The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [136]
In crafting mathematical methods for systematically sacrificing detail in favor of such higher-level aggregate understanding, physicists honed a wide range of techniques and developed a number of powerful concepts. One such concept, encountered briefly in earlier chapters, is entropy. Initially introduced in the mid-nineteenth century to quantify energy dissipation in combustion engines, the modern view, emerging from Ludwig Boltzmann’s work in the 1870s, is that entropy provides a characterization of how finely arranged—or not—the constituents of a given system need to be for it to have the overall appearance that it does.
To get a feel for this, imagine that Felix is frantic because he believes the apartment he shares with Oscar has been broken into. “They’ve ransacked us!” he tells Oscar. Oscar brushes him off—surely Felix is having one of his moments. To make his point, Oscar throws open the door to his bedroom, revealing clothing, empty pizza boxes, and crushed beer cans strewn everywhere. “It looks just like it always does,” Oscar barks. Felix isn’t swayed. “Of course it looks the same—ransack a pigsty and you get a pigsty. But look at my room.” And he throws open his own door. “Ransacked,” mocks Oscar; “it’s neater than a straight whiskey.” “Neat, yes. But the intruders have left their mark. My vitamin bottles? Not lined up in order of size. My collected works of Shakespeare? Out of alphabetical order. And my sock drawer? Look at this—some black pairs are in the blue bin! Ransacked, I tell you. Obviously ransacked.”
Putting Felix’s hysteria aside, the scenario makes plain a simple but essential point. When something is highly disordered, like Oscar’s room, a great many possible rearrangements of its constituents leave its overall appearance intact. Grab the twenty-six crumpled shirts that were scattered across the bed, floor, and dresser, and toss them this way and that, fling the forty-two crushed beer cans randomly here and there, and the room will look the same. But when something is highly ordered, like Felix’s room, even small rearrangements are easily detected.
This distinction underlies Boltzmann’s mathematical definition of entropy. Take any system and count the number of ways its constituents can be rearranged without affecting its gross, overall, macroscopic appearance. That number is the system’s entropy.* If there’s a large number of such rearrangements, then entropy is high: the system is highly disordered. If the number of such rearrangements is small, entropy is low: the system is highly ordered (or, equivalently, has low disorder).
For more conventional examples, consider a vat of steam and a cube of ice. Focus only on their overall macroscopic properties, those you can measure or observe without accessing the detailed state of either’s molecular constituents. When you wave your hand through the steam, you rearrange the positions of billions upon billions of H2O molecules, and yet the vat’s uniform haze looks undisturbed. But randomly change the positions and speeds of that many molecules in a piece of ice, and you’ll immediately see the impact—the ice’s crystalline structure will be disrupted. Fissures and fractures will appear. The steam, with H2O molecules randomly flitting through the container, is highly disordered; the ice, with H2O molecules arranged in a regular, crystalline pattern, is highly ordered. The entropy of the steam is high (many rearrangements will leave it looking the same); the entropy of the ice is low (few rearrangements will leave it looking the same).
By assessing the sensitivity of a system’s macroscopic appearance to its microscopic details, entropy is a natural concept in a mathematical formalism that focuses on aggregate physical properties. The Second Law of Thermodynamics developed this line of insight