Abstract
In previous publications I have suggested that opposite thermodynamic arrows of time could coexist in our universe. This letter responds to the comments of H.D. Zeh (elsewhere in this volume).
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Notes
- 1.
The idea of having opposite arrows has been taken up by other authors as well. Wiener [12] speculated on this subject, and Creswick [13] has looked into the possibility physically producing systems that in a sense evolve backward in time. Finally there have been recent works of science fiction [14, 15] that explored some of the consequences of these ideas.
- 2.
The time separation used in any particular two-time problem depends on what is being studied. For looking at Gold’s proposal I generally think of the earlier time as being (approximately) the era of recombination, when our present cosmic background radiation was emitted. In a time-symmetric cosmology, I take the other time to be a corresponding time interval before the big crunch (or oscillation minimum). These are times for which matter should be distributed (roughly) uniformly, representing an entropy maximum for a system dominated by short-range forces. As the universe expands and gravitational forces dominate, uniformity becomes an extremely unlikely circumstance, so that what was a maximum becomes a minimum. This justifies two-time low-entropy boundary conditions. For the opposite-arrow boundary value problem, I have in mind a smaller time interval (later and “earlier” than recombination) and regions of space smaller than the entire universe. Finally, for the quantum problems associated with finding “special states”. my time range is before and after the operation of a particular apparatus.
- 3.
The numbers given here differ slightly from those in [8]. Here I use the physically more realistic statistics of indistinguishable particles. These numbers also reflect more detailed state counting. Specifically, for an ideal gas the number of states is \(\mathcal{N}=\exp(S/k)=\exp (N\log{ [{(V/N)}e^{5/2}/{\lambda_{\mathrm{th}}^{3}}]})\), using the standard expression [21] for the entropy of an ideal gas in three dimensions. N is computed from the pressure using PV=NkT, k is the Boltzmann constant, and \(\lambda_{\mathrm{th}}= h/ {\sqrt{2\pi m k T}}\) is the thermal wavelength. The formula for \(\mathcal{N}\) can also be used directly to see the effect of volume changes, as discussed in the text.
- 4.
The terms “apparatus” and “system” do not imply that this scheme holds only for laboratory experiments. Any situation that could lead to superpositions of macroscopically different states will have this feature. Again, the present paper is not about quantum measurement theory. and for the many questions that may come to mind please consult [8].
- 5.
Postscript: I’ve long been suspicious of the alleged paradoxes that would arise in a Gödel universe by virtue of its closed timelike curves. I expect that there could be a reduction in the class of “initial value” problems that have a solution, as for other paradoxes mentioned in this article. (“Initial values” would also be final values and would presumably be on a single spacelike surface. They would involve test particles, not the matter giving rise to the metric itself.) Also, the usual paradoxes are macroscopic, implying the existence, at least locally, of an arrow of time. It’s not clear that such could exist. In the summer of 2010 I met another person with similar ideas about this problem, Noam Erez of the Weizmann Institute and my comments here are partly informed by our conversations.
- 6.
Bear in mind that this has meaning only with respect to a particular coarse graining.
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Acknowledgements
This work was supported by the United States National Science Foundation Grant PHY 00 99471.
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Schulman, L.S. (2014). Two-Way Thermodynamics: Could It Really Happen?. In: Albeverio, S., Blanchard, P. (eds) Direction of Time. Springer, Cham. https://doi.org/10.1007/978-3-319-02798-2_20
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