# Non-Equilibrium Statistical Mechanics Inspired by Modern Information Theory

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## Abstract

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## 1. Introduction

**What do we do?**—We calculate expressions for how much work one can extract in the context of a system, called the working medium, undergoing a transform of its state and Hamiltonian. You may think of the work as energy transferred to another system, the work reservoir, during the transform of the working medium system. We are in particular interested in deriving expressions for the optimal work given the initial and final conditions. This means the largest increase in the work reservoir we can attain or, in the case where work needs to be put in, the lowest decrease we can attain. We also consider cycles of such processes, like heat engines, as well as quantitative restrictions on how a system can change when it interacts with a heat-bath. We try to make as general statements as possible, assuming as little as possible about the initial and final conditions, and are now at a stage where neither the initial nor final state needs to be a thermal state and the initial and final Hamiltonians can be arbitrary. Accordingly, we can answer how much work a Maxwell’s daemon [7] type agent with insider information beyond that of the standard thermodynamical observer can extract from a system, as a function of its extra information. We can address that question because the information the agent has would be encoded in and thus represented by the state it assigns to the system. We also consider daemons with a quantum memory. In this case it is not the state of the system that represents the knowledge, as this is not well-defined, but rather the correlations in the joint system-memory system.

**Figure 1.**Example of a distribution $p\left(w\right)$ over energy w transferred to work-reservoir using some strategy $Str$. The distribution in this instance has two peaks. ${\langle W\rangle}_{Str}$ is the average of this distribution. ${W}_{Str}^{\epsilon}$ is the guaranteed work up to failure probability ε. ε is the probability of getting less work than ${W}_{Str}^{\epsilon}$. When distributions have a significant spread around the average, as is the case here, these two quantities can differ greatly. We argue ${W}_{Str}^{\epsilon}$ is a more useful quantity to know.

**Why?**—Let me begin with the motivation for considering the guaranteed work rather than the average. Suppose I apply for a job to lift boxes from the floor onto the table. The potential energy gain of the box, $mgh$ is the energy transferred to the work reservoir, assuming there is no additional kinetic energy. Now consider two different approaches I may use to lift the box: (I) I lift it in the natural way from the floor, just high enough so that it sits atop the table, succeeding say $99\%$ of the time; ($II$) I throw the box high in the air up to a higher shelf, succeeding say $1\%$ of the time, and otherwise crashing to the floor. Now actually these two approaches will, by a suitable choice of height of the table and shelf, give the same $\langle mgh\rangle $. I would suggest, however, that you would not find me a very useful lifter if I insist on using approach $II$. If, instead of using $\langle mgh\rangle $ as the measure of work, you used ${W}^{\epsilon}$, you could see that for example ${W}_{I}^{0.009}=mgh$ and ${W}_{II}^{0.009}=0$: the second approach suddenly looks very bad. This suggests that when there are significant fluctuations in the work output, the guaranteed work is more useful to know than the average work. In many physical scenarios, very small systems in particular, fluctuations of this type are indeed significant. Moreover thresholds like the table in the above example do appear frequently in nature, for example as activation energies and semi-conductor band-gaps. Distinguishing between guaranteed and average work in these systems is crucial.

## 2. Single-Shot Entropies (a.k.a Smooth Entropies)

#### 2.1. Min and Max Entropies

**Figure 2.**This depicts a probability distribution with support on 8 events and event 1 having the highest probability of occurring (with ${p}_{max}=0.4$). These two numbers are crucial aspects of a distribution more generally. Its support size may be called its width and the max probability its height. For some operational questions one matters and for some the other. The Shannon entropy cares about both, but the min entropy only cares about the height and the max entropy only cares about the width.

#### 2.2. Smoothing

**Figure 3.**Smoothing the max entropy amounts to taking the lowest probabilities of the distribution away, up until the point that their weights sum to ε, and then taking the max entropy of that new distribution. If there are many events with small probabilities the smooth entropy can thus be much lower than the non-smooth one.

#### 2.3. Conditional Entropy, Relative Entropy

## 3. Work Extraction Games: the Set-Ups under Which Optimal Work is Calculated

#### 3.1. The General Idea

#### 3.2. Single Szilard Engine

**Figure 4.**Szilard’s engine. There is a single particle in a box, and a heat bath at temperature T. The daemon/agent inserts a divider in the middle of the box, measures the position of the particle, L vs R, and hooks up the weight accordingly (or we may take the weight to always be on right, but give the agent the option of flipping the box). It can extract work isothermally.

