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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?

Qmechanic
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sayan chattopadhyay
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2 Answers2

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I'll give you a complete mathematical answer at the simpler level of symbolic dynamics. Proved by Ornstein and Weiss:

  1. 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 $(i_1i_2\cdots)\in\Sigma^+$;

  2. 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.

John B
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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.

Cornelius Klots
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