Phase space 相空間

「相空間」描述系統狀態的變化，原使用在物理、數學、化學等領域，後來也被其他領域借用，做為描述系統動態的工具。其實也跟整個電腦興起後所談的非線性空間相關，非線性空間也是一個相空間。這也是有關達爾文的後續生物學家的研究，用相空間來解釋演化的趨勢。

以下是維基的解釋

相空間 From Wikipedia

在數學與物理學中，相空間(phase space)是一個用以表示出一系統所有可能狀態的空間；系統每個可能的狀態都有一相對應的相空間的點。以力學系統來說，相空間通常是由位置變數以及動量變數所有可能值所組成。將位置變數與動量變數畫成時間的函數有時稱為相空間圖，簡稱「相圖」(phase diagram)。然而在物質科學(physical sciences)中，「相圖」這詞更常是留給一化學系統用以表示其熱力學相態多種穩定性區域的圖表，為壓力、溫度及化學組成等等之函數。

在一相空間中，系統的每個自由度或參數可以用多維空間中的一軸來代表。對於系統每個可能的狀態，或系統參數值允許的組合，可以在多維空間描繪成一個點。通常這樣的描繪點連接而成的線可以類比於系統狀態隨著時間的演化。最後相圖可以代表系統可以存在的狀態，而它的外型可以輕易地闡述系統的性質，這在其他的表示方法則不那麼顯明。一相空間可有非常多的維度。舉例來說，一氣體包含許多分子，每個分子在x、y、z方向上就要有3個維度給位置與3個維度給速度，可能還需要額外的維度給其他的性質。

在古典力學中，相空間坐標由廣義坐標qi以及其共軛的廣義動量pi所組成。研究由許多系統所構成的系綜在此空間中的運動是屬於古典統計力學的範疇。


From Wikipedia, the free encyclopedia

Phase space of a dynamical system with focal stability.In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. A plot of position and momentum variables as a function of time is sometimes called a phase plot or a phase diagram. Phase diagram, however, is more usually reserved in the physical sciences for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, which consists of pressure, temperature, and composition.

In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space. For every possible state of the system, or allowed combination of values of the system's parameters, a point is plotted in the multidimensional space. Often this succession of plotted points is analogous to the system's state evolving over time. In the end, the phase diagram represents all that the system can be, and its shape can easily elucidate qualities of the system that might not be obvious otherwise. A phase space may contain very many dimensions. For instance, a gas containing many molecules may require a separate dimension for each particle's x, y and z positions and velocities as well as any number of other properties.

In classical mechanics the phase space co-ordinates are the generalized coordinates qi and their conjugate generalized momenta pi. The motion of an ensemble of systems in this space is studied by classical statistical mechanics. The local density of points in such systems obeys Liouville's Theorem, and so can be taken as constant. Within the context of a model system in classical mechanics, the phase space coordinates of the system at any given time are composed of all of the system's dynamical variables. Because of this, it is possible to calculate the state of the system at any given time in the future or the past, through integration of Hamilton's or Lagrange's equations of motion. Furthermore, because each point in phase space lies on exactly one phase trajectory, no two phase trajectories can intersect.

For simple systems, such as a single particle moving in one dimension for example, there may be as few as two degrees of freedom, (typically, position and velocity), and a sketch of the phase portrait may give qualitative information about the dynamics of the system, such as the limit-cycle of the Van der Pol oscillator shown in the diagram.


Phase portrait of the Van der Pol oscillatorHere, the horizontal axis gives the position and vertical axis the velocity. As the system evolves, its state follows one of the lines (trajectories) on the phase diagram.

Classic examples of phase diagrams from chaos theory are the Lorenz attractor and Mandelbrot set.