Tell the story of probability (1)

Entropy and information are both concepts that fascinate me in modern science, and both are centered on the concept of probability. In order to avoid the article being too long, I intend to summarize the related concepts and personal thoughts into four parts.

Let's talk about the first part: let's talk about randomness first.

Random phenomena are considered unpredictable, but if you think about it carefully, one day you will definitely think of a question: Are random things really random? For example, the classic coin toss, if all the initial conditions are well controlled, the result is that the reaction should be controllable. I've seen artists play dice on TV and throw whatever they want.

Let's look at other random phenomena. In World of Warcraft, the blade drop rate of Essinos is 4.5 1%. How is this achieved? The general process (because I don't know the specific algorithm) may be simple: when you overthrow Illidan, the server assigns you a random number with the interval of [0, 1]. If it is less than 0.0455438+0, you can see the equipment at the moment you pick it up.

However, all computers, whether the earliest paper tape machine or today's PC, do the same thing: accept input->; Calculation-> output, so the computer will not "figure out" the real random number, it can only be realized by algorithm. Here is a video (you may need to climb over the wall, and if you can watch it, you can ignore the rest of this paragraph) which can explain it well. There are many algorithms, among which the linear congruence method is commonly used. The example in the video is the "middle square" algorithm invented by von Neumann in 1949. The process is simple: first, find an initial value (we call it random number seed, which is generally difficult to predict, for example, the current time of a minute is several milliseconds, and there seems to be an error in the video, which should be 183 milliseconds in the first minute of 9: 00: 03: 03, but it is not important), and the video takes seed =. Step 2, square the initial value to get 29929, and take the middle three numbers 992. The third step is to square 29929 to get 89574504 1, then take the middle three numbers 745 and repeat. This will produce a sequence: 992745857. ...

So if you can get the seed, then you can predict this sequence, so we say it is pseudo-random. The seed here is time, and once you have an event that can make the program run, this randomness is completely shattered. That's why an ancient rocking machine is still used in today's lottery.

Interestingly, we can deduce other randomness from one randomness. For example, the pseudo-random sequence just mentioned is actually a random number table. If mapping f: random number table-> is specified; Students in the class, so that random roll call can be realized.

So how do we generate random numbers in the range we want from the random counting table? Let's look at the practice of C language.

There are two main functions related to it in C language: srand (seed) and rand(void). The function of srand () is just like the example just now, accepting a seed and generating a random sequence. The rand () function is derived: the mapping from the random number table to the interval [0, RAND_MAX] (RAND_MAX is generally defined as 32767), so a random number 1 ~ 32767 is returned. Everything will be easy after that. For example, if you want to play dice, then rand () will make a remainder of 6. Because you make several remainders, there will be several patterns in the result, so it can be completely projected to our target random integer set. What if the target number set is a real number [0, 1]? Very simple, just divide rand () by RAND_MAX. This paragraph just wants to show that randomness can be derived from each other, and it is also a derivation in the process of computer simulation of random sampling, which is essential for Monte Carlo calculation.

Most of the space above is used to pull the computer, but actually my plan is not like this (? ▽? ) "... just want to make people think, how many things are really unpredictable? In fact, most so-called random events are not inherently random, just like tossing coins and dice. We say these events are random because we are lazy about these system information. Ordinary people don't spend time being a "dice artist", and you won't build machines (vacuuming to avoid air influence, automatic control by single chip microcomputer, etc.). ) to flip a coin "predictably".

In fact, in classical statistical physics, the so-called randomness is not essential, it comes from our ignorance of system information.

So is there any essential randomness? Under the current cognitive framework, the random phenomenon caused by the superposition principle of quantum mechanics can be considered to be random in nature. This principle means that a particle can be in the superposition of multiple microscopic States at the same time according to a certain probability distribution. When you measure it, it will collapse to an eigenstate according to the probability distribution, that is, the physical law itself requires it to be random. Therefore, it is very important that with the emergence of quantum mechanics, our understanding of "prediction" will change: * * predictability does not mean that we can give an accurate answer, but giving a probability distribution (wave function) is also a kind of prediction * *. However, this is only a question of definition. In order to facilitate communication, we should be as consistent as possible.

So in the later quantum statistics, we need to deal with the superposition of these two randomness at the same time. For convenience, we introduce the density matrix.

There is another situation-chaos. Chaos means that the law itself is predictable, but it is so sensitive to the initial conditions that we can't predict it under the existing (perhaps forever) technology-_-#. This is called the butterfly effect. The most typical population model logistic series:

When r is greater than about 3.5699, chaos appears, but there is a stable island. See Wikipedia for details. So, is chaos predictable or unpredictable? As a frontier, this sub-discipline still has great differences.

To sum up, most random phenomena are not inherently random. At present, only randomness caused by superposition principle in quantum mechanics is considered to be essential, while chaos remains to be discussed.

Next, let's talk about probability.