Page 4 - Science Focus (Issue 19)

Before the Froth Fades Away:
Beer Froth and Its Mathematics
ظثऊ̰ʘۃjਭৢظثٙᅰኪ
By Ranting Huang 黃冉婷
You are in a bar trying to get some
beer. The bartender comes and serves
you a large glass of Guinness. The dark
caramel colored liquid pours into the
glass, producing a large head of foam.
Before the froth fades away, you
take a good sip of the beer, feeling
completely invigorated.
Arnd Leike, a physicist from
Ludwig Maximilian University
of Munich focusing on
renewable energies and
particle physics, drinks too,
but he gets more than just
that. In 2002, Leike won
the Ig Nobel Prize for
demonstrating
archeology. With the law of decay known, half-life of
carbon-14 can be deduced, and archeologists use
this knowledge to estimate the age of fossils and other
organic materials, such as bone and wood.
The story of the beer froth is not over yet. After Leike’s
publication, studies of the beer froth have continued.
A research team from the University of Bremen, “called
in question the results presented by Leike”, stating that
that the volume of beer froth obeys the mathematical the foam volume does not follow a simple exponential
law of exponential decay. The Nobel Laureate in law and includes more complicated factors [3]. And
chemistry, Dudley Herschbach, awarded him the it is these clashes of thoughts that help sciences to
prize at Sanders Theater, Harvard University. During advance like a never-stopping wheel. People study
the ceremony, Leike was given some beer served in a what was done before, learn from the successes and 這裡所涉及到的數學概念是指數衰減。如果某一個量的
graduated cylinder, which was exactly what he used failures, and derive new ideas. 衰減速度和它的值成一定比例，我們則稱之為指數衰減。
in his experiment. One may say that the Ig Nobel Prize should not 這種現象可以用一條簡單的微分方程表示，而稍加計算就
可以得到一個更簡單的函數：y = ae -bt 。想像一條向下彎
The initial goal of the research is to show his students be recognized in the same way as other prestigious
how to check the consistency of theoretical models prizes, yet just as the event’s organizer Marc Abrahams 的曲線，其尾部無限接近於零，這便是表示指數衰減的圖
with experimental data. In the experiment, Leike used said, “Every winner was chosen for work that first makes 像。從數學上講，這樣一個簡潔的公式使我們能夠掌握複雜
3 types of beer – Erdinger Weissbier, Augustinerbräu people laugh and then makes people think.” [4] 的物理系統，進而可以預測它們的未來並對其進行改動。
München and Budweiser Budvar. For data collection, 指數衰減和指數增長無處不在。從快速增長的細菌種群
a graduated cylinder is filled with a freshly opened 到金融學裡的複利息，從牛頓的冷卻定律到大氣壓力隨高
bottle of beer and the height of the froth is measured
15 times over a period of six minutes [1]. Later analysis 度的變化，指數變化隱藏在地球上許多現象的背後。指數
of the data suggested that the rate of decay not only 衰減的例子之一是考古學中的放射性碳定年法。在知道了
is exponential but also depends on the type of beer 衰減的規律後，我們就可以推算出碳-14 的半衰期，考古
and can be used to differentiate beers [2]. 學家則利用這些資訊來推算化石和其他例如骨頭和木頭
等有機物的年齡。
The mathematical idea here is exponential decay.
We say that a quantity is subject to exponential decay 啤酒泡沫的故事尚未結束。在 Leike 的論文發表後，相
if it decreases at a rate proportional to its value. 關的研究仍在繼續。不來梅大學的一個研究團隊對 Leike
This phenomenon can be expressed in a 的研究結果表示質疑，他們指出啤酒泡沫的消散並不僅僅
simple differential equation, which is later 設想你在酒吧，正打算來點啤酒喝。酒保給你上了一大杯健 遵循簡單的指數規律，而是包含了更複雜的因素 [3]。正因
showed to be an even simpler function, 力士，你看著深焦糖色的液體被傾倒在杯子裡，激起了好些泡 為有了這些思想碰撞的火花，科學才能像永不停息的輪子
y = ae -bt . Imagine a curve bending 沫。在泡沫消失之前，你啜了一口，不禁感到神清氣爽。 一樣前進。人們研究前人的成果，從他們的成功和失敗中
downwards with its tail going infinitely 學習，催生出新的想法。
close to zero yet never reaching 路德維希 - 馬克西米利安 - 慕尼黑大學的物理學家 Arnd
zero, and this is a simple graph of Leike 專注於研究可再生能源和粒子物理，他也喝酒，但並 也許有人會說，搞笑諾貝爾獎不應該與其他正經獎項一樣
exponential decay. Mathematically, 未止步於喝酒。2002 年，Leike 因為證明了啤酒泡沫的消散 被嚴肅對待，但正如它的籌辦者 Marc Abrahams 所言：
such a concise formula enables us 符合指數衰減規律而獲得搞笑諾貝爾獎。諾貝爾化學獎得主 「每個獎項都是頒發給那些乍看之下令人發笑，繼而卻能
to master complex physical systems Dudley Herschbach 在哈佛大學桑德斯劇院授予了 Leike 引人深思的研究。」 [4]
in the sense that we can predict 這一獎項。在頒獎典禮上，Leike 還喝到了裝在量筒裡的啤
their futures as well as altering 酒，而量筒正是他在實驗中使用的器材。
them.
這項研究最初的目標是向他的學生展示如何檢查理論 References 參考資料：
Exponential decay, and [1] Physicsworld. (2002). Beer paper wins Ig Nobel physics prize.
exponential growth, are all around 模型與實驗資料的一致性。Leike 使用了三種啤酒來進行 Retrieved from https://physicsworld.com/a/beer-paper-wins-ig-
us. From a rapidly growing bacterial 實驗，分 別 是 Erdinger Weissbier，Augustinerbräu nobel-physics-prize/
population to compound interest in finance, München 和 Budweiser Budvar。在收集資料的階段， [2] Leike, A. (2001). Demonstration of the exponential decay law
using beer froth. European Journal of Physics, 23, 21-26.
from Newton’s law of cooling to variation 剛開瓶的啤酒會被倒進標有刻度的量筒，並在六分鐘內分 [3] Sauerbrei, S., Haß, E. C., & Plath, P. J. (2006). The Apollonian
in atmospheric pressure with altitude, this 15 次測量啤酒泡沫的高度 [1]。後來針對資料的分析表 decay of beer foam bubble size distribution and the lattices of
mathematical formula lies behind many 明，啤酒泡沫的消散速率不僅是指數級的（exponential）， young diagrams and their correlated mixing functions. Discrete
Dynamics in Nature and Society, 2006, 1-35.
phenomena on Earth. One of the scenarios where 還會因啤酒的類型不同而有所分別，而這一點恰好可用於 [4] Nadis, S. (2002). Bubble won't burst for spoof Nobels. Nature,
exponential decay applies is radiocarbon dating in 區分啤酒 [2]。 419(6907), 549.