The fundamental laws of mechanics — Newton’s, but also Maxwell’s and Schrödinger’s — are time-reversible: if you film an elastic collision between two marbles and play the reel backwards, you see a sequence that is still perfectly compatible with . Why, then, do we immediately recognise as “wrong” a film in which a broken glass reassembles itself, or in which the smoke from a cigarette goes back into the stub?
The answer given by Boltzmann and Gibbs at the end of the nineteenth century is surprising: the arrow of time does not live in the equations of motion, but in an extremely special initial condition. Our universe started out — for reasons cosmology still debates — in a state of very low entropy; since then, every macroscopic process has been moving towards more probable macrostates, simply because they are vastly more numerous. The glass breaks because the “scattered shards” microstates are far more numerous than the “whole glass” microstates: not because a law forbids reassembly, but because that reassembly would require a microscopic conspiracy so improbable that it would not happen even over cosmic timescales.
The second law thus becomes a statistical truth, not a deterministic one: it does not say “it is impossible for entropy to decrease”, it says “it is enormously improbable”. The distinction is subtle but profound. For a mole of gas, the probability of spontaneously seeing all the molecules concentrate in half the container is of the order of , a number so small that human language struggles to express it.
This is where thermodynamics and statistical mechanics make peace: irreversibility is not a new law of nature, but the way a system with an enormous number of degrees of freedom explores its own state space, ending up almost certainly in the regions most crowded with microstates.
Collegamenti
Argomenti: Entropy and the second law Concetti: Second law of thermodynamics · Entropy Competenze: Micro–macro interpretation
Esercizi collegati: Worked exercise — the coffee cools down · Hot shower: ΔS of the universe · Heat from hot to cold: proof