timedependence

Timedependence refers to the characteristic of a process, quantity, or system whose value changes over time. It is the opposite of time invariance, where a quantity would remain the same if observed at different times under equal conditions. A quantity is time-dependent if it is written as a function of time, Q(t), or if its evolution is governed by dynamical laws that produce changes over time.

In physics and engineering, time dependence appears in evolving states such as the position x(t) of a particle, the expectation value of an observable under a time-dependent Hamiltonian, or the response of a system to a driving force. In chemistry and biology, reaction concentrations c(t), population sizes N(t), and other state variables commonly follow time-dependent laws. The classic example is radioactive decay, where N(t) = N0 e^{-λ t}, or a damped oscillator x(t) = A e^{-β t} cos(ω' t). Many processes exhibit explicit time dependence through rate equations, such as dc/dt = -k c leading to c(t) = c0 e^{-kt}.

Time dependence also appears in statistics and biomedicine, where time-dependent covariates affect survival analysis via hazard functions h(t|X(t)). In materials science, properties like creep and diffusion depend on time, reflecting aging and non-equilibrium dynamics. Modeling time dependence typically uses differential equations, stochastic processes, or time series analysis to describe how observables evolve and to identify characteristic timescales.

Understanding timedependence is essential for predicting system behavior, designing experiments, and distinguishing intrinsic temporal changes from external or seasonal variation.