# From Maximum Entropy to Maximum Entropy Production: A New Approach

## Abstract

**:**

## 1. Introduction

## 2. Background: The Maximum Entropy Production Principle and Some Open Problems

#### 2.1. An Example: Two-Box Atmospheric Models

**Figure 1.**A diagram showing the components of the two-box atmospheric heat transport model. Labels written next to arrows indicate rates of flow of energy, whereas labels in boxes represent temperature.

#### 2.2. Generalisation: Negative Feedback Boundary Conditions

#### 2.3. The System Boundary Problem

#### 2.4. Some Comments on Dewar’s Approach

## 3. Thermodynamics as Maximum Entropy Inference

#### 3.1. A New Argument for the MEPP

#### 3.2. Application to the Steady State with a Fixed Gradient

## 4. A Possible Solution to the System Boundary Problem

**Figure 2.**Two possible ways in which negative feedback boundary conditions could be realised. $\left(a\right)$ Heat flows into $\mathrm{A}$ from a much larger reservoir ${\mathrm{A}}^{\prime}$, and out of $\mathrm{B}$ into another large reservoir ${\mathrm{B}}^{\prime}$. $\left(b\right)$ A reversible heat engine is used to extract power by transferring heat between two external reservoirs ${\mathrm{A}}^{\u2033}$ and ${\mathrm{B}}^{\u2033}$, and this power is used to transfer heat reversibly out of $\mathrm{B}$ and into $\mathrm{A}$. In this version the only entropy produced is that produced inside $\mathrm{C}$ due to the heat flow Q.

#### 4.1. Application to Atmospheres and Other Systems

## 5. Discussion

#### 5.1. The Need for Experimental Study

#### 5.2. The Relationship Between Thermodynamics and Kinetics

## 6. Conclusions

## Acknowledgements

## References

- Kleidon, A.; Lorenz, R.D. Entropy production by earth system processes. In Non-Equilibrium Thermodynamics and The Production of Entropy: Life, Earth, and Beyond; Kleidon, A., Lorenz, R.D., Eds.; Springer Verlag: Berlin, Germany, 2005. [Google Scholar]
- Paltridge, G.W. Climate and thermodynamic systems of mximum dissipation. Nature
**1978**, 279, 630–631. [Google Scholar] [CrossRef] - Lorenz, R.D.; Lunine, J.I.; Withers, P.G. Titan, Mars and Earth: entropy production by latitudinal heat transport. Geophys. Res. Lett.
**2001**, 28, 415–151. [Google Scholar] [CrossRef] - Dewar, R.C. Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states. J. Phys. A: Math. Gen.
**2003**, 36, 631–641. [Google Scholar] [CrossRef] - Dewar, R.C. Maximum entropy production and the fluctuation theorem. J. Phys. A: Math. Gen.
**2005**, 38, 371–381. [Google Scholar] [CrossRef] - Martyushev, L.M.; Seleznev, V.D. Maximum enrtopy production principle in physics, chemistry and biology. Phys. Rep.
**2006**, 426, 1–45. [Google Scholar] [CrossRef] - Attard, P. Theory for non-equilibrium statistical mechanics. Phys. Chem. Chem. Phys.
**2006**, 8, 3585–3611. [Google Scholar] [CrossRef] [PubMed] - Niven, R.K. Steady state of a dissipative flow-controlled system and the maximum entropy production principle. Phys. Rev. E
**2009**, 80, 021113. [Google Scholar] [CrossRef] - Zupanovic, P.; Botric, S.; Juretic, D. Relaxation processes, MaxEnt formalism and Einstein’s formula for the probability of fluctuations. Croatia. Chemica. Acta.
**2006**, 79, 335–338. [Google Scholar] - Essex, C. Radiation and the irreversible thermodynamics of climate. J. Atmos. Sci.
**1984**, 41, 1985–1991. [Google Scholar] [CrossRef] - Jaynes, E. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Jaynes, E. Information theory and statistical mechanics II. Phys. Rev.
**1957**, 108, 171–190. [Google Scholar] [CrossRef] - Jaynes, E. Gibbs vs Boltzmann entropies. Am. J. Phys.
**1965**, 33, 391–398. [Google Scholar] [CrossRef] - Jaynes, E. Where do we stand on maximum entropy? In The Maximum Entropy Formalism; Levine, R.D., Tribus, M., Eds.; MIT Press: Cambridge, MA, USA, 1979. [Google Scholar]
- Jaynes, E. The minimum entropy production principle. Ann. Rev. Phys. Chem.
**1980**, 31, 579–601. [Google Scholar] [CrossRef] - Jaynes, E. Macroscopic prediction. In Complex Systems – Operational Approaches; Haken, H., Ed.; Springer-Verlag: Berlin, Germany, 1985. [Google Scholar]
- Jaynes, E. Probability Theory: the Logic of Science; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Paltridge, G.W. Stumbling into the MEP racket: an historical perspective. In Non-equilibrium Thermodynamics and the Produciton of Entropy; Kleidon, A., Lorenz, R.D., Eds.; Springer Verlag: Berlin, Germany, 2005; Chapter 3. [Google Scholar]
- Odum, H.T.; Pinkerton, R.C. Time’s speed regulator: The optimum efficiency for maximum output in physical and biological systems. Am. Sci.
**1955**, 43, 331–343. [Google Scholar] - Jaynes, E. Prior Probabilities. IEEE Trans. Syst. Sci. Cybern.
**1968**, 4, 227–241. [Google Scholar] [CrossRef] - Shimokawa, S.; Ozawa, H. Thermodynamics of the ocean circulation: A global perspective on the ocean system and living systems. In Non-equilibrium Thermodynamics and the Produciton of Entropy; Kleidon, A., Lorenz, R.D., Eds.; Springer Verlag: Berlin, Germany, 2005; Chapter 10. [Google Scholar]

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**MDPI and ACS Style**

Virgo, N. From Maximum Entropy to Maximum Entropy Production: A New Approach. *Entropy* **2010**, *12*, 107-126.
https://doi.org/10.3390/e12010107

**AMA Style**

Virgo N. From Maximum Entropy to Maximum Entropy Production: A New Approach. *Entropy*. 2010; 12(1):107-126.
https://doi.org/10.3390/e12010107

**Chicago/Turabian Style**

Virgo, Nathaniel. 2010. "From Maximum Entropy to Maximum Entropy Production: A New Approach" *Entropy* 12, no. 1: 107-126.
https://doi.org/10.3390/e12010107