These are some ideas stimulated by my research; I write them to make sure I think about / understand certain questions.

Some sections will be quite rough / unintelligible.

\label{sec:t2}

I want to extend the Second Law of Thermodynamics, to include not only heat and information reservoirs, but also reservoirs of other conserved quantities such as angular momentum (see Vaccaro et al. (2011)). What are the implications for the Second Law? Can we derive new statement of the Second Law from a Hamiltonian framework, as in Deffner et al. (2013)?

AG says such ideas are ‘pathological’ – they are not in the spirit of statistical mechanics, which never considers angular momentum.

What would be the consequences of *reversible computation* on the form of the Second Law given in Deffner et al. (2013)? A reversible computation could involve no information erasure and hence no heat dissipation. What would the consequences be for Maxwell’s Demon?

According to Landauer, computer states evolve irreversibly (many-to-one processes). This means the “informative degrees of freedom” (IDF) entropy decreases. BUT, why are the IBF so important? They aren’t physically meaningful, only meaningful to us. The *physical* quantity would surely be the entropy of all the DoF combined. So, the entropy of IDF is said to increase because the operation is irreversible, but could this be a value judgement?

See here and here for reversible computing info.

I don’t see how you can *not* lose energy when erasing a bit – even for reversible computation, surely you just dissipate twice as much heat as before (if you go forwards then backwards)? The path is important, not just the end states.

Erasing information is irreversible. Full cycle of MD requires erasing any information that the demon wrote down. **Bennet 1982 – quantify information**. Thermodynamic entropy different from information theory entropy, as latter invokes no temperature or reservoir.

More reading: Mandal et al. (2013)

What is the effect of renormalising on the work of the Demon? On the nature of information?

Does it cost to *process* information? The answer can be surmised from the realisations of MD.

\label{sec:gibbs}

All this work has been moved to a new document, which can be found at URL https://www.authorea.com/users/1329/articles/2725/_show_article.

Introns versus exons. Subject to different selection pressures hence different statistics. Farach etal. 1994, Schmitt 1997.

Introns pasted together from disparate chunks.

Most naive assessment of entropy finds difference between exons and introns (Knopa and Owens 1989).

A list interesting / useful papers, with synopses and thoughts.

**Maxwell’s Demon**

Hosoya et al. (2011) MD and data compression

Strasberg et al. (2013) model the connection between Maxwell’s Demon to the system it interferes with. Interesting for section \ref{sec:t2}, but also may hold clues to how to quantify information accuracy etc.

Shizume (1995) - Landauer microscopic derivation

Piechocinska (2000) – Landauer microscopic derivation

**Gibbs’ Paradox**

Versteegh et al. (2011) makes case against “indistingusihable particle” invocation in classical theory. Repudiates claims of unpleasant consequences.

Bérut et al. (2012) perform an experimental test of Landauer’s principle using a colloid in a double optical well. They find agreement with bound on heat generation.

**Entropy estimation**

Schmitt and Herzel 1997 – DNA

**Information theory**

Ostrowski (2010) suggests that the minimum energy to (reversibly) copy one bit of information is \(\ln4/\beta\). Uses a quantum system, and assumes low signal-to-noise.

Bennett (2003) – reversible computation, Landauer

**Foundations**

Jaynes (1965) discusses the Boltzmann and Gibbs entropy functions (defined in terms of \(6N\)-dimensional distribution functions). \(H_{\rm G}\) seems to be more general, though more difficult to manage, as it accounts for interactions between particles limiting the weight in some regions of phase space. \(H_{\rm G}\) is shown to be correct after all. It will agree with experimental values, up to an additive constant. Section IV: volume of “reasonably probable” phase space is independent of “reasonably” in the thermodynamic limit. Upshot: \(H_{\rm G}\) corresponds to Boltzmann’s equation, at least in the thermodynamic limit. Entropy defined as \(S=k\ln W\) is a generalised entropy, applicable to nonequilibrium states; it comes from Liouville’s theorem and doesn’t need any canonical distributions etc. Really interesting insights about anthropomorphic entropy – not a property of a physical system, but of the experiments we choose to perform.

*What is the specific question we are trying to answer?*.

J. Bergli, Y. M. Galperin, N. B. Kopnin. Information flow and optimal protocol for Maxwell’s demon single electron pump.

*ArXiv e-prints*(2013).S. Deffner, C. Jarzynski. Information processing and the second law of thermodynamics: an inclusive, Hamiltonian approach.

