Le Monde

An ice cube in hot coffee. The coffee loses heat: its entropy falls, and that is perfectly allowed. The ice melts and then warms: its entropy rises, and by more. The sum is always positive, never by miracle but by arithmetic — the same heat $Q$ is worth more entropy when poured in at a low temperature than it costs when leaving a high one, because $\Delta S=Q/T$ and $T$ is smaller down there. Drag the sliders and watch the three terms. The only negative one is the coffee's; it never wins.

A deck of fifty-two cards admits $52!$ orders, about $8.07\cdot10^{67}$. The base-two logarithm of that number is $225.58$: that, exactly, is the information a well-shuffled deck carries, and the entropy your wrist has just created. The figure is large enough to be worth saying another way: there are more possible orders of a deck of cards than there are atoms in the Earth, by a wide margin. You will sometimes see $237$ bits instead of $225.58$; that is not an error, it is a $54$-card deck with the jokers — the page computes both, and the gap is exactly $\log_2 53+\log_2 54$.

Here is the fact that settles everything, and that is almost always stated backwards. The Earth's energy budget balances to within a fraction of a per cent: as many watts arrive as leave. If life fed on energy it would starve. What arrives and does not leave is quality: a solar photon comes from a surface at $5778$ K, and to shed the same energy the Earth must emit from $255$ K — therefore emit about twenty-three times as many. Twenty-three lukewarm photons for one scorching one: that is the entropy flux the biosphere exports, and inside which it lives. Schrödinger first wrote that life feeds on “negative entropy”, then corrected himself in a later edition: the right term is free energy.

Sur la Terre comme machine thermique et son bilan d'entropie : H. Ozawa, A. Ohmura, R. D. Lorenz, T. Pujol, The second law of thermodynamics and the global climate system, Reviews of Geophysics 41 (2003) — production d'entropie d'environ $0{,}9$ W·K⁻¹·m⁻², dont plus de neuf dixièmes par simple absorption du rayonnement solaire. Rapportée à la surface du globe, cela fait $4{,}6\cdot10^{14}$ W/K ; d'autres bilans, qui ne comptent pas tout à fait les mêmes termes, donnent $6{,}5\cdot10^{14}$ W/K. La page affiche les deux plutôt que de choisir.

Bekenstein and Hawking found that a black hole's entropy depends neither on its mass nor on its volume but on the area of its horizon: $S=kA/4\ell_P^2$, where $\ell_P$ is the Planck length. The result is brutal. A star like the Sun carries about $10^{58}$ in units of $k$; collapse exactly the same matter into a black hole and you get $10^{77}$ — nineteen orders of magnitude gained without adding a single atom. That is why black hole formation is the most irreversible process in the universe, and why they dominate any entropy budget they appear in. Drag the mass: the radius follows $M$, the entropy follows $M^2$, the Hawking temperature follows $1/M$ and the evaporation time follows $M^3$. All of it is computed here from the formulas, not recited.

In 2010 Egan and Lineweaver redid the full entropy budget of the observable universe with up-to-date measurements of the supermassive black hole mass function. The result: $S_{\text{obs}}=3.1\cdot10^{104}\,k$, dominated by supermassive black holes — an order of magnitude above what was estimated before them. The cosmic microwave background, that radiation filling all space and always quoted as the great entropy reservoir, weighs $10^{88}$: sixteen orders below, which is to say nothing. And if you count the cosmic event horizon itself you find $2.6\cdot10^{122}\,k$, which dwarfs everything inside it. Look at the scale: it is logarithmic, and every division is a factor of ten.

C. A. Egan & C. H. Lineweaver, A larger estimate of the entropy of the universe, Astrophysical Journal 710 (2010) 1825. Total observable $3{,}1^{+3{,}0}_{-1{,}7}\times10^{104}\,k$ ; horizon des événements cosmologique $2{,}6\pm0{,}3\times10^{122}\,k$ ; matière noire $10^{88\pm1}\,k$. Les barres portent ces incertitudes telles quelles.

Push the slider and watch the universe age. We are at $1.4\cdot10^{10}$ years, which on this scale is the very beginning. Around $10^{14}$ years the last star goes out for want of hydrogen and the sky goes dark for good. Around $10^{67}$ years a stellar-mass black hole finishes evaporating — that number the page computed for you in the previous chapter, from $t\propto M^3$. Around $10^{100}$ years the largest ones have gone too, and nothing is left but ever colder radiation in ever emptier space. This is not the end of time: entropy approaches its maximum without ever quite arriving, and nothing forbids a fluctuation from undoing it all — Poincaré's theorem even guarantees it, provided you wait a time that will not fit on this scale, or on any other.

Sur l'idée qu'une pensée pourrait durer indéfiniment dans un univers qui se refroidit, et sur ce que l'expansion accélérée en fait : F. Dyson, Time without end (1979) — l'intelligence éternelle de Dyson.

Back to the beginning: LE COMPTE, where all of this was still just beads in a box.