We have pointed out that the random walk is closely similar in itsmathematics to the coin-tossing game we considered at the beginning of the chapter.If we imagine the direction of each step to be in correspondence with theappearance of heads or tails in a coin toss, then D is just NH−NT , the difference in the number of heads and tails. Since NH+NT=N, the total number of steps (and tosses), we have D=2NH−N. We have derived earlier an expression for the expected distributionof NH (also called k ) and obtained the result of Eq. (6.5).Since N is just a constant, we have the corresponding distribution for D. (Since for every head more than N/2 there is a tail “missing,” we have the factor of 2 between NH and D .) The graph of Fig. 6–2 representsthe distribution of distances we might get in 30 random steps (where k=15 is to be read D=0 ; k=16 , D=2 ; etc.).
在本章的开始，我们思考了抛硬币的数学，现在，我们指出，随机行走的数学与抛硬币的数学，非常近似。如果我们想象每一步的方向，都与抛硬币中的正面或反面相应，那么，D就正是NH−NT，正面次数与负面次数之差。由于NH+NT=N，即总步数（总抛掷数），我们就有D=2NH−N。对于被期待的NH（也被称为 k）的分布，我们导出过一个表达，并获得了结果，即方程（6.5）。由于N正是一个常数，所以对D，我们也有一个相应的分布。（由于对于每一个正面超过N/2的局，那么负面就“丢失了”，所以，NH和 D之间，我们有一个2的因子。）图6-2中的曲线，代表了在30次随机的步伐中，我们能得到的距离的分布(这里k=15，将被解读为 D=0 ; k=16 , D=2 ；等等。)。

The variation of NH from its expected value N/2 is
NH距离它的被期待的值N/2的差异是：

The rms deviation is
rms偏差是

According to our result for Drms , we expect that the “typical” distance in 30 steps ought to be 30−−√≈5.5 , or a typical k should be about 5.5/2=2.75 units from 15 . We see that the “width” of the curve in Fig. 6–2,measured from the center, is just about 3 units, in agreement with this result.
根据我们从Drms得到的结果，我们期待，在30步中，“典型的”距离应该是30的平方根，或者,一个典型的k值应该距15约为5.5/2=2.75个单位。在图6.2中，我们看到，从中心量的曲线的“宽度”，大约是3个单位，与这个结果一致。

Fig. 6–6.The fraction of the tosses that gaveheads in a particular sequence of N tosses of a penny. 一枚硬币，抛掷N遍，给出的具体次序中的正面的分数。{横坐标：抛掷的次数；纵坐标：正面的分数}
In Fig. 6–6 we haveplotted the fraction NH/N for the coin tosses reported earlier in this chapter. We see the tendencyfor the fraction of heads to approach 0.5 for large N . Unfortunately, for any given run or combination of runs there is no guaranteethat the observed deviation will be even near the expecteddeviation. There is always the finite chance that a large fluctuation—a longstring of heads or tails—will give an arbitrarily large deviation. All we cansay is that if the deviation is near the expected 1/2N−−√ (say within a factor of 2 or 3 ), we have no reason to suspect the honesty of the coin. If it is muchlarger, we may be suspicious, but cannot prove, that the coin is loaded (orthat the tosser is clever!).
本章早些时候，我们谈到了抛硬币，图6-6，我们画了正面的分数与抛硬币的次数的曲线。我们看到，对于大的N，正面的分数趋于0.5的趋势。不幸的是，对于任何被给与的运行或运行的组合，无法保证，被观察到的偏差，将会更接近被期待的偏差。一个大的波动，即一长串的正面或反面，将会提供一个任意大的偏差，但这种机会，总是有限的。我们所能说的只是，如果偏差接近被期待的值1/（2根号N），（比如说，在因子2或3之间），那么，我们就没有理由怀疑硬币的诚实性。如果偏差要大的多，我们可以怀疑但却没法证明：硬币是有利于某一方的（或抛硬币的人很聪明！）

