How are information and chaos defined and measured in entropy theory?

Entropy. entropy

Describes one of the important state functions of a thermodynamic system. The magnitude of entropy reflects the stability of the state of the system, and the change of entropy indicates the direction in which the thermodynamic process proceeds, and entropy provides a quantitative expression for the second law of thermodynamics.

In order to quantitatively state the second law of thermodynamics, we should look for a state function that remains constant in a reversible process and changes monotonically in an irreversible process. When Clausius studied the Carnot heat engine, according to Carnot's theorem, he concluded that for any cyclic process, there is , where q is the tiny heat absorbed by the system from a heat source with temperature t, and the equal and unequal signs correspond to reversible and irreversible processes respectively. Reversible cyclic indicates the existence of a state function entropy, defined as:

For the adiabatic process q 0, so s 0, that is, the entropy of the system does not change in the reversible adiabatic process, and increases monotonically in the irreversible adiabatic process. This is the principle of entropy increase. Since all changes in an isolated system have nothing to do with the outside world and must be adiabatic processes, the principle of entropy increase can also be expressed as:

The entropy of an isolated system never decreases. It shows that the entropy of an isolated system increases monotonically as it moves from a non-equilibrium state to an equilibrium state, and reaches a maximum when the system reaches an equilibrium state. The change in entropy and the maximum value determine the direction and limit of the process of an isolated system, and the principle of entropy increase is the second law of thermodynamics.

Energy is a measure of the movement of matter, in various forms, and can be converted into each other. A form of energy, such as more internal energy, indicates greater potential for conversion. The literal meaning of entropy is transformation, which describes the direction and degree to which the transformation is completed by the spontaneous transformation of internal energy with other forms of energy.

As the transition progresses, the system tends to equilibrium and the entropy becomes larger, indicating that while the total energy remains the same during this process, there is less and less energy available for utilization or conversion. Internal energy, entropy and thermodynamics.

The first and second laws provide a comprehensive and complete understanding of the basic characteristics of the energy conversion process associated with thermal motion.

Microscopically speaking, entropy is a measure of the disorder of a large number of microscopic particles that make up a system, and the more disordered and chaotic the system, the greater the entropy. The microscopic nature and statistical significance of the irreversibility of thermodynamic processes is that the system tends from ordered to disordered, from a state with a low probability to a state with a high probability.

In information theory, entropy can be used as a measure of the uncertainty of an event. The greater the amount of information, the more regular the architecture, and the more complete the functionality, the less entropy. Using the concept of entropy, the measurement, transmission, transformation, and storage of information can be studied theoretically.

In addition, entropy also has certain applications in cybernetics, probability theory, number theory, astrophysics, life sciences and other fields.

Can entropy be measured manually?

It shows that the greater the uncertainty of the matter. The higher the entropy, the greater the uncertainty of the event. The greater the uncertainty of the information, the higher the value of that information.

It is calculated as follows: h = p(x) log (1 p(x)), where p(x) is the probability. Layman's Explanation:

0. The units are compared, such as 3kg, and the letter bird is compared with 1kg, so the most basic uncertainty is defined as the standard.

1. The unit of entropy is bitbits, and the most basic definition of the rough situation is: do two equal probability events.

This level of uncertainty is defined as 1bits, i.e., log2.

3. Then the probability is obviously 1 4, then 4 1 times (reciprocal) will only appear once, and the entropy is log4=2.

4. The probability is 2 5, and it will only appear once if you do 5 2 times, and the entropy is log5 2.

Popular explanation: 1. Represents the usefulness of information, the more useful information entropy.

The bigger it is, the more negative it is, and if I say something that doesn't affect others, it can affect myself.

2. Represents the compression size of information, if there is repetition in a paragraph, removing the repetition is equivalent to compression, and the limit of this compression is information entropy.

The greater the information entropy, the higher the nucleus of the component of the flickering, and the more it is said, the more the uncertainty of the information increases.

The information entropy is very small, and the problem is straightforward, and it can be explained in one or two sentences.

Understand the amount of information and the entropy of information in one minute.

In information theory, entropy is used to measure the expected value of the occurrence of a random variable. It represents the amount of information lost during signal transmission before it is received, also known as information entropy. Information entropy is also known as source entropy and average self-information.

