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Measure-preserving
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most dissipative system, non-dissipative systems) as well as systems in thermodynamic equilibrium. Definition A measure-preserving dynamical system is defined as a probability space and a Invariant measure, measure-preserving transformation on it. In more detail, it is a system :(X, \mathcal, \mu, T) with the following structure: *X is a set, *\mathcal B is a sigma-algebra, σ-algebra over X, *\mu:\mathcal\rightarrow[0,1] is a probability measure, so that \mu (X) = 1, and \mu(\varnothing) = 0, * T:X \rightarrow X is a measurable function, measurable t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixing (mathematics)
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''. The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including ''strong mixing'', ''weak mixing'' and ''topological mixing'', with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity). Informal explanation The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, drinks, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
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 in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conservative System
In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink over time. Precisely speaking, they are those dynamical systems that have a null wandering set: under time evolution, no portion of the phase space ever "wanders away", never to be returned to or revisited. Alternately, conservative systems are those to which the Poincaré recurrence theorem applies. An important special case of conservative systems are the measure-preserving dynamical systems. Informal introduction Informally, dynamical systems describe the time evolution of the phase space of some mechanical system. Commonly, such evolution is given by some differential equations, or quite often in terms of discrete time steps. However, in the present case, instead of focusing on the time evolution of discrete points, one shifts atte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Recurrence Theorem
In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. This time may vary greatly depending on the exact initial state and required degree of closeness. The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume. The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems. The theorem is named after Henri Poincaré, who discussed it in 1890 and proved by Constantin Carathéodory using measure theory in 1919. Precise formulation Any dynamical system de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermalization
In physics, thermalisation is the process of physical bodies reaching thermal equilibrium through mutual interaction. In general the natural tendency of a system is towards a state of equipartition of energy and uniform temperature that maximizes the system's entropy. Thermalisation, thermal equilibrium, and temperature are therefore important fundamental concepts within statistical physics, statistical mechanics, and thermodynamics; all of which are a basis for many other specific fields of scientific understanding and engineering application. Examples of thermalisation include: * the achievement of equilibrium in a plasma. * the process undergone by high-energy neutrons as they lose energy by collision with a moderator. The hypothesis, foundational to most introductory textbooks treating quantum statistical mechanics, assumes that systems go to thermal equilibrium (thermalisation). The process of thermalisation erases local memory of the initial conditions. The eigenstate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
