Physics
Entropy
A macrostate is what you can see: how many particles are on the left. A microstate is the full detail: which ones. Entropy just counts how many microstates wear the same macrostate, S = k ln Ω, and that counting alone is enough to make time run one way.
1 step / 2 s5000 / s
Other arrangements that look the same from outside
What you see—
Ways to arrange it Ω—
Entropy S/k = ln Ω—
Largest possible S/k—
Share of all arrangements—
Steps taken0
Back to all-left after—
…at the current speed—
One step moves a single randomly chosen particle to the other side. The rule has no preferred direction, and no particle knows where it should be. Everything you see comes from counting.
What to observe
- Slow the speed right down to a step every second or two and watch single moves. Each one is a coin flip with no bias at all, yet the pile on the left keeps shrinking. Nothing pushes the particles right; there are just far more ways to be mixed than to be sorted.
- Compare the two rows above the chart. The microstate strip changes at every single step. The macrostate beside it, just a count, barely moves. Entropy is the size of the gap: how many different strips give the same count.
- Look at the two curves in the chart. The bars are the entropyS(n) = ln Ω(n), a broad gentle hill. The line is the probabilityΩ(n)/2N, a razor-thin spike. Probability is the exponential of entropy, so a mild bump in S becomes an overwhelming bias in P. That single fact is the second law.
- Drop N to 4 or 6. Entropy now visibly falls as often as it rises, and the system returns to all-left every so often. The second law is not a prohibition, it is a statement about odds, and the odds are only unbeatable when N is large.
- Raise N back to 200 and read the recurrence time. Nothing forbids every particle from crowding left again, it simply takes longer than the universe has existed. Irreversibility is about counting, not about the laws of motion.
- With the overlay on, the measured visit frequency settles onto the Ω(n)/2N curve. Time spent in a macrostate is proportional to how many microstates it owns: entropy is a census, and the dynamics just takes it.
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