timeevolution

Time evolution refers to how the state of a system changes as time progresses. It is the process by which initial conditions at one time determine the configuration of the system at later times, according to the governing laws of the theory being used. The precise mathematical description depends on the type of system and the physical framework.

In classical mechanics, evolution is described by Newton's laws or, equivalently, Hamilton's equations. Given initial positions and momenta, the system follows deterministic trajectories in phase space. Time serves as the independent parameter along these trajectories.

In quantum mechanics, the state is described by a wavefunction or a density operator. Time evolution is generated by the system’s Hamiltonian. In the Schrödinger picture, states evolve unitarily according to the Schrödinger equation, with the evolution operator U(t) = exp(-iHt/ħ). For open or interacting systems, evolution may be non-unitary and is described by master equations or completely positive maps, accounting for decoherence and dissipation.

Relativity brings a geometric view of time. In special relativity, time is frame-dependent and intertwined with space. In general relativity, evolution proceeds along worldlines in curved spacetime, with proper time as the invariant parameter along a particle’s history. In cosmology, the expansion of the universe is described by the scale factor a(t), governing evolution over cosmic time.

Time evolution is also studied statistically, via ensemble dynamics like Liouville or Boltzmann equations, and is central to numerical simulations, where discrete time integration schemes (for example, Euler or Runge-Kutta methods) approximate the evolution of complex systems.