Butterfly effect – 蝴蝶效应

The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. Small variations of the initial condition of a dynamical system may produce large variations in the long term behavior of the system. This is sometimes presented as esoteric behavior, but can be exhibited by very simple systems: for example, a ball placed at the crest of a hill might roll into any of several valleys depending on slight differences in initial position. Quantum chaos is the study of the butterfly effect in semiclassical physics and quantum mechanics.

It is a common subject in fiction when presenting scenarios involving time travel and with “what if” scenarios where one storyline diverges at the moment of a seemingly minor event resulting in two significantly different outcomes.

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Theory

Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather), since it is impossible to measure the starting atmospheric conditions completely accurately.

Origin of the concept and the term

The term “butterfly effect” itself is related to the work of Edward Lorenz, and is based in chaos theory and sensitive dependence on initial conditions, first described in the literature by Jacques Hadamard in 1890[1] and popularized by Pierre Duhem’s 1906 book. The idea that one butterfly could eventually have a far-reaching ripple effect on subsequent historic events seems first to have appeared in a 1952 short story by Ray Bradbury about time travel (see Literature and print here) although Lorenz made the term popular. In 1961, Lorenz was using a numerical computer model to rerun a weather prediction, when, as a shortcut on a number in the sequence, he entered the decimal .506 instead of entering the full .506127 the computer would hold. The result was a completely different weather scenario.[2] Lorenz published his findings in a 1963 paper for the New York Academy of Sciences noting that “One meteorologist remarked that if the theory were correct, one flap of a seagull’s wings could change the course of weather forever.” Later speeches and papers by Lorenz used the more poetic butterfly. According to Lorenz, upon failing to provide a title for a talk he was to present at the 139th meeting of the American Association for the Advancement of Science in 1972, Philip Merilees concocted Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas as a title.

Although a butterfly flapping its wings has remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have varied widely.[3]

The phrase refers to the idea that a butterfly’s wings might create tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in a certain location. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale alterations of events. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different. While the butterfly does not “cause” the tornado in the sense of providing the energy for the tornado, it does “cause” it in the sense that the flap of its wings is an essential part of the initial conditions resulting in a tornado, and without that flap that particular tornado would not have existed.

Examples in semiclassical and quantum physics

The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in semiclassical and quantum physics including atoms in strong fields and the anisotropic Kepler problem. Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments; however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by Martin Gutzwiller and Delos and co-workers.

Other authors suggest that the butterfly effect can be observed in quantum systems. Karkuszewski et al. consider the time evolution of quantum systems which have slightly different Hamiltonians. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians. Poulin et al. present a quantum algorithm to measure fidelity decay, which “measures the rate at which identical initial states diverge when subjected to slightly different dynamics.” They consider fidelity decay to be “the closest quantum analog to the (purely classical) butterfly effect.” Whereas the classical butterfly effect considers the effect of a small change in the position and/or velocity of an object in a given Hamiltonian system, the quantum butterfly effect considers the effect of a small change in the Hamiltonian system with a given initial position and velocity. This quantum butterfly effect has been demonstrated experimentally. Quantum and semiclassical treatments of system sensitivity to initial conditions are known as quantum chaos.

original link

http://en.wikipedia.org/wiki/Butterfly_effect

蝴蝶效应

蝴蝶效應是指在一個動力系統中，初始條件下微小的變化能帶動整個系統的長期的巨大的連鎖反應。這是一種混沌現象。

“蝴蝶效應”在混沌學中也常出現。

蝴蝶效應的由來

[蝴蝶效應]來源於美國氣象學家勞侖次60年代初的發現。在《混沌學傳奇》與《分形論——奇異性探索》等書中皆有這樣的描述：“1961年冬季的一天，勞侖次（E．Lorenz）在皇家麥克比型電腦上進行關於天氣預報的計算．為了考察一個很長的序列，他走了一條捷徑，沒有令電腦從頭運行，而是從中途開始．他把上次的輸出直接打入作為計算的初值，但由於一時不查，他無意間省略了小數點後六位的零頭，然後他穿過大廳下樓，去喝咖啡．一小時後，他回來時發生了出乎意料的事，他發現天氣變化同上一次的模式迅速偏離，在短時間內，相似性完全消失了．進一步的計算表明，輸入的細微差異可能很快成為輸出的巨大差別．這種現象被稱為對初始條件的敏感依賴性．在氣象預報中，稱為‘蝴蝶效應’．……”“勞侖次最初使用的是海鷗效應．”“勞侖次1979年12月29日在華盛頓的美國科學促進會的演講：‘可預言性：一隻蝴蝶在巴西扇動翅膀會在德克薩斯引起龍捲風嗎?’”

蝴蝶效應的含義

某地上空一隻小小的蝴蝶扇動翅膀而擾動了空氣，長時間後可能導致遙遠的彼地發生一場暴風雨，以此比喻長時期大範圍天氣預報往往因一點點微小的因素造成難以預測的嚴重後果．微小的偏差是難以避免的，從而使長期天氣預報具有不可預測性或不準確性．這如同打檯球、下棋及其他人類活動，往往“差之毫釐，失之千里”、“一著不慎，滿盤皆輸”。

長時期大範圍天氣預報是對於地球大氣這個複雜系統進行觀測計算與分析判斷。它受到地球大氣溫度、濕度、壓強諸多隨時隨地變化的因素的影響與制約，可想其綜合效果的預測是難以精確無誤的、蝴蝶效應是在所必然的．我們人類研究的物件還涉及到其他複雜系統（包括“自然體系”與“社會體系”），其內部也是諸多因素交相制約錯綜複雜，其“相應的蝴蝶效應”也是在所必然的．“今天的蝴蝶效應”或者“廣義的蝴蝶效應”已不限於當初勞侖次的蝴蝶效應僅對天氣預報而言，而是一切複雜系統對初值極為敏感性的代名詞或同義語，其含義是：對於一切複雜系統，在一定的“閾值條件”下，其長時期大範圍的未來行為，對初始條件數值的微小變動或偏差極為敏感，即初值稍有變動或偏差，將導致未來前景的巨大差異，這往往是難以預測的或者說帶有一定的隨機性。

相似 骨牌效應 又名 多米諾骨牌效應

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