Information entropy: It is used to measure the uncertainty of an event, the greater the uncertainty, the greater the entropy.

For any random variable x, its entropy is defined as:

Conditional entropy: There are two random variables x and y, and the magnitude of the uncertainty of x after the y event is determined is called the conditional difference return entropy. Conditional entropy is defined as follows:

Mutual informationThe function of information is to eliminate the uncertainty of events, and mutual information is used as a measure of the correlation between two events (x,y), that is, the amount of information provided to eliminate the uncertainty of the other event x on the premise that one of the events y is determined. Mutual information is defined as follows:

In the figure above, the red circle represents the entropy h(x) of event x, the blue circle represents the entropy of event y, the area of the two circles together represents the joint entropy h(x, y), and the solid color represents the conditional entropy (minus the reduction in entropy due to a known event), and the intersection of the intersection in the middle i(x; y) indicates the correlation of events x and y.

Relative entropy (i.e., kl divergence).It is important to note that the kl divergence is asymmetrical, i.e

Reference: The Beauty of Mathematics by Wu Jun.

Information is a very abstract concept. People often say that there is a lot of information, or that there is little information, but it is difficult to say exactly how much information there is. For example, Yirong.

How much information is there in this 500,000-word Chinese book.

It was not until 1948 that Shannon proposed the concept of "information entropy" that the problem of quantitative measurement of information was solved. The word information entropy was borrowed from thermodynamics by e Shannon. Thermoentropy in thermodynamics is a physical quantity that indicates the degree of confusion in the state of a molecule.

Shannon uses the concept of information entropy to describe the uncertainty of a source.

Claude Elwood Shannon, the father of information theory, was the first to elucidate the relationship between probability and information redundancy in mathematical language.

Father of information theory C e.In his 1948 book, "A Mathematical Theory of Communication," Shannon pointed out that there is redundancy in any piece of information, and the magnitude of the redundancy is related to the probability or uncertainty of each symbol (number, letter, or word) in the message.

Drawing on the concept of thermodynamics, Shannon called the average amount of information without redundancy "information entropy" and gave a mathematical expression for calculating information entropy.

Understand the amount of information and the entropy of information in one minute.

Not contradictory. Both the premise is omitted in the previous and latter sentences, which are as follows:

1. When the ordered state of the system is consistent, the entropy value of the data is smaller, and the more scattered the data, the larger the entropy value. This is described in terms of the completeness of the information.

2. When the amount of data is consistent, "the more orderly the system, the lower the entropy value; The more chaotic or decentralized the system, the higher the entropy value. This is described in terms of the orderliness of the information.

There is no necessary relationship between orderliness and data volume, and it cannot be said that the larger the data volume, the worse or better the orderliness, nor can it be said that the more orderly the system, the larger or smaller the data volume. These two are two different levels of description, so they are not contradictory.

In physics, entropy entrophy is a measure of the degree of "chaos", so "the more ordered the system, the lower the entropy; The more chaotic or decentralized the system, the higher the entropy value.

In information theory, information is negative entropy. Therefore, the more concentrated the "information data", the smaller the entropy, and the more dispersed the "information data", the greater the entropy. There is no contradiction between the two.

However, it should be noted that the amount of information is not a physical quantity, and the information data is measured in binary binary, which is not the same concept as the decimal measure of physics.

Entropy is the quotient of heat energy divided by temperature, which indicates the degree to which heat is converted into work. In information theory, entropy is used to represent a measure of uncertainty and disorganization, and Shannon believes that information is the reduction of uncertainty, that is, less entropy, and information growth is a process of inverse entropy (which is not possible in thermodynamics). This is the explanation of the latter sentence, I don't know where the source of the previous sentence is, I have to read it in context to understand.

I think that the more centralized the data, the more complete the information and less uncertainties. However, the information of network data is also different from the traditional understanding, such as Bozeman's belief that information should lead to action, and news is a kind of entertainment more than information.

Information is a very abstract concept. We often say that there is a lot of information, or there is little information, but it is difficult to say exactly how much information there is. How much information is more than a 500,000-word Chinese book.

Entropy represents a physical quantity (denoted s) of the state of a material system, which indicates the degree to which that state is likely to occur. In thermodynamics, it is a relatively abstract physical quantity used to explain the irreversibility of thermal processes. The processes that actually take place in an isolated system necessarily increase its entropy.