Brownian Motor Applet
Brownian Motor Java Applet [external link]

ブラウン運動モーターの Java Applet が このページ にありました[英語]。
このページは、Applet の解説を日本語訳したものです。
あと、このページに解説があります。 気になる科学探検隊「マクロスケールの常識を覆す分子モーター」

   Brownian Motor (ブラウン運動モーター)

In his famous lectures Richard Feynman discussed the impossibility to violate the second law of thermodynamics by a ratchet mechanism.
The simplest model for a ratchet is an overdamped Brownian particle in an asymmetric but spatially periodic potential (with asymmetry α and period L).
もっとも簡単なラチェットのモデルは、ブラウン運動粒子を非対称な、空間的な周期性を持ったポテンシャル中に置いたものだ (非対称性α、周期L)。

Due to the fluctuating force caused by the pushing molecules of the surrounding fluid or gas the Brownian particle may overcome the potential barrier moving to the left or to the right.
The probabilities for both directions are equal.
Thus on average the particle does not move.
Hence building a motor which turns thermal energy into mechanical work from a single heat bath is impossible.
But the ratchet can be turned into a so-called a Brownian motor that seems to violate the second law of thermodynamics.
The idea is to turn the ratchet potential periodically on and off with a frequency 1/2τ.
そのアイデアとは、ラチェットポテンシャルを周期的に、周波数 1/2τ でON/OFFすることだ。
Under certain circumstances this may yield directed motion even against an applied force f.
It is indeed a device doing work.
Below you can play with a Java applet which simulates such a system.
以下で、このシステムをシミュレートした Javaアプレットを動かすことができる。
Vary the parameters and figure out the condition for a drift from right to left even for f > 0.
パラメーターをいろいろ変えてみて、たとえ力f > 0 がかかっていても、右に動くような条件を探してみよう。
If you want to know why a Brownian motor isn't a perpetuum mobile of the second kind click on "here".

[* Javaアプレットはオリジナル・サイトを見てね *]


* Parameters can be changed either by manuipulating the corresponding scroll bar or by directly putting a number into the number field.
There are the following parameters:

N = number of particles.
N = 粒子の数.
kT = thermal energy kT in units of the potential barrier.
kT = ポテンシャル障壁のユニット内の熱エネルギー kT  (k:ボルツマン定数、T:温度)
alpha = asymmetry α of the ratchet potential.
alpha = ラチェットの非対称の度合いを表す数値 α.
tau = switching time τ.
tau = スイッチ切替時間 τ.
f = applied force f in units of the potential barrier divided by L.
f = ポテンシャル障壁のユニット内に働く外力f、Lで割ったもの。

* The start/stop button starts or stops the animation in the animation area.
start/stop ボタンは、アニメーションを開始、または停止する。
* The reset button resets the particle positions.
* The animation area shows:
1. The result of the simulation. シミュレーション結果
If the number of particle is less than or equal 20 the particles are shown by circles.
For a larger ensemble the particle positions are shown as a function of time (time direction is pointing upwards).
The black line shows the averaged particle position.
In the upper left corner the time and the averaged velocity (assuming L = 1) are shown.
左上の隅に、時刻と平均速度(L = 1 として)が示してある。
2. The potential. ポテンシャル
The actual potential is shown in black.
When the saw-toothed part is switch off it is still shown in light gray.
The potential is tilted due to the external force f.
3. The potential energy. ポテンシャル・エネルギー
The averaged potential energy is shown as a green bar.

   Why is a Brownian motor not a perpetuum mobile of the second kind?

As long as the ratchet potential is off the particle will move diffusively according to a (biased) random walk,
leading to a variance in position of Δx = √2Dγ and a mean position of <x> = fτ/γ,
位置の分散 Δx = √2Dγ と、位置の平均 <x> = fτ/γ から導かれるように。
where D = kT / γ is the diffusion constant.
ここで D = kT / γ は拡散定数を表している。
When the ratchet potential is switched on, the particle gets trapped in one of the potential minima.
If αL >= Δx >= (1-α)L for the variance holds, the particle on average gets trapped into the minimum left to the starting point.
もし分散を αL >= Δx >= (1-α)L というように保てば、粒子は平均してスタート地点より右側の最小値に捕らえられるだろう。
The maximum flux is obtained if the switching time τ is large enough to assure that the particle can adjust in the trapping minimum ('adiabatic adjustment time') and also is small enough to fulfill the above requirement for the variance.
One can say roughly that a net flux to the left always occurs, when thermal energy is significantly smaller than the potential maximum,
the external force is chosen not too big and the driving frequency matches the adiabatic adjustment time needed for the particle to move into a potential minima.

Where does the energy come from leading to a drift against the external force?
The energy does not come from the heat bath but from the ratchet potential when it is switched on.
At that moment the potential energy of the particle will be suddenly increased.
In the simulation this can be seen by a sudden increase of the energy bar.
But most of the energy pushed into the system will be just dissipated into the heat bath
due to the relaxation of the particle into a potential minima.
Only a tiny portion will be used for doing work.
Thus a Brownian motor does not violate any law of thermodynamics it only turns one type of work into another one.
Nevertheless the fluctuating force due to the heat bath is essential for a Brownian motor.

For more details and possible applications in biology and chemistry read the following review article:
R.D. Astumian: Thermodynamics and Kinetics of a Brownian Motor, Science 276, p. 917-922 (1997).