Сұйықтардағы бөлшектер қозғалысы және графтардағы ағын

Motion of particles in a fluid
flows in graph theory

In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and Anosov flows. Flows may also be defined for systems of random variables and stochastic processes, and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps the Bernoulli flow.

It is very common in many fields, including engineering, physics and the study of differential equations, to use a notation that makes the flow implicit. Thus, x(t) is written for \varphi^t(x 0), and one might say that the variable x depends on the time t and the initial condition 1= x = x0. Examples are given below. In the case of a flow of a vector field V on a smooth manifold X, the flow is often denoted in such a way that its generator is made explicit. For example,

Given x in X, the set is called the orbit of x under φ. Informally, it may be regarded as the trajectory of a particle that was initially positioned at x. If the flow is generated by a vector field, then its orbits are the images of its integral curves.

Let f: \R \to X be a time dependent trajectory which is a bijective function. Then a flow can be defined by

The flows of time independent and time dependent vector fields are defined on smooth manifolds exactly as they are defined on the Euclidean space \R^n and their local behavior is the same. However, the global topological structure of a smooth manifold is strongly manifest in what kind of global vector fields it can support, and flows of vector fields on smooth manifolds are indeed an important tool in differential topology. The bulk of studies in dynamical systems are conducted on smooth manifolds, which are thought of as "parameter spaces" in applications. Formally: Let be a differentiable manifold. Let denote the tangent space of a point Let be the complete tangent manifold; that is, Let

be a time dependent vector field on ; that is, f is a smooth map such that for each and , one has that is, the map maps each point to an element of its own tangent space. For a suitable interval containing 0, the flow of f is a function that satisfies

Ergodic dynamical systems, that is, systems exhibiting randomness, exhibit flows as well. The most celebrated of these is perhaps the Bernoulli flow. The Ornstein isomorphism theorem states that, for any given entropy H, there exists a flow φ(x, t), called the Bernoulli flow, such that the flow at time 1=t = 1, i. e. φ(x, 1), is a Bernoulli shift. Furthermore, this flow is unique, up to a constant rescaling of time. That is, if ψ(x, t), is another flow with the same entropy, then , for some constant c. The notion of uniqueness and isomorphism here is that of the isomorphism of dynamical systems. Many dynamical systems, including Sinai's billiards and Anosov flows are isomorphic to Bernoulli shifts.