Qualitative: the falling glass

Problem

A glass falls off the table and shatters on the floor. Can it spontaneously reassemble, putting the intact pieces back on the table? Discuss in five lines in terms of entropy and number of microstates.

Solution

No. The final state “shards scattered on the floor plus heat dissipated in the impact” corresponds to an enormous number of microstates (many ways to arrange fragments and thermal agitation); the state “whole glass on the table” corresponds to very few ordered microstates. So breaking means moving to a much higher $S$, as required by the second law.

Spontaneous reassembly would require a macroscopic decrease in entropy $\Delta S<0$. Its probability is of the order of $P\sim e^{\Delta S/k_B}=e^{-\abs{\Delta S}/k_B}$ with $\abs{\Delta S}$ enormous compared to $k_B$: a number indistinguishable from zero. It would require a microscopic conspiracy of all the molecules, never observed even on cosmic timescales. $\ev{\text{no: prob.}\sim e^{-\abs{\Delta S}/k_B}\approx 0}$

Linked atoms

Topics: Entropia e secondo principio Concepts: Entropia · Secondo principio della termodinamica Skills: Interpretazione micro-macro