There is an implication in such an expression that there is a“true” or “correct” probability which could be computed if we knew enough,and that the observation may be in “error” due to a fluctuation. There is,however, no way to make such thinking logically consistent. It is probablybetter to realize that the probability concept is in a sense subjective, that itis always based on uncertain knowledge, and that its quantitative evaluation issubject to change as we obtain more information.
在这样一种表达中，有一个暗示，那就是，如果我们知道的足够多，那么，就有一个“真实的”或“正确的”概率在那里，它是我们可以计算出来的，且由于波动，可能会导致观察中出现错误。然而，却没有什么方法，可以让这种思考在逻辑上一致。我们最好要意识到，概率概念在某种意义上是主观的，它总是基于不确定的知识；当我们得到更多的信息时，就会带来的变化，而对它的定量估值，总是从属于这种变化。

这一节公式多，好在微信公众号的编辑器，与word类似，能更清楚点。
https://mp.weixin.qq.com/s/TlBixzslSEPppTxF5c5bYw

6–4A probability distribution 6-4 概率分布
Let us return now to the random walk andconsider a modification of it. Suppose that in addition to a random choice of thedirection (+ or − ) of each step, the length of each step also varied in someunpredictable way, the only condition being that on the average the steplength was one unit. This case is more representative of something like thethermal motion of a molecule in a gas. If we call the length of a step S, then S may have any value at all, but most often will be “near” 1 . To be specific, we shall let ⟨S2⟩=1 or, equivalently, Srms=1 . Our derivation for ⟨D2⟩ would proceed as before except that Eq. (6.8)would be changed now to read
现在让我们回到随机行走，并考虑对它的一个修改。假设除了每一步的方向是随机选择之外，每步的长度，也以某种不可预测的方式在变，唯一的条件就是，平均来说，每步的长度都是一个单位。这种情况，更像是气体中分子的热运动。如果我们称步长为S，那么，S就可以取任意值，但是，大部分会“接近于”1。特别地，我们将让 ⟨S2⟩=1，或者等价地说，让Srms=1。我们对于 ⟨D2⟩的偏差，将会像以前那样进展，除非方程（6.8）现在将会被变成这样：

We have, as before, that
正如以前一样，我们会有：

What would we expect now for the distribution of distances D? What is, for example, the probability that D=0 after 30 steps? The answer is zero! The probability is zero that Dwill be any particular value, since there is no chance at allthat the sum of the backward steps (of varying lengths) would exactly equal thesum of forward steps. We cannot plot a graph like that of Fig. 6–2.
现在，对于距离的分布，我们该期待什么呢？例如，30步后D=0的概率是什么？答案是零。概率是零。那个D会是任何具体的值，由于（长度变化的）向后的步数，恰好等于向前的步数，根本没有这种机会。应该呢，完全与向前的部署一样。像图6-2那样的曲线, 我们无法画出。
{这里有个断句问题:概率是零。为一句。不然意思不对}

We can, however, obtain a representationsimilar to that of Fig. 6–2, if weask, not what is the probability of obtaining D exactly equal to 0 , 1 , or 2 , but instead what is the probability of obtaining D near 0 , 1 , or 2 . Let us define P(x,Δx) as the probability that D will lie in the interval Δx located at x (say from x to x+Δx ). We expect that for small Δx the chance of D landing in the interval is proportional to Δx , the width of the interval. So we can write
P(x,Δx)=p(x)Δx.(6.17)
The function p(x) is called the probability density.
然而，如果我们所问的，不是得到D准确地等于 0、1或2的概率，而是得到D接近于 0、1或2的概率，那么，一个类似于图6-2的表达，我们可以得到。让我们把 P(x,Δx)定义为：D落在x处的Δx范围之内（即从x到x+Δx）的概率。我们期待，D落在小的Δx范围之内的机会，与Δx的宽度正比。于是我们有：
P(x,Δx)=p(x)Δx.(6.17)
函数p(x)被称为概率密度。

The form of p(x) will depend on N , the number of steps taken, and also on the distribution ofindividual step lengths. We cannot demonstrate the proofs here, but forlarge N , p(x) is the same for all reasonable distributions in individual steplengths, and depends only on N . We plot p(x) for three values of N in Fig. 6–7. Youwill notice that the “half-widths” (typical spread from x=0 ) of these curves is N−−√ , as we have shown it should be.
p(x)的形式，依赖于所走的步数N，同时也依赖于个别步长的分布。这里我们没法演示证据，但是，对于大的N来说，对于所有个别步长的合理分布， p(x)都是一样的，且只依赖于N。在图6-7中，我们为三个N值，画了其 p(x)曲线。你将会注意到，这些曲线的“半宽度”（从0开始的典型展开）是根号N，正如我们所指出那样：它应该如此。

