Dick Bedeaux patiently traces the evolution of the second law of thermodynamics from its formulation in the mid-19th century through today, from the perspective of statistical mechanics.

In its earliest form, as laid out by Rudolf Clausius, the law states that the entropy of the world always increases. This proposition in some sense launched the field of thermodynamics, according to Bedeaux: “It got going in order to understand exactly where the laws came from.” There was particular interest in exploring entropy in terms of the motions of particles. Scientists began refining theories around the behavior of gas in equilibrium, looking at density, velocity distribution, potential energy and heat conductivity. After Maxwell and Boltzmann appeared to have succeeded at a proof of the second law, other scientists questioned its validity: If Newton’s equations are reversible, they reasoned, why can’t a system in some sense reverse velocities and return along the same path? Others worked out the recurrence paradox, that “if you take any kind of motion in phase space and follow trajectories, that trajectory after a sufficiently long time will come arbitrarily close to the original point…” These two paradoxes posed a fundamental challenge to proofs of the second law.

The debate continued through the end of the 19th century, into the 20th, with additional efforts to refine the notion of entropy using concepts of probability -- courtesy of the burgeoning discipline of statistical mechanics, according to Bedeaux. This field enabled better descriptions of equilibrium and non-equilibrium states. There was “a lot of progress,” says Bedeaux: Einstein explained Brownian motion in 1905, and closer to our own era, scientists “punched a hole in the argument about the recurrence paradox,” using probabilistic descriptions. Nevertheless, in our own times, “in non-equilibrium statistical mechanics, there is as yet no fully satisfactory derivation of the second law.” To meet this challenge, a proof should provide a simple mechanical example, “like hard discs between reflecting walls, in order to be convincing.” While equations of motion on the microscopic level incorporating the idea of irreversibility demonstrate entropy production, it’s uncertain “whether Mother Nature believes this herself -- we must do experiments to verify,” concludes Bedeaux.