The rise of stochasticity in physics

Abstract In the last 175 years, the physical understanding of nature has seen a revolutionary change. Until about 1850, Newton’s theory and the mechanical world view derived from it provided the dominant view of the physical world, later supplemented by Maxwell’s theory of the electromagnetic field. That approach was entirely deterministic and free of probabilistic concepts. In contrast to that conceptual edifice, today many fields of physics are governed by probabilistic concepts. Statistical mechanics in its classical or quantum version and random-matrix theory are obvious examples. Quantum mechanics is an intrinsically statistical theory. Classical chaos and its quantum manifestations also require a stochastic approach. The paper describes how a combination of discoveries and conceptual problems undermined the mechanical world view, led to novel concepts, and shaped the modern understanding of physics. As essential causes, I list Clausius’ formulation of the second law of thermodynamics, the very large number of molecules in a macroscopic gas container which caused Maxwell to formulate his statistical kinetic theory, Boltzmann’s statistical approach which attempted to reconcile irreversibility as manifest in the second law of thermodynamics with Newtonian mechanics, the discovery of radioactivity and of discrete spectral lines in the emission of light from stars and atoms that could not be understood in classical terms, Poincare’s discovery of unstable orbits in the astronomical three-body problem, and black-body radiation that led Planck to the discovery of the quantum of action. A further cause was the need to compensate for incomplete knowledge. Wigner introduced random-matrix theory because the nuclear Hamiltonian was largely unknown. That theory now governs the statistical theory of nuclear reactions. It has spread to many fields of physics, from the theory of electron transport through mesoscopic samples to quantum chromodynamics, to Andreev scattering, to the SYK model used in astrophysics and condensed-matter theory, to two-dimensional gravity, to random quantum circuits, and even to number theory.