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Time Evolution in Physics
In physics “evolution” names the change of a system’s state over time as a rule gives it: from the state at one moment and the law of motion, the state at any later moment follows. The word says how the change is given, not what shape the history takes, so it covers reversible and irreversible change alike, and physics disputes with itself which of the two its evolution is at root.
Rule-given change
In classical mechanics the state of a system is the positions and momenta of its parts, and Hamilton’s equations carry it forward in time; physicists speak of the Hamiltonian flow and of a system’s evolution in its space of states. In quantum mechanics the state is a vector whose change is given by Schrödinger’s equation, and the map that takes the state at one time to the state at another is called the time-evolution operator. The evolution it describes is unitary: it preserves the information in the state and can be run backwards, so that nothing in the equation distinguishes past from future. The Schrödinger and Heisenberg pictures differ over whether the states or the observables are said to evolve, and agree in their predictions.
John von Neumann’s Mathematical Foundations of Quantum Mechanics (1932) set a second kind of change beside the first: alongside the continuous, reversible evolution of the Schrödinger equation, the discontinuous and irreversible change that occurs on measurement. How the two relate is the measurement problem, and one of the reasons the question of reversibility has never closed in physics.
Irreversible evolution
The word is used just as readily for change that cannot be run backwards. The master equations of statistical physics, the Fokker–Planck equation for the spreading of probability, and the evolution of open quantum systems all describe the evolution of a system in contact with an environment, in which information leaks away and the change has a direction.
Which shape at root
Physics’ own dispute is whether its fundamental evolution is reversible, with the arrow of time arising at larger scales, or irreversible from the start. Ludwig Boltzmann’s H-theorem (1872) derived the growth of entropy from the motion of molecules, and Josef Loschmidt objected (1876) that from time-symmetric laws no one-way result can follow without an assumption that breaks the symmetry; the exchange is set out under The laws and the reduction. One line of answer keeps the fundamental laws reversible and locates the arrow in the special low-entropy state of the early universe, the “past hypothesis” of David Albert’s Time and Chance (2000), a position Roger Penrose and Sean Carroll also hold; for quantum systems it adds decoherence, developed by H. Dieter Zeh and Wojciech Zurek, the spreading of correlations into an environment that makes quantum superpositions effectively irreversible.
Ilya Prigogine held the opposite: that irreversibility is a fundamental feature of nature, and that dynamics must be reformulated to include it, the argument of From Being to Becoming (1980). His school’s work on systems far from equilibrium, where order arises through fluctuations in flows of energy, also carried the physics toward biology. From 1972 the school applied it to the organisation of living systems and to prebiotic self-organisation, where it met Eigen’s theory of molecular selection (see Chemical evolution). Prigogine’s position on fundamental irreversibility is set out, with its reception, under Away from equilibrium.
See also: Evolution (the subject landing) · Thermodynamics · The laws and the reduction · Away from equilibrium · Chemical evolution · Prigogine · Schrödinger · Boltzmann · Penrose · Carroll · Relational quantum mechanics