Physics

The development of the second law of thermodynamics was intimately tied to the kinetic theory of gases, and carried with it the rebirth of atomism and the founding of statistical mechanics. Despite the fact that Sadi Carnot believed that caloric was not lost when it traveled from the hot body to the cool body of an engine, he recognized that the work delivered depended on the temperature difference between the two bodies and that this difference constantly decreased. This observation was clarified by the German physicist Rudolf Clausius. In 1851, a few years after the acceptance of the first law of thermodynamics, Clausius recognized the need for a second law, to account for the fact that energy was often irrecoverably lost by a system. In a paper published in 1865, Clausius analyzed thermodynamic cycles with a quantity that he dubbed the "entropy" and found that it usually went up or at best (for a reversible process) was zero.

The Austrian Ludwig Boltzmann read Clausius's paper and set about developing a mechanical interpretation of the second law. In a first attempt, published in 1866, he used Hamilton's principle to analyze the development of a thermodynamic system made up of discrete particles. After Joseph Stefan (1835–1893) alerted Boltzmann to James Clerk Maxwell's probabilistic approach, Boltzmann made refinements to Maxwell's ideas and incorporated them into his mechanical interpretation. In 1872, he published a paper that made use of a transport equation (now called the "Boltzmann equation") to describe the evolution of a probability distribution of particles. As the atoms of a gas collided and eventually reached an equilibrium velocity distribution, the entropy was maximized.

Boltzmann's ideas were met with a number of objections. One objection argued that because Newton's laws were reversible, thermodynamic processes described by the motion of atoms could be reversed in time to yield processes that deterministically went to states of lower entropy, thus contradicting the second law. Boltzmann's response highlighted the statistical nature of his interpretation, arguing that, given particular initial conditions, any thermodynamic system has a vastly greater number of final states available to it with relatively high entropy. An increase of entropy means that the system has become randomized as the available energy is spread around to its constituent atoms. In 1877 Boltzmann published a paper that incorporated this idea and defined the entropy as a log of a quantity measuring the number of states available to a system. In doing his calculations, Boltzmann used the device of counting energy in discrete increments, which he took to zero at the end of his calculation. This method, a harbinger of the quantization of energy, influenced Planck and Einstein, over twenty years later.

Boltzmann had less success answering a second set of objections regarding atomism. The British physicist William Thomson (1824–1907) and Scottish physicist Peter Tait (1831–1901) rejected atomism as a result of their adherence to the dynamical theory of matter, which rejected the existence of a void. Similarly, Ernst Mach put forward empiricist counterarguments, which rejected Boltzmann's adherence to entities that could not be confirmed by direct observation.