#### 3.3. Multiple Szilard Engines

#### 3.4. Two-level Quantum System

**Figure 5.**Landauer-style erasure/resetting of the state of a two-level quantum system with the Hamiltonian H=0 both initially and finally. An optimal protocol is to raise the second level isothermally and quasi-statically towards infinity, at a cost of $kTln2$, followed by a decoupling from the heat bath and a lowering of the second level back to 0 at zero work cost or gain.

#### 3.5. More General Game on Many Energy Levels

#### 3.6. Expressions for Optimal Work

#### 3.7. Quantum Memory

- (1)
- Extract ${W}_{\text{out}}=2kTln2$ work from both Sys and M.
- (2)
- Reset Sys to $|0\rangle $ by using $W=kTln2$ work.

**Figure 6.**A unitary implementing Shor’s algorithm is implemented in circuit model computation. Not all qubits are measured in the end to get the output. We may extract work from correlations between the output qubits and the rest. After the proposed protocol the reduced state on the output qubits is invariant so the computation output is not affected. The energy extracted comes from the computer and its surroundings, so the computer is cooled.

#### 3.8. Conceptual Questions

## 4. Conclusions and Outlook

## Acknowledgments

## Conflicts of Interest

## References

- Dahlsten, O.C.O.; Renner, R.; Rieper, E.; Vedral, V. Inadequacy of von Neumann entropy for characterizing extractable work. New J. Phys.
**2011**, 13, 053015. [Google Scholar] [CrossRef] - Rio, L.; Aberg, J.; Renner, R.; Dahlsten, O.; Vedral, V. The thermodynamic meaning of negative entropy. Nature
**2011**, 474, 61–63. [Google Scholar] [CrossRef] [PubMed] - Aberg, J. Truly work-like work extraction. Nat. Commun.
**2013**, 4. [Google Scholar] [CrossRef] [PubMed] - Horodecki, M.; Oppenheim, J. Fundamental limitations for quantum and nano thermodynamics. Nat. Commun.
**2013**, 4. [Google Scholar] [CrossRef] [PubMed] - Egloff, D.; Dahlsten, O.C.O.; Renner, R.; Vlatko, V. Laws of thermodynamics beyond the von neumann regime. 2012; arXiv:1207.0434. [Google Scholar]
- Faist, P.; Dupuis, F.; Oppenheim, J.; Renner, R. A quantitative Landauer’s principle. 2012; arXiv:1211.1037. [Google Scholar]
- Maroney, O. Information processing and thermodynamic entropy. Available online: http://plato.stanford.edu/entries/information-entropy/ (accessed on 17 August 2013).
- Baldo, M. Introduction to Nanoelectronics; MIT OpenCourseWare, License; Creative Commons BY-NC-SA; Massachusetts Institute of Technology: Cambridge, MA, USA, 2012; Available online: http://ocw.mit.edu (accessed on 30 July 2013).
- Frank, M.P. Approaching the Physical Limits of Computing. In Proceedings of IEEE 35th International Symposium on Multiple-Valued Logic, Piscataway, NJ, USA, 19–21 May 2005; pp. 168–185.
- Jarzynski, C. Nonequilibrium equality for free energy differences. Phys. Rev. Lett.
**1997**, 78, 2690–2693. [Google Scholar] [CrossRef] - Lloyd, S. Quantum-mechanical Maxwell’s demon. Phys. Rev. A
**1997**, 56, 3374–3382. [Google Scholar] [CrossRef] - Gemmer, J.; Mahler, G. Quantum Thermodynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems; Lecture Notes in Physics; Springer: Berlin, Germany, 2004. [Google Scholar]
- Allahverdyan, A.E.; Balian, R.; Nieuwenhuizen, T.M. Maximal work extraction from finite quantum systems. Europhys. Lett.
**2004**, 67, 565–571. [Google Scholar] [CrossRef] - Linden, N.; Popescu, S.; Skrzypczyk, P. How small can thermal machines be? The smallest possible refrigerator. Phys. Rev. Lett.
**2010**, 105, 130401. [Google Scholar] [CrossRef] [PubMed] - Brandão, F.G.S.L.; Horodecki, M.; Oppenheim, J.; Renes, J.M.; Spekkens, R.W. The resource theory of quantum states out of thermal equilibrium. 2011; arXiv:1111.3882. [Google Scholar]
- Jennings, D.; Rudolph, T.; Hirono, Y.; Nakayama, S.; Murao, M. Exchange fluctuation theorem for correlated quantum systems. 2012; arXiv:1204.3571. [Google Scholar]
