state exist, too, and you can get inside the phase space and roam over its landscape â provided you know the right spells, secret entrances and other magical paraphernalia. L-space is a case in point. On Roundworld, we can pretend that phase space exists, and we can imagine exploring its geography. This pretence has turned out to be extraordinarily insightful.
Associated with any physical system, then, is a phase space, a space of the possible. If youâre studying the solar system, then the phase space comprises all possible ways to arrange one star, nine planets, a considerable number of moons and a gigantic number of asteroids in space. If youâre studying a sand-pile, then the phase space comprisesthe number of possible ways to arrange several million grains of sand. If youâre studying thermodynamics, then the phase space comprises all possible positions and velocities for a large number of gas molecules. Indeed, for each molecule there are three position coordinates and three velocity coordinates, because the molecule lives in three-dimensional space. So with N molecules there are 6 N coordinates altogether. If youâre looking at games of chess, then the phase space consists of all possible positions of the pieces on the board. If youâre thinking about all possible books, then the phase space is L-space. And if youâre thinking about all possible universes, youâre contemplating U-space. Each âpointâ of U-space is an entire universe (and you have to invent the multiverse to hold them all â¦)
When cosmologists think about varying the natural constants, as we described in Chapter 2 in connection with the carbon resonance in stars, they are thinking about one tiny and rather obvious piece of U-space, the part that can be derived from our universe by changing the fundamental constants but otherwise keeping the laws the same. There are infinitely many other ways to set up an alternative universe: they range from having 101 dimensions and totally different laws to being identical with our universe except for six atoms of dysprosium in the core of the star Procyon that change into iodine on Thursdays.
As this example suggests, the first thing to appreciate about phase spaces is that they are generally rather big. What the universe actually does is a tiny proportion of all the things it could have done instead. For instance, suppose that a car park has one hundred parking slots, and that cars are either red, blue, green, white, or black. When the car park is full, how many different patterns of colour are there? Ignore the make of car, ignore how well or badly it is parked; focus solely on the pattern of colours.
Mathematicians call this kind of question âcombinatoricsâ, and they have devised all sorts of clever ways to find answers. Roughly speaking, combinatorics is the art of counting things without actually counting them. Many years ago a mathematical acquaintance of ours came across a university administrator counting light bulbs in the roofof a lecture hall. The lights were arranged in a perfect rectangular grid, 10 by 20. The administrator was staring at the ceiling, going â49, 50, 51 â¦â
âTwo hundred,â said the mathematician.
âHow do you know that?â
âWell, itâs a 10 by 20 grid, and 10 times 20 is 200.â
âNo, no,â replied the administrator. âI want the exact number.â 2
Back to those cars. There are five colours, and each slot can be filled by just one of them. So there are five ways to fill the first slot, five ways to fill the second, and so on. Any way to fill the first slot can be combined with any way to fill the second, so those two slots can be filled in 5 Ã 5 = 25 ways. Each of those can be combined with any of the five ways to fill the third slot, so now we have 25 Ã 5 = 125 possibilities. By the same reasoning, the total number of ways to fill the whole car park is 5 Ã 5
Associated with any physical system, then, is a phase space, a space of the possible. If youâre studying the solar system, then the phase space comprises all possible ways to arrange one star, nine planets, a considerable number of moons and a gigantic number of asteroids in space. If youâre studying a sand-pile, then the phase space comprisesthe number of possible ways to arrange several million grains of sand. If youâre studying thermodynamics, then the phase space comprises all possible positions and velocities for a large number of gas molecules. Indeed, for each molecule there are three position coordinates and three velocity coordinates, because the molecule lives in three-dimensional space. So with N molecules there are 6 N coordinates altogether. If youâre looking at games of chess, then the phase space consists of all possible positions of the pieces on the board. If youâre thinking about all possible books, then the phase space is L-space. And if youâre thinking about all possible universes, youâre contemplating U-space. Each âpointâ of U-space is an entire universe (and you have to invent the multiverse to hold them all â¦)
When cosmologists think about varying the natural constants, as we described in Chapter 2 in connection with the carbon resonance in stars, they are thinking about one tiny and rather obvious piece of U-space, the part that can be derived from our universe by changing the fundamental constants but otherwise keeping the laws the same. There are infinitely many other ways to set up an alternative universe: they range from having 101 dimensions and totally different laws to being identical with our universe except for six atoms of dysprosium in the core of the star Procyon that change into iodine on Thursdays.
As this example suggests, the first thing to appreciate about phase spaces is that they are generally rather big. What the universe actually does is a tiny proportion of all the things it could have done instead. For instance, suppose that a car park has one hundred parking slots, and that cars are either red, blue, green, white, or black. When the car park is full, how many different patterns of colour are there? Ignore the make of car, ignore how well or badly it is parked; focus solely on the pattern of colours.
Mathematicians call this kind of question âcombinatoricsâ, and they have devised all sorts of clever ways to find answers. Roughly speaking, combinatorics is the art of counting things without actually counting them. Many years ago a mathematical acquaintance of ours came across a university administrator counting light bulbs in the roofof a lecture hall. The lights were arranged in a perfect rectangular grid, 10 by 20. The administrator was staring at the ceiling, going â49, 50, 51 â¦â
âTwo hundred,â said the mathematician.
âHow do you know that?â
âWell, itâs a 10 by 20 grid, and 10 times 20 is 200.â
âNo, no,â replied the administrator. âI want the exact number.â 2
Back to those cars. There are five colours, and each slot can be filled by just one of them. So there are five ways to fill the first slot, five ways to fill the second, and so on. Any way to fill the first slot can be combined with any way to fill the second, so those two slots can be filled in 5 Ã 5 = 25 ways. Each of those can be combined with any of the five ways to fill the third slot, so now we have 25 Ã 5 = 125 possibilities. By the same reasoning, the total number of ways to fill the whole car park is 5 Ã 5
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