Everyday entropy: space

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Flying ping-pong balls

The genius of statistical mechanics was to recognise that the entropy of a material was defined by the space available to the particles in a system, not their temperature.  This makes entropy calculable without reference to thermal equilibrium, and allows for the application of entropy to all physical systems where momentum, number and available space are observable.  The important thing to understand is that space is as important as energy and mass in understanding the outcome of a given process.  The number of possible configurations is an exponential function of the space available to the parts of the system, which is why entropy is defined in terms of a logarithm.  A linear change in volume results in an exponential change in the available configurations.  The base of the logarithm is e because the probability of not finding a particle in a given state resembles a Bernoulli trial due to the large number of particles and locations.

The problem, obviously, is that statistical mechanics makes entropy very nearly incomprehensible and entirely unsexy.  Energy is sexy; entropy looks like flaccid, white toast.  But if you think of entropy as the relationship between matter, energy and the available space, it starts to gain some curvaceous shape.  Matter takes the liberty of the space it can access with the energy available to it.  In these terms, it would be tempting to say that energy is masculine and entropy feminine is a yin-yang kind of interplay, but that’s just wrong.  In statistical mechanics, entropy encompasses everything.  Energy is a conveniently limited set of momenta.  The classical gender tropes only fit into the classical thermodynamic paradigm of entropy/energy.

Let’s say you have a million btu of energy, if the available space is a million square miles, that energy can’t help you no matter how concentrated it is in terms of material or particles.  The stuff will try to occupy every available location and whatever you thought you had will be nothing but information.  More space increases entropy, and this relationship explains the energy inefficiency of low-density development.  People feel free when they have acres of space at their disposal, just like gas molecules, but they end up using energy without noticing any benefits because there is no pressure to stop them.  It explains why the population of middle America, having spent heavily to spread themselves as thinly as possible, are now frustrated by the lack of return on investment.

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