In
mathematics,
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
, a deterministic system is a system in which no
randomness is involved in the development of future states of the system. A deterministic
model will thus always produce the same output from a given starting condition or initial state.
Dynamical systems
at Scholarpedia
In physics
Physical laws that are described by differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
represent deterministic systems, even though the state of the system at a given point in time may be difficult to describe explicitly.
In quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
, the Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
, which describes the continuous time evolution of a system's wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
, is deterministic. However, the relationship between a system's wave function and the observable properties of the system appears to be non-deterministic.
In mathematics
The systems studied in chaos theory are deterministic. If the initial state were known exactly, then the future state of such a system could theoretically be predicted. 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 with Lyapunov exponents.
Markov chains and other random walks are not deterministic systems, because their development depends on random choices.
In computer science
A deterministic model of computation
In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes h ...
, for example a deterministic Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algor ...
, 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.
A deterministic algorithm
In computer science, a deterministic algorithm is an algorithm that, given a particular input, will always produce the same output, with the underlying machine always passing through the same sequence of states. Deterministic algorithms are by far ...
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. 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 a pseudorandom number generator, but one may also use some external physical process, such as the last digits of the time given by the computer clock.
A ''pseudorandom number generator'' is a deterministic algorithm, that is designed to produce sequences of numbers that behave as random sequences. A hardware random number generator, however, may be non-deterministic.
Others
In economics, the Ramsey–Cass–Koopmans model is deterministic. The stochastic equivalent is known as real business-cycle theory.
See also
* Deterministic system (philosophy)
* Dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
* Scientific modelling
Scientific modelling is a scientific activity, the aim of which is to make a particular part or feature of the world easier to understand, define, quantify, visualize, or simulate by referencing it to existing and usually commonly accepted ...
* Statistical model
* Stochastic process
References
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System
Dynamical systems