Fig. 6–7.The probability density for endingup at the distance D from the starting place in a random walk of N steps. (D is measured in units of the rms step length.) 图6-7 在一个N步的随机行走中，距离开始位置为D处的概率密度。（D是用rms步长的单位来测量的。）

You may notice also that the value of p(x)near zero is inversely proportional to N−−√ . This comes about because the curves are all of a similar shape andtheir areas under the curves must all be equal. Since p(x)Δxis the probability of finding D in Δx when Δx is small, we can determine the chance of finding D somewhere inside an arbitrary interval from x1 to x2 , by cutting the interval in a number of small increments Δxand evaluating the sum of the terms p(x)Δx for each increment. The probability that D lands somewhere between x1 and x2 , which we may write P(x1<D<x2) , is equal to the shaded area in Fig. 6–8. Thesmaller we take the increments Δx , the more correct is our result. We can write, therefore,
你可能也会注意到，零附近p(x)的值，与根号N成反比。之所以如此，乃是因为，曲线的形状都相似，且它们在曲线下的面积，应该全是相等的。由于p(x)Δx是Δx比较小时，在Δx中找到D的概率，我们可以这样，来规定找到D的机会，即通过把从x1到x2这一间隔，分成更小的增量Δx，并对每一个增量，计算其p(x)Δx的总和。D落在x1和x2之间某处的概率，我们可以写成P(x1<D<x2)，它等于图6-8中的阴影的面积。我们所选的增量Δx越小，我们的结果就越准确。因此我们就有：

Fig. 6–8.The probability that thedistance D traveled in a random walk is between x1 and x2 is the area under the curve of p(x) from x1 to x2 . 图 6-8 在一个从x1到x2的任意行走中，所走的距离D的概率，就是从x1到x2间的p(x)曲线下的面积

The area under the whole curve is the probabilitythat D lands somewhere (that is, has some value between x=−∞ and x=+∞ ). That probability is surely 1 . We must have that
整个曲线下面的面积，就是D落在某处的概率（也就是说，在x=−∞ 和 x=+∞之间的某个值）。这个概率当然是1。我们应该就有：
Since the curves in Fig. 6–7 getwider in proportion to N−−√ , their heights must be proportional to 1/N−−√ to maintain the total area equal to 1 .
由于图6-7中的曲线，与根号N成正比，随之增大而变宽，所以，它们的高度就应该与1/（根号N）成正比，以保持整个面积等于1。

The probability density function we havebeen describing is one that is encountered most commonly. It is known as the normalor Gaussian probability density. It has the mathematical form
我们一直在讲的概率密度公式，是最经常会遇到的公式。通常又被称为常规概率密度或高斯概率密度。其数学形式为：

where σ is called the standard deviation and is given, in our case,by σ=N−−√ or, if the rms step size is different from 1 , by σ=N−−√Srms .
这里σ被称为标准偏差，在我们这种情况，它是通过σ=根号N，而被给予的，或者，如果rms步长的大小，并不是1，那么就是通过σ=根号N*Srms，而被给予。

