Editing H-theorem (section)

=== Spin echo ===

As a demonstration of Loschmidt's paradox, a modern counterexample (not to Boltzmann's original gas-related ''H''-theorem, but to a closely related analogue) is the phenomenon of [[spin echo]].<ref>{{Cite journal | last1 = Rothstein | first1 = J. | title = Nuclear Spin Echo Experiments and the Foundations of Statistical Mechanics | doi = 10.1119/1.1934539 | journal = American Journal of Physics | volume = 25 | issue = 8 | pages = 510–511 | year = 1957 |bibcode = 1957AmJPh..25..510R }}</ref> In the spin echo effect, it is physically possible to induce time reversal in an interacting system of spins.

An analogue to Boltzmann's ''H'' for the spin system can be defined in terms of the distribution of spin states in the system. In the experiment, the spin system is initially perturbed into a non-equilibrium state (high ''H''), and, as predicted by the ''H'' theorem the quantity ''H'' soon decreases to the equilibrium value. At some point, a carefully constructed electromagnetic pulse is applied that reverses the motions of all the spins. The spins then undo the time evolution from before the pulse, and after some time the ''H'' actually ''increases'' away from equilibrium (once the evolution has completely unwound, the ''H'' decreases once again to the minimum value). In some sense, the time reversed states noted by Loschmidt turned out to be not completely impractical.