- Toyabe, S.; Sagawa, T.; Ueda, M.; Muneyuki, E.; Sano, M. Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nat. Phys.
**2010**, 12, 988–992. [Google Scholar] [CrossRef] - Renner, R. Security of quantum key distribution. 2005; arXiv:quant-ph/0512258. [Google Scholar]
- Cover, T.M.; Thomas, J.A. Elements of Information Theory; Wiley Series in Telecommunications and Signal Processing; Wiley: New York, NY, USA, 1991. [Google Scholar]
- Tomamichel, M.; Colbeck, R.; Renner, R. A fully quantum asymptotic equipartition property. IEEE Trans. Inf. Theory
**2009**, 55, 5840–5847. [Google Scholar] [CrossRef] - Datta, N. Min- and max-relative entropies and a new entanglement monotone. IEEE Trans. Inf. Theory
**2009**, 55, 2816–2826. [Google Scholar] [CrossRef] - König, R.; Renner, R.; Schaffner, C. The operational meaning of min- and max-entropy. IEEE Trans. Inf. Theory
**2009**, 55, 4337–4347. [Google Scholar] [CrossRef] - Szilárd, L. Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Zeitschrift für Physik
**1929**, 53, 840–856. (in German). [Google Scholar] [CrossRef] - Landauer, R. Irreversibility and heat generation in the computing process. IBM J. Res. Dev.
**1961**, 5, 183–191. [Google Scholar] [CrossRef] - Browne, C. EPSRC summer project, Oxford 2012, Discussions with J. Aberg. 2012. [Google Scholar]
- Zurek, W.H. Maxwell’s demon, Szilard’s engine and quantum measurements. 2003; arXiv:quant-ph/0301076. [Google Scholar]
- Bennett, C.H. Notes on Landauer’s principle, reversible computation, and Maxwell’s demon. Stud. Hist. Philos. Sci. B
**2003**, 34, 501–510. [Google Scholar] [CrossRef] - Alicki, R.; Horodecki, M.; Horodecki, P.; Horodecki, R. Thermodynamics of quantum information systems—Hamiltonian description. Open Syst. Inf. Dyn.
**2004**, 11, 205–217. [Google Scholar] [CrossRef] - McDiarmid, C. On the Method of Bounded Differences; London Mathematical Society Lecture Note Series 141; Oxford University: Oxford, UK, 1989; pp. 148–188. [Google Scholar]
- Donald, M. Free energy and the relative entropy. J. Stat. Phys.
**1987**, 49, 81–87. [Google Scholar] [CrossRef] - Ruch, E.; Mead, A. The principle of increasing mixing character and some of its consequences. Theor. Chim. Acta
**1976**, 41, 95–117. [Google Scholar] [CrossRef] - Ruch, E. The diagram lattice as a structural principle. Theor. Chim. Acta
**1975**, 38, 167–183. [Google Scholar] [CrossRef] - Mead, C.A. Mixing character and its application to irreversible processes in macroscopic systems. J. Chem. Phys.
**1977**, 66, 459–467. [Google Scholar] [CrossRef] - Gour, G.; Müller, M.P.; Narasimhachar, V.; Spekkens, R.W.; Yunger Halpern, N. The resource theory of informational nonequilibrium in thermodynamics. 2013; arXiv:1309.6586. [Google Scholar]

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Dahlsten, O.C.O.
Non-Equilibrium Statistical Mechanics Inspired by Modern Information Theory. *Entropy* **2013**, *15*, 5346-5361.
https://doi.org/10.3390/e15125346

**AMA Style**

Dahlsten OCO.
Non-Equilibrium Statistical Mechanics Inspired by Modern Information Theory. *Entropy*. 2013; 15(12):5346-5361.
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**Chicago/Turabian Style**

Dahlsten, Oscar C. O.
2013. "Non-Equilibrium Statistical Mechanics Inspired by Modern Information Theory" *Entropy* 15, no. 12: 5346-5361.
https://doi.org/10.3390/e15125346