April 13, 2024

I, Science

The science magazine of Imperial College

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

He reached toward the ash tray on the desk, selected the shortest – and hence best – cigarette butt, dabbed it against the ceramic surface until it began to burn, then lifted it to his lips. Puffing shreds of tobacco into it, he sat staring fixedly out the office window
– Philip K. Dick, “Your Appointment Will Be Yesterday”[1]

At The Movies

Imagine that we sit in a cinema and watch a film. It’s quite a boring film, and it’s this: The green baize of a snooker table, seen from above. You can’t see the edges of the table, just a uniform green surface that fills the screen. Nothing happens for a few seconds. Then suddenly from the left and right, two snooker balls roll towards each other. They collide elastically somewhere near the middle of the screen and roll away out of shot. And that’s it, that’s the end of the film. Nothing else happens.[2]

And now, a question: was the film playing forwards or backwards?

While you think about this, let’s watch another one. It’s set in space, but unfortunately it’s not a sci-fi blockbuster – in fact, it’s even more boring than the first. It shows simply a star floating in the blackness, around which a planet is slowly orbiting. After watching for a few minutes, you see that the planet has moved some way along its orbital path. It’s moving in an anti-clockwise direction. The End.[3]

Again (assuming you’re not queuing at the box office to demand a refund), the same question: was the film playing forwards or backwards?

To answer this, let’s think about what the two films would look like when run the opposite way. In the first, two balls roll into shot, collide, and roll away again. In the second, the planet moves slowly along its path, only this time in a clockwise direction. Which screening, the first or the second, showed the “correct” sequence of events?

I suggest that – armed only with the knowledge of what you saw on the screen – there is actually no way of telling which was playing forwards and which backwards. We see equally valid physical processes no matter which way the films run. These apparently trivial examples reflect a curious feature of classical mechanics: the equations describing such phenomena do not imply a preferred direction in time.

The word used to describe situations that still look the same when something changes is “symmetrical”. Here we have two situations that are physically acceptable when we “run time backwards” and we say that they are symmetrical under time reversal. Mathematically, what is meant by “time reversal” amounts to changing the sign of time in the equations – if we change every “t” that occurs to “- t”, we still have a mathematically consistent framework.[9] It’s not just classical mechanics that possesses this property – Maxwell’s laws of electromagnetism and Einstein’s Special and General Relativity are also symmetric under time reversal. So too is Schrödinger’s Equation, the fundamental equation of quantum mechanics.

In fact, modern particle physics reveals that Nature respects a more general symmetry called CPT symmetry, in which the “T” stands for time.[5] We’ll ignore this, on the assumption that it has little impact on the classical systems we’re considering – and likewise the various issues associated with time-asymmetry in the quantum mechanical “measurement process”.[6,7,8]

Now let’s watch a third and final film. Again, it shows the green baize of a snooker table (it’s a sequel to the first). This time, a single ball rolls into shot from the left, slows down and stops. Fin.

Again – is it playing forwards or backwards? Well, this time you might say it’s easy – obviously it’s running forwards. Why? Because if we ran it backwards, we’d see an initially motionless ball start to accelerate, which is something that snooker balls never do in real life.

What’s the difference between this film and the first two? All three show phenomena that can be described by the laws of classical physics, and we have said that these laws are equally valid under time reversal. So why does the third film seem so opposed to our common sense when played backwards?

More generally, if the laws that govern the Universe operate equally well backwards as well as forwards, then why don’t we see snooker balls spontaneously starting to move? If you drop an egg in your kitchen, why do you reach for the dustpan and a dishcloth, rather than just waiting for it to meld together and jump back into your hands? Why don’t cigarettes reassemble themselves from diffuse smoke particles? In short, why – following the astrophysicist Arthur Eddington, who coined the phrase in 1929[4] – does the Universe possess an “arrow of time”?

A Quiver of Arrows

It’s actually possible to identify several arrows of time.[5] One relates to electromagnetic radiation and essentially concerns the fact that we see electromagnetic waves spreading out into space when a charge oscillates, but never see waves converging from all directions onto a single charge.[6] These two situations correspond respectively to the “forwards” and “backwards” time solutions of Maxwell’s equations. [5,6]

Another arrow, this one cosmological, arises due to the expansion of the universe[5] – since the universe is expanding, its volume isn’t constant in time. This is a consequence of the Universe’s origin in the Big Bang, of which more later.

Here we’ll only really consider the so-called “thermodynamic” arrow. We’ll see that it is the origin of most of the time-asymmetric behaviour we observe in everyday situations. The precise relation it bears to the other arrows is still somewhat unclear, although various physicists have suggested that it is somehow the most “fundamental”, in the sense that it underpins the others.[5,6] It has an intimate connection to the 2nd Law of Thermodynamics, which we’ll explore in more detail next.

Continues in PART 2.

PART 1 | PART 2 | PART 3

Text and illustrations © Jack Jelfs 2010


[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