# Research Notes, Autumn 2013

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.

# Extending the Second Law of Thermodynamics

\label{sec:t2}

## Alternative Reservoirs

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.

## Reversible Computation and Maxwell’s Demon

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.

## Misc

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.

# Notes on DNA

• 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).

# Some Relevant Papers

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

Maxwell’s Demon

Entropy estimation

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?.

• Jaynes (1980)