One of the ideas involved in the concept of entropy is that nature tends from order to disorder in isolated systems. But we even know that Poincare recurrence time also is a particular time after which a system of particles get back to their original position,and entropy is how can a system of particles be arranged. So are these two related?
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$\begingroup$ Isn't Poincare recurrence time essentially the time it takes for the entropy decrease function of a system to occur? $\endgroup$– V.H. BelvadiCommented Nov 15, 2014 at 8:31
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$\begingroup$ Also, the order-to-disorder analogy is not accurate; in other words, entropy decrease is not prohibited in nature. Statistically, the likelihood is so small that we nonchalantly claim it never occurs. However, it can and will occur when the Poincare recurrence time is infinitely long, so to speak. The familiar box with gas molecules is the simplest example of this. After time, there is a small chance that the molecules will bunch up, decreasing entropy, the way they started. $\endgroup$– V.H. BelvadiCommented Nov 15, 2014 at 8:43
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$\begingroup$ Is there any mathematical way to represent this?? $\endgroup$– sayan chattopadhyayCommented Nov 15, 2014 at 10:05
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2 Answers
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I'll give you a complete mathematical answer at the simpler level of symbolic dynamics. Proved by Ornstein and Weiss:
if $\sigma^+|\Sigma^+$ is a one-sided topological Markov chain and $\mu^+$ is an ergodic $\sigma^+$-invariant probability measure on $\Sigma^+$, then $$ \lim_{k\to\infty} \frac{\log\inf\{n\in\mathbb N:(i_{n+1}\cdots i_{n+k})=(i_1\cdots i_k)\}}{k}=h_{\mu^+}(\sigma^+) $$ for $\mu^+$-almost every $(i_1i_2\cdots)\in\Sigma^+$;
if $\sigma|\Sigma$ is a two-sided topological Markov chain, and $\mu$ is an ergodic $\sigma$-invariant probability measure in~$\Sigma$, then $$ \lim_{k\to\infty} \frac{\log\inf\{n\in\mathbb N:(i_{n-k}\cdots i_{n+k})=(i_{-k}\cdots i_k)\}}{2k+1}=h_\mu(\sigma) $$ for $\mu$-almost every $(\cdots i_{-1}i_0i_1\cdots)\in\Sigma$.
For the general case, I recommend the book "Dimension and Recurrence in Hyperbolic Dynamics", by Barreira.
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4$\begingroup$ Could you define some of the terms used here? This is almost impossible to parse unless one is already familiar with the subject. $\endgroup$– RococoCommented Feb 15, 2018 at 18:08
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$\begingroup$ Could you please be more specific? The terms (not the results) in my answer are really at the rudiments of dynamics (which is the canonical area of Poincaré recurrence). Because of this, my view (apparently of 2016 already) is that one either knows the basic terms or one really needs to study them with time. Still, I will try my best if you tell me exactly what is missing. $\endgroup$– John BCommented Feb 15, 2018 at 18:37
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3$\begingroup$ That may be, but you should not assume that your reader is familiar with symbolic dynamics. What is $h$- is it the entropy? What does it mean to be $\sigma$ invariant? What is a topological Markov chain and how is it relevant to the question? What in your answer relates to a recurrence time? $\endgroup$– RococoCommented Feb 15, 2018 at 22:55
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Yes, entropy and recurrence times are related, indeed proportionally so. The logarithm of the entropy of a well-defined (eg at a single temperature) system contains the density-of-states, the number of quantum levels in the vicinity of the average energy. Entropy increase accompanies an increase in that density- i.e. more states for the system to get lost in. Meanwhile, the recurrence time is also proportional to that same density of states.
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$\begingroup$ Spontaneous drops in entropy (disorder) seem plausible, what with disorder being as erratic as it's usually considered to be, and account for the possibility (described by Columbia University's Brian Green) that drops in the cosmological constant (dark energy), as envisaged by Leonard Susskind, might "neuter" Poincare recurrence, & thereby incorporate the recurrences (which are not regularly periodic) more consistently with quantum cosmology. (See p.382-384 in Greene's 2020 book titled "Until the End of Time".) $\endgroup$– EdouardCommented May 27, 2022 at 4:45