[Paper Review] Compactified Time and likely Entropy -- World Inside Time Machine: Closed Time-like Curve --

H. B. Nielsen, Masao Ninomiya|ArXiv.org|Jan 9, 2006

Neural Networks and Applications3 references16 citations

TL;DR

This paper investigates a classical and quantum mechanical model of a universe with compactified time, where time is periodic with period $T$. By enforcing periodicity, the authors show that entropy is not maximized but instead distributed smoothly across macrostates, leading to a constant average entropy over time—implying that such a closed time-like curve (CTC) universe cannot naturally explain the second law of thermodynamics, which requires increasing entropy.

ABSTRACT

If a macroscopic (random) classical system is put into a random state in phase space, it will of course the most likely have an almost maximal entropy according to second law of thermodynamics. We will show, however, the following theorem: If it is enforced to be periodic with a given period $T$ in advance, the distribution of the entropy for the otherwise random state will be much more smoothed out, and the entropy could be very likely much smaller than the maximal one. Even quantum mechanically we can understand that such a lower than maximal entropy is likely. A corollary turns out to be that the entropy in such closed time-like loop worlds remain constant.

Motivation & Objective

• To explore whether a time-symmetric model with compactified time can explain the second law of thermodynamics.
• To investigate the statistical distribution of entropy in a system forced to be periodic with period $T$.
• To determine whether such a periodic system can support a non-maximal, constant entropy, contradicting the second law's prediction of increasing entropy.
• To examine the implications of time machine-like structures (closed time-like curves) for entropy and thermodynamic arrow of time.
• To assess whether periodic boundary conditions in time can serve as a foundation for a deeper understanding of thermodynamic irreversibility.

Proposed method

• The model enforces a fixed period $T$ on a classical or quantum mechanical system, treating time as a compact $S^1$ manifold.
• The system's time evolution is governed by a Hamiltonian $H$, and periodicity requires $e^{-iTH} = 1$, leading to quantized energy eigenvalues $E_n = 2ar{h}\pi n / T$.
• Macrostates are defined by a set of conserved quantities $\{I_i\}$, with entropy $S(\{I_i\}) = \log \dim \mathcal{H}_{\{I_i\}}$, where $\mathcal{H}_{\{I_i\}}$ is the Hilbert subspace of states with those quantum numbers.
• The analysis shows that only macrostates with $I_1^\prime = 2\pi n / T$ satisfy the periodicity condition, restricting the allowed states.
• The entropy is computed as the logarithm of the dimension of the subspace satisfying the periodicity constraint, leading to a smooth distribution of entropy across macrostates.
• Quantum mechanical treatment uses the Hamiltonian as a macroscopic variable, and the periodicity condition is derived from the time-translation operator $e^{-iTH}$.

Experimental results

Research questions

• RQ1 Can a time-symmetric model with compactified time naturally produce increasing entropy, as required by the second law of thermodynamics?
• RQ2 What is the statistical distribution of entropy in a system forced to be periodic with a fixed period $T$?
• RQ3 Does the imposition of periodicity in time lead to a suppression of maximal entropy states, and if so, why?
• RQ4 Can a closed time-like curve (CTC) universe exhibit a constant entropy over time, and what does this imply for the arrow of time?
• RQ5 Is the second law of thermodynamics compatible with time-reversal invariant laws when time is compactified?

Key findings

• The entropy in a system with enforced periodicity $T$ is not maximized; instead, it is distributed smoothly across macrostates, with no preference for maximal entropy.
• Only macrostates with energy eigenvalues $E_n = 2\pi n / T$ satisfy the periodicity condition, drastically restricting the allowed states.
• The time translation operator $e^{-iTH}$ acts as the identity on allowed states, confirming that the system returns to its initial state after time $T$, confirming the CTC structure.
• The entropy remains approximately constant over the time loop, with no net increase, contradicting the second law’s prediction of increasing entropy.
• The probability distribution of entropy is uniform across allowed macrostates, despite large variations in phase space volume $e^{S}$, indicating a smooth statistical behavior.
• The model fails to reproduce the second law of thermodynamics, suggesting a fundamental incompatibility between time-reversal invariance and the observed increase of entropy.