*ArXiv e-prints*(2013).H. Ge, H. Qian. Maximum Entropy Principle, Equal Probability a Priori and Gibbs Paradox.

*ArXiv e-prints*(2011).V. Ihnatovych. Study of the possibility of eliminating the Gibbs paradox within the framework of classical thermodynamics.

*ArXiv e-prints*(2013).V. Ihnatovych. The logical foundations of Gibbs’ paradox in classical thermodynamics.

*ArXiv e-prints*(2013).C.-L. Lee, Y.-R. Chen. Gibbs paradox and a possible mechanism of like-charge attraction in colloids.

*ArXiv e-prints*(2012).D. Mandal, H. T. Quan, C. Jarzynski. Maxwell’s Refrigerator: An Exactly Solvable Model.

*ArXiv e-prints*(2013).M. Ostrowski. Minimum energy required to copy one bit of information.

*ArXiv e-prints*(2010).R. Padyala. A note on Gibbs paradox.

*ArXiv e-prints*(2013).H. Peters. Demonstration and resolution of the Gibbs paradox of the first kind.

*ArXiv e-prints*(2013).Dmitri V. Averin, Mikko Möttönen, Jukka P. Pekola. Maxwells demon based on a single-electron pump.

*Physical Review B***84**American Physical Society, 2011. LinkCharles H. Bennett. Notes on Landauers principle, reversible computation, and Maxwells Demon.

*Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics***34**, 501-510 Elsevier, 2003. LinkAntoine Bérut, Artak Arakelyan, Artyom Petrosyan, Sergio Ciliberto, Raoul Dillenschneider, Eric Lutz. Experimental verification of Landauer’s principle linking information and thermodynamics.

*Nature***483**, 187-189 Nature Publishing Group, 2012. LinkDavid S. Corti. Comment on “The Gibbs paradox and the distinguishability of identical particles,” by M. A. M. Versteegh and D. Dieks [Am. J. Phys. 79, 741–746 (2011)].

*American Journal of Physics***80**, 170 American Association of Physics Teachers, 2012. LinkHui Dong, ChengYun Cai, ChangPu Sun. A quantum solution to Gibbs paradox with few particles.

*Science China Physics, Mechanics and Astronomy***55**, 1727-1733 Springer-Verlag, 2012. LinkYun Gao, Ioannis Kontoyiannis, Elie Bienenstock. Estimating the Entropy of Binary Time Series: Methodology, Some Theory and a Simulation Study.

*Entropy***10**, 71-99 MDPI AG, 2008. LinkAkio Hosoya, Koji Maruyama, Yutaka Shikano. Maxwells demon and data compression.

*Physical Review E***84**American Physical Society, 2011. LinkE. T. Jaynes. Gibbs vs Boltzmann Entropies.

*American Journal of Physics***33**, 391 American Association of Physics Teachers, 1965. LinkE T Jaynes. The Minimum Entropy Production Principle.

*Annual Review of Physical Chemistry***31**, 579-601 Annual Reviews, 1980. LinkV. P. Maslov. Mixture of new ideal gases and the solution of the Gibbs and Einstein paradoxes.

*Russian Journal of Mathematical Physics***18**, 83-101 Pleiades Publishing, 2011. LinkV. P. Maslov. Solution of the gibbs paradox using the notion of entropy as a function of the fractal dimension.

*Russian Journal of Mathematical Physics***17**, 288-306 Pleiades Publishing, 2010. LinkHjalmar Peters. Statistics of Distinguishable Particles and Resolution of the Gibbs Paradox of the First Kind.

*Journal of Statistical Physics***141**, 785-828 Springer-Verlag, 2010. LinkBarbara Piechocinska. Information erasure.

*Physical Review A***61**American Physical Society, 2000. LinkKousuke Shizume. Heat generation required by information erasure.

*Physical Review E***52**, 3495-3499 American Physical Society, 1995. LinkPhilipp Strasberg, Gernot Schaller, Tobias Brandes, Massimiliano Esposito. Thermodynamics of a Physical Model Implementing a Maxwell Demon.

*Physical Review Letters***110**American Physical Society, 2013. LinkJ. A. Vaccaro, S. M. Barnett. Information erasure without an energy cost.

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences***467**, 1770-1778 The Royal Society, 2011. LinkMarijn A. M. Versteegh, Dennis Dieks. The Gibbs paradox and the distinguishability of identical particles.

*American Journal of Physics***79**, 741 American Association of Physics Teachers, 2011. Link