. One way to catalogue these different combinations is as follows: imagine a “space” – an ordinary three dimensional space, for example a big box, will do. Now divide this space into 3 x 1064 tiny volumes, where each volume corresponds to one specific arrangement of the tiles. We’ll call the space in its entirety “Arrangement Space”, i.e. a space that contains all possible arrangements of the tiles.

Imagine also that whenever the tiles are in a certain arrangement, the volume corresponding to that particular arrangement is illuminated. As you play the game, a light moves around the space, depending on which arrangement the tiles are in. If one of these little volumes corresponds to the “S” state, the aim of the game becomes to make that volume the one that’s illuminated.

What does this game have to do with our snooker ball? First, I’ll introduce the concept of a “system”. A system is simply a collection of objects under consideration – essentially we draw an imaginary “bag” around the things we’re interested in, and call everything inside it the “system”. So, let’s just draw an invisible cube, with sides of say 20cm, around the decelerating ball. Everything inside it is our system. Since the ball is smaller than this cube, a piece of the table and the surrounding air are included too.

Because atoms are so tiny, our system comprises an enormous number of particles, all of which will be in constant jostling motion. Now suppose we own an incredible device that allows us to take a “snapshot” of what every particles is doing – where they are and how fast they’re moving – at a particular instant. If we took another snapshot a moment later we’d find that, because they were constantly fidgeting about, the microscopic arrangement of the particles had changed. A third would yield another, yet again different, arrangement.

We can now catalogue the possible arrangements – snapshots – of the microscopic ingredients of our system in a similar way to before. Imagine another space similar to the Arrangement Space above (actually of many more dimensions than three, but let’s not worry about that). Now each single point in the space corresponds to a particular arrangement of the particles of the system at any one time. So the first snapshot we took corresponds to one point in the space, the second to another – nearby – point and so on. The technical name for this is a “phase space”. [5,7,8]

If, as above, we imagine that for each possible arrangement of the particles the corresponding point lights up, then we’ll see a little spot of light moving through phase space as the microscopic state of the system evolves. The precise way in which this spot meanders about will be determined by the laws that govern the motion of the particles in our system.

Continues in PART 3.

PART 1 | PART 2 | PART 3

Text and illustrations © Jack Jelfs 2010

NOTES

[1] Philip K. Dick – We Can Remember It For You Wholesale – The Collected Stories of Philip K. Dick Volume Four, Orion Books, 1987.

[2] The image of two billiard balls colliding as an example of a time reversible process occurs in John Gribbin – White Holes: Cosmic Gushers In The Universe, Delacorte Press, 1977.

[3] The image of a planet orbiting a star as an example of a time reversible process occurs in R.P. Feynman – The Character of Physical Law, British Broadcasting Corporation, 1965.

[4] Arthur Eddington – Nature of the Physical World, Cambridge University Press, 1929.

[5] H.D. Zeh – The Physical Basis of the Direction of Time, Springer–Verlag, 1989.

[6] Paul Davies – About Time, Viking, 1995.

[7] Roger Penrose – The Emperor’s New Mind, Oxford University Press, 1989.

[8] Roger Penrose – The Road to Reality, Vintage Books, 2004.

[9] Gilbert N. Lewis – The Symmetry of Time in Physics, Science 71, 1930. Reprinted in P.T Lansberg (ed.) – The Enigma of Time, Adam Hilger, 1982

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