time derivative — from formula anatomyThe time derivative is a mathematical operation that measures how quickly a quantity changes with respect to time. In physics, it appears everywhere: velocity is the time derivative of position, acceleration is the time derivative of velocity, and the rate of change of energy, momentum, or any other observable is expressed through its time derivative. It is the language of dynamics, encoding the rules of motion and evolution for physical systems.
The concept of the derivative was developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Newton called it the 'fluxion' and used it in his laws of motion.
Formally, the time derivative of a function f(t) is defined as the limit of the ratio [f(t+Δt) − f(t)]/Δt as Δt approaches zero. This gives the instantaneous rate of change at a single moment. In practice, physicists use it in differential equations, such as Newton's second law (F = m·d²x/dt²) or the Schrödinger equation, where the time derivative of the quantum state determines how the state evolves.
Time derivatives are used in every simulation of physical systems, from predicting planetary orbits to modeling electrical circuits and weather patterns.
In quantum mechanics, the time derivative of an observable is not always simply the rate of change of its expectation value—it also involves the commutator with the Hamiltonian.
| ρ | variable | density matrix of the open quantum system — density matrix |
| t | variable | time — time second |
| H | variable | Hamiltonian of the system — hamiltonian joule |
| L_n | variable | Lindblad operators (jump operators) for each dissipation channel — lindblad operator |
| i | constant | imaginary unit, square root of -1 — imaginary unit = 1 |
| ħ | constant | Planck constant divided by 2π — reduced planck constant = 1.054571817e-34 joule second |
| d/dt | operator | derivative with respect to time — time derivative |
| [H, ρ] | operator | commutator of H and ρ — commutator |
| {L_n† L_n, ρ} | operator | anticommutator of L_n† L_n and ρ — anticommutator |
| ∑_n | operator | sum over all dissipation channels — sum over channels |
| q | variable | Deceleration parameter — deceleration parameter |
| a | variable | Scale factor of the universe — scale factor |
| t | variable | Cosmic time — time second |
| d/dt | operator | Computes the rate of change over time — time derivative |
| d^2/dt^2 | operator | Computes the acceleration over time — second time derivative |
| r | variable | radial coordinate (distance from center) — radius metre |
| M(r) | variable | enclosed mass within radius r — mass kilogram |
| k | variable | constant of motion (related to energy per unit mass) — specific energy metre squared per second squared |
| G | constant | Newtonian constant of gravitation, fundamental strength of gravity — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| ˙ | operator | derivative with respect to time — time derivative |
| L_H | variable | Instantaneous non-thermal emission (luminosity) — luminosity watt |
| L_S | variable | Plasma heating rate (energy per time) — power watt |
| \frac{d}{dt} | operator | Computes the rate of change with respect to time — time derivative |
| I | variable | information flow between subsystems — information flow bit per second |
| σ | variable | entropy production rate — entropy production rate joule per kelvin second |
| A | variable | activity (a measure of dynamical activity in the system) — activity per second |
| β | variable | inverse temperature (1/k_B T) — inverse temperature per joule |
| |·|² | operator | squared magnitude — absolute value squared |
| d/dt | operator | derivative with respect to time — time derivative |
| concentration | variable | local concentration of the diffusing species — concentration mole per cubic metre |
| t | variable | time — time second |
| flux | variable | diffusive flux vector — diffusive flux mole per square metre per second |
| d/dt | operator | rate of change with respect to time — time derivative |
| divergence | operator | measures the net outflow of a vector field per unit volume — divergence operator |
