Arrow of time derived from Markov like model?

[Gerenuk]

I want to find a solid argument why entropy always increases and even more important why it is
$$S=\sum_i p_i \ln p_i$$
I've seen some more or less sophisticated arguments. What I'd find most convincing is an argument based on general converging Markov processes, that show that the above expression will increase.
For example for a state vector $$\textbf{A}=(p_0,p_1,...)$$
$$\textbf{A}_n=\textbf{M}^n\textbf{A}_0$$
$$\therefore S(\textbf{A}_{n+1})>S(\textbf{A}_n)$$
under special circumstances.

Does such kind of argument exist?

 

[Aaron Bex]

Yes, such an argument does exist. The argument is based on the idea of a Markov process, which is a mathematical system that models how a system changes over time. A Markov process is defined as a series of states (or "states of the system") and transitions between those states that are determined by a set of probabilities. In this case, the probabilities are determined by the probability distribution of the initial state vector \textbf{A}_0.

In such a Markov process, the entropy of the system will always increase with each iteration. This is because, for any given transition in the Markov process, the probability of moving from one state to another is always less than or equal to the probability of staying in the same state. This means that, over time, the system will become more and more likely to stay in its current state and therefore the entropy of the system will increase. In other words, the entropy of the system increases as the probability of the system staying in the same state increases.

The mathematical proof for this is based on the fact that the entropy of the system is given by the expression S=\sum_i p_i \ln p_i. Since the probability of any given transition is always less than or equal to the probability of staying in the same state, it follows that p_i \ln p_i is always greater than or equal to 0. This means that the sum of these

 

[charng]

The concept of the arrow of time and the increase of entropy are fundamental principles in the study of thermodynamics and statistical mechanics. The arrow of time refers to the one-directional flow of time, from past to future, which is observed in our universe. Entropy, on the other hand, is a measure of the disorder or randomness in a system, and it is known to always increase with time.

The relationship between the arrow of time and entropy has been a subject of debate and research for many years. One possible explanation for the increase of entropy is through the use of a Markov-like model. Markov processes are stochastic processes where the future state depends only on the current state and not on the past states.

In this context, it is possible to show that the entropy of a system will always increase over time. This can be seen through the use of a state vector, such as \textbf{A}=(p_0,p_1,...), and a transition matrix \textbf{M}. The state vector represents the probabilities of the system being in each state, and the transition matrix represents the probabilities of transitioning from one state to another.

Through the use of a general converging Markov process, it can be shown that the entropy of the system, given by the expression S=\sum_i p_i \ln p_i, will always increase over time. This is because as the system evolves, the probabilities of being in each state will change, leading to a change in the entropy of the system.

While there may be other arguments for the increase of entropy, the use of a Markov-like model provides a solid and convincing explanation. This is because it is based on the fundamental principles of thermodynamics and statistical mechanics, and it can be mathematically proven under certain conditions. Therefore, the argument based on general converging Markov processes offers a strong and compelling explanation for the increase of entropy and the arrow of time.