We remarked earlier that the motion of amolecule, or of any particle, in a gas is like a random walk. Suppose we open abottle of an organic compound and let some of its vapor escape into the air. Ifthere are air currents, so that the air is circulating, the currents will alsocarry the vapor with them. But even in perfectly still air, the vaporwill gradually spread out—will diffuse—until it has penetrated throughout theroom. We might detect it by its color or odor. The individual molecules of the organicvapor spread out in still air because of the molecular motions caused bycollisions with other molecules. If we know the average “step” size, and thenumber of steps taken per second, we can find the probability that one, orseveral, molecules will be found at some distance from their starting pointafter any particular passage of time. As time passes, more steps are taken andthe gas spreads out as in the successive curves of Fig. 6–7. In alater chapter, we shall find out how the step sizes and step frequencies arerelated to the temperature and pressure of a gas.
早先我们曾经说过，气体中的一个分子或任何粒子的运动，就像是一个随机行走。假设我们打开一瓶有机化合物，让其蒸汽跑到空气中。如果有气流，那么，空气就会循环传播，这个气流将会带着这个蒸汽。但是，即使在完全静止的空气中，这个蒸汽也会逐渐扩散—弥漫，直至完全渗透整个房间。我们可以通过颜色和气味，来探测它。独立的有机蒸汽的分子，会在静止的空气中扩散，乃是因为，与其他分子的碰撞，会带来分子运动。如果我们知道平均“步伐”的大小，以及每秒所走的步数，那么，我们就可以找到，一个或数个分子，在任何具体的时间段之后，距离起始点某一距离的概率。随着时间的流逝，会走更多的步，而气体的扩散，也就正如图6-7中成功的曲线一样。在后面的一章中，我们将会找出，步伐的大小和频率，与气体的温度和压力，有何关系。

We often think of the curve of Fig. 6–9 in asomewhat different way. If we consider the molecules in a typical container (witha volume of, say, one liter), then there are a very large number N of molecules present (N≈1022 ). Since p(v)Δv is the probability that one molecule will have its velocityin Δv , by our definition of probability we mean that the expectednumber ⟨ΔN⟩ to be found with a velocity in the interval Δv is given by
⟨ΔN⟩=Np(v)Δv. (6.21)
We call Np(v) the “distribution in velocity.” The area under the curve between twovelocities v1 and v2 , for example the shaded area in Fig. 6–9,represents [for the curve Np(v) ] the expected number of molecules with velocities between v1 and v2 . Since with a gas we are usually dealing with large numbers ofmolecules, we expect the deviations from the expected numbers to be small(like 1/N−−√ ), so we often neglect to say the “expected” number, and say instead:“The number of molecules with velocities between v1 and v2 is the area under the curve.” We should remember, however, thatsuch statements are always about probable numbers.
通常我们是以一种不同的方式，来思考图6-9中的曲线。如果我们考虑一个典型容器中的分子（比如说体积为一升），那么，就会有大量的数以N计的分子数在场(N≈1022 )。由于对于一个分子，p(v)Δv是其速度在Δv之间的概率，那么，通过我们关于概率的定义，我们所意味的就是，待发现的、速度在Δv之间的、被期待的数目 ⟨ΔN⟩，它通过下式给出：
⟨ΔN⟩=Np(v)Δv. (6.21)
我们称Np(v)为“在速度中的分布”。曲线下、速度v1和v2之间的面积，例如图6-9中的阴影面积，（[对于曲线 Np(v)]）代表着速度在v1和v2之间的被期待分子数。由于对一种气体来说，我们总是要面对大量的分子数，我们期待，距离被期待的数目的偏差，应该小（比如1/(根号)），于是，我们通常会由于疏忽，不是说“被期待”的数目，而是说：“速度在v1和v2之间的分子数目，就是曲线下的面积。”然而我们应该记住，这种声明所关于的，只是可能的数目。

6–5The uncertainty principle 6-5 测不准原理
The ideas of probability are certainlyuseful in describing the behavior of the 1022 or so molecules in a sample of a gas, for it is clearly impracticaleven to attempt to write down the position or velocity of each molecule. When probabilitywas first applied to such problems, it was considered to be a convenience—away of dealing with very complex situations. We now believe that the ideas ofprobability are essential to a description of atomic happenings.According to quantum mechanics, the mathematical theory of particles, there isalways some uncertainty in the specification of positions andvelocities. We can, at best, say that there is a certain probability that anyparticle will have a position near some coordinate x .
一个样本气体中，大概有1022个分子，在描述这样一个样本的表现时，概率的想法，当然有用，因为，要记录每个分子的位置和速度，显然很不现实。当概率最初被应用到这种问题上时，它被认为是很方便的，即是一种处理这种非常复杂的情况的方法。我们现在相信，概率概念，对于描述原子{事件}的发生，是非常本质的。依据量子力学，粒子的数学理论，在对位置和速度的说明中，总有不确定性。我们至少可以说，任何粒子，处于坐标x处，是有一定概率的。