Concept

# Time evolution

Summary
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called stateful systems). In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies is governed by the principles of classical mechanics. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by Newton's laws of motion. These principles can also be equivalently expressed more abstractly by Hamiltonian mechanics or Lagrangian mechanics. The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a Turing machine can be regarded as the time evolution of the machine's control state together with the state of the tape (or possibly multiple tapes) including the position of the machine's read-write head (or heads). In this case, time is discrete. Stateful systems often have dual descriptions in terms of states or in terms of observable values. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant in quantum mechanics where the Schrödinger picture and Heisenberg picture are (mostly) equivalent descriptions of time evolution. Consider a system with state space X for which evolution is deterministic and reversible. For concreteness let us also suppose time is a parameter that ranges over the set of real numbers R. Then time evolution is given by a family of bijective state transformations Ft, s(x) is the state of the system at time t, whose state at time s is x. The following identity holds To see why this is true, suppose x ∈ X is the state at time s. Then by the definition of F, Ft, s(x) is the state of the system at time t and consequently applying the definition once more, Fu, t(Ft, s(x)) is the state at time u. But this is also Fu, s(x).
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications (15)

## Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

Fabrizio Minganti

The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the
VEREIN FORDERUNG OPEN ACCESS PUBLIZIERENS QUANTENWISSENSCHAF2022

Related units

Related concepts (29)
Time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called stateful systems). In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies is governed by the principles of classical mechanics. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by Newton's laws of motion.
Path-ordering
In theoretical physics, path-ordering is the procedure (or a meta-operator ) that orders a product of operators according to the value of a chosen parameter: Here p is a permutation that orders the parameters by value: For example: If an operator is not simply expressed as a product, but as a function of another operator, we must first perform a Taylor expansion of this function. This is the case of the Wilson loop, which is defined as a path-ordered exponential to guarantee that the Wilson loop encodes the holonomy of the gauge connection.
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
Related courses (14)
ME-273: Introduction to control of dynamical systems
Cours introductif à la commande des systèmes dynamiques. On part de quatre exemples concrets et on introduit au fur et à mesure un haut niveau d'abstraction permettant de résoudre de manière unifiée l
MATH-561: Spectral theory
This course is an introduction to the spectral theory of linear operators acting in Hilbert spaces. The main goal is the spectral decomposition of unbounded selfadjoint operators. We will also give el
MATH-478: Dispersive PDEs
This course will give an introduction to some aspects of nonlinear dispersive partial differential equations. These are time evolution problems that arise in many contexts in physics, such as quantum
Related lectures (94)
Introduction to Quantum Chaos
Covers the introduction to Quantum Chaos, classical chaos, sensitivity to initial conditions, ergodicity, and Lyapunov exponents.
Ergodicity & Mixing: Understanding Chaos
Explores ergodicity and mixing in dynamical systems to understand chaos and system behavior.
Observability and Governability
Explores observability, governability, state regulators, and canonical forms in dynamic systems.
Related MOOCs (3)
Path Integral Methods in Atomistic Modelling
The course provides an introduction to the use of path integral methods in atomistic simulations. The path integral formalism allows to introduce quantum mechanical effects on the equilibrium and (ap
Thermodynamics
Ce cours vous apportera une compréhension des concepts fondamentaux de la thermodynamique du point de vue de la physique, de la chimie et de l’ingénierie. Il est scindé un deux MOOCs. Première partie: