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Inmathematics,computer science andphysics, adeterministic system is a system in which norandomness is involved in the development of future states of the system.[1] A deterministicmodel will thus always produce the same output from a given starting condition or initial state.
Physical laws that are described bydifferential equations represent deterministic systems, even though the state of the system at a givenpoint in time may be difficult to describe explicitly. Inquantum mechanics, theSchrödinger equation, which describes the continuoustime evolution of a system'swave function, is deterministic. However, the relationship between a system's wave function and theobservable properties of the system appears to be non-deterministic.[citation needed]
The systems studied inchaos theory are deterministic. If the initial state were known exactly, then the future state of such a system couldtheoretically bepredicted. However, in practice, knowledge about the future state is limited by the precision with which the initial state can be measured, and chaotic systems are characterized by a strong dependence on the initial conditions. This sensitivity to initial conditions can be measured withLyapunov exponents.Markov chains and otherrandom walks are not deterministic systems, because their development depends on random choices.[citation needed]
A deterministicmodel of computation, for example adeterministic Turing machine, is a model of computation such that the successive states of the machine and the operations to be performed are completely determined by the preceding state. Adeterministic algorithm is an algorithm which, given a particular input, will always produce the same output, with the underlying machine always passing through the same sequence of states.[citation needed]
There may be non-deterministic algorithms that run on a deterministic machine, for example, an algorithm that relies on random choices. Generally, for such random choices, one uses apseudorandom number generator, but one may also use some external physical process, such as the last digits of the time given by the computer clock. Apseudorandom number generator is a deterministic algorithm, that is designed to produce sequences of numbers that behave as random sequences. Ahardware random number generator, however, may be non-deterministic.[citation needed]
In economics, theRamsey–Cass–Koopmans model is deterministic. The stochastic equivalent is known asreal business-cycle theory. As determinism relates to modeling in the natural sciences, a deterministic model[2] uses existing data to model the future behavior of a system. The deterministic model is useful for systems that do not experience frequent or unexpected behavior - unless that behavior is already present in the system via existing data. This type of modeling is distinct from stochastic modeling[2] or forward modeling.[3] Stochastic modeling uses random data in the model while forward modeling uses a given model to predict future behavior in a system. Deterministic models are used across the natural sciences, including geology, oceanography,[4] physics, and other disciplines.