Take away the wall and the gas fills the box. Every collision obeys laws that work just as well backwards, yet a gas never gathers itself back into one half. Reverse every velocity and watch it do exactly that, then see how little it takes to spoil it.
The wall has just been pulled out. Let the gas spread for a few seconds, then press Reverse time.
Each dot is a gas particle bouncing off the walls and off each other with no friction, so no energy is lost. They all start in the left half. The top graph tracks what fraction is on the left; it drops to one half and then just jitters there.
The bottom graph is the entropy. Count the ways to choose which n of the N particles sit on the left: that is C(N, n). Boltzmann's entropy is S = k ln W, the log of that count. An even split has by far the most ways, so that is where the gas spends nearly all of its time.
The laws behind every collision here run equally well forwards and backwards. A film of any single bounce looks fine in reverse. In 1876 Josef Loschmidt used this to challenge Boltzmann: if each step is reversible, how can entropy only go up?
Boltzmann's answer was counting. Mixed arrangements outnumber sorted ones so hugely that almost every path leads toward mixing. Going back needs every velocity reversed with perfect precision, and chaos turns the tiniest error into a total one. The deeper puzzle is why the universe started in such a low-entropy state at all.
The same counting that keeps your gas mixed sets hard limits on engines, fridges, computers and life, and even tells us how much information a black hole holds.
Sadi Carnot showed in 1824 that an engine running between a hot and a cold temperature can turn at most 1 − Tcold/Thot of its heat into work, with temperatures in kelvin. A steam plant at 600 °C dumping heat at 30 °C can never beat about 65%. Real ones manage around 40%. The rest has to leave as heat to carry away entropy.
In the demo: spreading out is free; herding particles back takes work.Heat never flows from cold to hot on its own, any more than the gas un-mixes. A fridge or heat pump forces it uphill with a compressor and pays in electricity. Because it moves heat rather than making it, a good heat pump delivers 3 to 4 units of heat for each unit of electricity.
In the demo: putting the wall back sorts nothing; sorting needs work.Rolf Landauer showed in 1961 that erasing one bit of information must release at least kT ln 2 of heat, about 3 × 10−21 joules at room temperature. Erasing squeezes two possible states into one, like pushing a particle into one half of the box. Experiments confirmed the limit in 2012; today's chips still use far more than this.
In the demo: one particle, two halves: ln 2 of entropy per bit.In 1948 Claude Shannon defined the entropy of a message using the same formula. It is the fewest bits per symbol any lossless code can average. ZIP and PNG squeeze repetitive data toward that floor, and truly random data, with maximum entropy, cannot be compressed at all.
In the demo: "all on the left" takes one line to describe; a mixed gas needs a bit per particle.Earth sends back into space about as much energy as it gets from the Sun. What it keeps is the low entropy. Sunlight arrives as relatively few high-energy photons from a 5,800 K surface and leaves as roughly 20 times as many infrared ones. Plants run on that difference, and everything else eats plants. A charged battery is a store of the same kind of order.
In the demo: life is a local, paid-for return toward the left side.Jacob Bekenstein argued in 1972, and Stephen Hawking confirmed in 1974, that a black hole has entropy proportional to the area of its horizon: one quarter of the area in Planck units. A black hole with the Sun's mass would hold about 1077 k of entropy, around 1019 times more than the Sun itself.
In the demo: entropy is a count of hidden arrangements, even for a black hole.