July 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

Special Boxes

Although every point in phase space corresponds to a different, unique arrangement of the particles, we can impose some organization on the space as a whole. Let’s imagine dividing it into a number of “boxes”. The rule for how we do so is as follows: each “box” contains points corresponding to states that are macroscopically indistinguishable from each other – i.e. every point in a particular box represents an arrangement of particles that “looks the same” from our everyday point of view.[5,7]

Some of these boxes will be bigger – much bigger – than others. It turns out that by far and away the hugest box of all – the one for which there are the most number of microscopic arrangements that are macroscopically indistinguishable – corresponds to the system being in thermal equilibrium.[7] In this state, all of the energy in the system is distributed evenly throughout – everything is effectively at the same temperature and there is no further exchange of heat.

Remember what we’re trying to understand – why doesn’t the snooker ball start to roll once it’s stopped? For this to occur, a huge number of the particles in our system need to all be moving in a particular way – precisely, their motion must be in the opposite direction to that caused by the friction of the rolling ball. If every particle in the table and the air moves like this, the motionless ball will start to move.

Such a specific arrangement corresponds to an unbelievably tiny subset of all possible arrangements. There will actually be more than one such microscopic arrangement that causes the ball to move in the same way, but the corresponding points will all be inside the same “box”. The volume of this box will, unsurprisingly, represent a mind-bogglingly miniscule portion of the entire phase space.

Let’s think of what our little illuminated spot does. The boxes corresponding to the ball being in motion will occupy a tiny volume of the total space. The spot will move out of these into larger and larger boxes. Eventually it will wander into the enormous box that corresponds to thermal equilibrium. Once in thermal equilibrium, the precise arrangement of particles of the system will continue to change as the particles jiggle around, so the illuminated spot will meander randomly.[8]

However – and this is the key point – because of the overwhelming hugeness of the “thermal equilibrium” box compared to all the others, the chances of it meandering into a smaller box are unimaginably tiny.[8] This is why the ball doesn’t spontaneously start rolling – it’s the same reason why the game from earlier is so un-winnable, at least in any realistic timescale. If we take the boxes corresponding to states in which the ball spontaneously starts to move and label these as “Special” (like the “S” state in the game) then the chance of the spot randomly wandering into any of these is so small as to be utterly negligible. This is actually another way of expressing the famous 2nd Law of Thermodynamics, which says that any system that’s “isolated”, in the sense that no extra energy or matter is supplied to it, will ultimately reach thermal equilibrium.

In The Beginning …

We now have an insight into why we don’t observe strange phenomena like cigarettes reassembling out of smoke, or eggs gluing themselves together and jumping back onto the counter – it’s not that such things are impossible, it’s just that the odds of them actually occurring are hugely one-sided.

Note however that our central question actually remains unanswered – we’ve just formulated it in a new way. Why are the odds so one-sided? This is still an unresolved issue. The ideas outlined below are mostly due to the mathematical physicist Roger Penrose, who has written extensively on this subject.[5,7,8]

We reached our conclusions above by considering which box in phase space our system was in once the ball had stopped rolling. Let’s now imagine a new phase space, in which the system is the entire Universe. The question then becomes, which box in this phase space does our Universe currently occupy – and why?

Now, you can tell from the fact that you’re reading this, from the fact that there are stars in the sky and so on, that the Universe is not yet in a state of thermal equilibrium – our illuminated spot is still far from wandering into the largest box. Let’s try to establish the path it’s taken to get to where it currently is. By the arguments given above, it’s overwhelmingly unlikely to wander from a larger box into a smaller one. To arrive at the box we’re in now, it must therefore have started out in a smaller box.[7]

Extending this reasoning as far back as possible eventually leads to the Big Bang itself. We are driven to the conclusion that the Universe must have somehow begun in an unimaginably tiny box in phase space. Penrose estimates that the volume of this initial box was a ridiculously small 1/1010123 of the total phase space! [8] Such a miniscule box corresponds to an incredibly specific initial configuration. This may be the explanation of why we have a thermodynamic arrow of time. [7,8]

The exact way in which the Big Bang was “special” in this sense is still not fully understood, and seems to be intimately associated with the deep features of gravitation.[5,7,8] Nonetheless, the implications are profound. For when we watch a snooker ball sit motionless, or an egg smash on the floor, or a cigarette burn to ash, we bear witness to a fundamental feature of reality – a direct consequence of the moment of creation itself.

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