Boltzmann Brains — 1. Chaos in the DVD
Celestograph, 1893. August Strinberg.
Irà Sheptûn May 2, 2024
“If you want to make an apple pie from scratch, you must first invent the universe.” Carl Sagan
Let’s pretend it’s 2006. You’ve just come back from the kitchen to continue watching a movie on DVD. The screen has now faded to black and the DVD logo is bouncing slowly from wall to wall in the box of the screen. You observe over time, the logo hits many different points within the rectangle of the box. One wouldn’t be alone in wondering how often the DVD logo will lock perfectly into one of the corners of the screen before bouncing back again. Naturally, this varies depending on the pixel size of the screen as well as the logo itself and its fixed velocity, but for a standard NTSC-format DVD Player with 4:3 aspect ratio, we can approximate the phenomenon to occur roughly every 500 – 600 bounces, or once every 3 hours. Now, suppose we blew up our screen slowly to the size of the observable universe, how often would we score a perfect corner lock-in then? The idea becomes completely absurd: we can all safely agree the probability is inconceivably, astonishingly tiny. But not impossible.
Celestograph, 1893. August Strinberg.
In the latter half of the Nineteenth Century, physicists were busy laying the foundations of what we now understand as statistical thermodynamics – the art of using the rules that govern the very small components of a system that are probabilistic in nature, to build up a clear picture of the system’s behaviour in general. These small components (individual particles amongst many) have position and momenta that are constantly changing. It is useful to think of a closed system as a new deck of cards, with each particle represented by a single card in the deck. A central concept of statistical thermodynamics is entropy, often synonymous with disorder, where higher entropy systems are subject to greater randomness and uncertainty in their behaviour. If I were to take a random card and place it somewhere else in our new deck as you observed, you’d probably have an easier time reverting the deck back to its original order than if I shuffled it thoroughly. You might say the one-card rearrangement is of lower entropy than the thoroughly shuffled deck, due to the degree of disorder inherent to the shuffling of the cards from their standard order, right? Well, you’d be correct! There are many more possible configurations in our higher entropy shuffled deck that are all equally likely compared to any one-card manoeuvre. However, this description of entropy as a measure of disorder can often be misleading, as we’ll soon discover.
Austrian physicist Ludwig Boltzmann, who alongside his contemporaries, was concerned with understanding how the properties that define the state of a closed system (such as average energy, temperature, or pressure) could be interpreted from the more probabilistic behaviour of the individual particles that make up a given thermodynamic process. The Second Law of Thermodynamics says that the entropy of a closed system can only stay constant or increase over time. Boltzmann contested that perhaps entropy could be thought of more as a statistical property – the chances of staying the same or increasing are high, however the chances of decreasing are not zero, subject to behavioural fluctuations in the particles of the system. Recall our shuffled deck of cards: it’s highly unlikely to return to its original order through constant random shuffling, but the possibility must occur with infinite time!
“The system experiences a fluctuation to lower entropy; an ordered state arising from a more disordered one.”
One might now see how the description of entropy as a measure of disorder can cause some problems. Let’s say we have a deck of cards arranged by suit, and another arranged by increasing number. Which is more inherently entropic? Both decks are undeniably examples of a certain order within their own closed systems. If we accept that both are rearranged from the same initial structure, one must have higher entropy than the other to account for the number of rearrangements to get the structure it is now in. In this way, we can properly define entropy as a measure of the number of ways you can arrange these particles without changing the overall state of the system. In other words, how many ways can I shuffle the deck without adding new cards or taking any away?
Celestograph, 1893. August Strinberg.
But wait! Aren’t these so-called entropic fluctuations to order not a gross violation of the Second Law? If entropy is a statistical property, these fluctuations become inherent to the nature of the Second Law and not a violation, because the net direction of entropy will still tend to increase. Returning to our DVD Player (assuming that the little DVD logo travels around the box with random motion) we can say our closed system will increase in entropy as with every wall bounce the DVD logo makes as it loses predictability in its path. Over infinite time, all possible positions of the DVD logo in the two-dimensional box will be true, and shall increase steadily in entropy with every bounce.
But what of our exciting statistically improbable instances where the logo perfectly locks into a corner? According to Boltzmann, in these instances, the system experiences a fluctuation to lower entropy; an ordered state arising from a more disordered one. However, this is all still outweighed by the increasing net entropy of our DVD Player; the energy used up to power our little dancing logo on its journey of increasing uncertainty, and the heat expended by its humble efforts’ ad infinitum. With this same reasoning, Boltzmann introduced new ideas around the nature of our early universe and how it came to be well before the formulation of Lemaître’s theory of the Primeval Atom, or as it’s now known, The Big Bang Theory.
Now, try to imagine a vast cosmos, infinite in age. All the different regions of this system have more or less reached an equal share of energy: it is uniformly distributed, or what we call in Thermal Equilibrium. At this stage in the lifecycle of a cosmos it has reached maximal entropy, or Heat Death. As we know, it’s not impossible for such infinitely large systems to experience large fluctuations. Given the infinite age of this cosmos, any statistically unlikely event (no matter how improbable) must be true. It is not impossible that one of these local regions in space and time might fluctuate just enough from maximal entropy to form a new ‘world’, if only for a short period of aeons. This new ‘world’ might be indistinguishable from the world that you and I live in; a lower-entropy state of unspent energy and potential, arising by sheer improbability, from an otherwise vast and dead cosmos, trapped in maximal entropy.
There is a chance, then, that we are ‘the moment’ in the truest sense of the term. Could we be special and lucky enough to exist within such an unlikely ‘new world’? Could ours be the moment the DVD logo locks perfectly into the corner of the screen - a Boltzmann Universe?
