# The Maximum Entropy Production Principle: Its Theoretical Foundations and Applications to the Earth System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. What is Entropy and Entropy Production?

#### 2.1. Equilibrium States

**Figure 1.**Entropy at equilibrium in an isolated system. Diagrams A and B represent a rigid box that is isolated from the rest of the universe in that it is impermeable to energy and matter. A similarly impermeable partition bisects the box. To the left of the partition are a number of gas molecules. Diagram A shows the situation immediately after a dividing partition is removed (removed instantaneously and without disturbing any of the molecules). Given the particular properties of the individual gas molecules, there will be a particular number of ways that they can rearrange themselves so that all of them remain in the left hand side of the box. This number will be much less than the different arrangements of having an equal number of molecules in both sides of the box as shown in diagram B which is the expected distribution of molecules at equilibrium.

#### 2.2. Non-Equilibrium States

**Figure 2.**Entropy in non-isolated systems. A rigid box that contains a number of gas molecules is connected to a hot and cold reservoir. In diagram A, an insulating partition separates the hot reservoir and the box. At equilibrium the molecules will be in a state of maximum entropy. In diagram B the insulating partition is partially raised so that an amount of heat flows from the hot reservoir into the box. This sets up a temperature gradient which results in a decrease in the entropy of the gas molecules. The dissipation of heat gradients keeps the gas molecules away from the maximum entropy equilibrium state. This non-equilibrium state may be a steady state with respect to the temperatures of the hot and cold reservoirs and the configurational entropy of the gas molecules.

## 3. Entropy production and the MEP principle

**Figure 3.**The rate of change of entropy production of a system over time, is a function of the entropy produced within the system and the entropy that is imported and exported into its surroundings. If Reservoir 1 where hotter than Reservoir 2, there would be a flux of heat through the system from hot to cold. $NE{E}_{1}$ would import entropy into the system and $NE{E}_{2}$ would export entropy while σ would be determined by the temperature gradient and the rate of heat flux.

**Figure 4.**Two box climate model. A simple two box climate model is shown. The equator receives more energy from the sun (I for insolation) than the poles; ${I}_{e}>{I}_{p}$. Longwave emissions, E, are also larger; ${E}_{e}>{E}_{p}$. The difference in insolation sets up a temperature gradient where the equator is hotter than the poles; ${T}_{e}>{T}_{p}$. A certain amount of heat, F, flows over this gradient with a diffusivity term, D, parameterising how easily this heat flows polewards. Over decadal time scales, the Earth’s climate is in steady state: energy emitted equals energy absorbed.

#### 3.1. Ambiguities of and Objections to the MEP Principle

## 4. Science and Falsification

_{1}→ TT

_{1}→ EE

_{1}→ PS

_{2}

## 5. Information, Probability and Inference

#### 5.1. Inference and Information

## 6. MaxEnt and the MEP Principle

_{1}→ P

_{1}→ O

_{1}→ I

_{2}

## 7. MEP Principle and the Earth System

#### 7.1. Earth System Processes away from Thermodynamic Equilibrium

**Figure 5.**Illustration of how variables should change as a system is maintained further and further away from thermodynamic equilibrium. The state of thermodynamic equilibrium is characterized by global variables (e.g., $T,p,\rho $) that are constant in space and time. The further the system is maintained away from equilibrium, the more the state should be associated with larger and larger gradients in space and time. The characteristic spatial scale $\Delta x$ at which the assumption of thermodynamic equilibrium applies should therefore decrease correspondingly, resulting in local variables (illustrated by ${T}_{i},{p}_{i},{\rho}_{i}$).

#### 7.2. Increasing the Resolution of Earth System Models: a Practical Application of the MEP Principle

**Figure 6.**Illustration of (a) a global grid used in climate models. Such grids are used for a discrete representation of variables, such as temperature, and implicitly assume a state of thermodynamic equilibrium within the grid. Subgrid scale heterogeneity, as found for instance in form of pattern formation of vegetation found in semiarid regions (b), illustrate that subgrid scale processes can operate far from thermodynamic equilibrium. MEP could help to scale up subgrid scale heterogeneity so that this is adequately represented at the grid scale, as for instance shown by [22]. Photo credit: Stephen Prince.

## 8. Discussion

## Acknowledgements

## References

- Ozawa, H.A.; Ohmura, A.; Lorenz, R.D.; Pujol, T. The second law of thermodynamics and the global climate system: A review of the maximum entropy production principle. Rev. Geophys.
**2003**, 41, 1018. [Google Scholar] [CrossRef] - Martyushev, L.M.; Seleznev, V.D. Maximum entropy production principle in physics, chemistry and biology. Phys. Rep.
**2006**, 426, 1–45. [Google Scholar] [CrossRef] - Kleidon, A.; Lorenz, R.D. Non-Equilibrium Thermodynamics and the Production of Entropy: Life, Earth, and Beyond; Springer, Berlin, 2005; Chapter 1. [Google Scholar]
- Kleidon, A. Non-equilibrium thermodynamics and maximum entropy production in the Earth system: applications and implications. Naturwissenschaften
**2009**, 96, 635–677. [Google Scholar] [CrossRef] - Paltridge, G.W. The steady-state format of global climate systems. Quart. J. Royal Meterological Soc.
**1975**, 104, 927–945. [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, 415–418. [Google Scholar] [CrossRef] - Lorenz, R.D. Planets, life and the production of entropy. Int. J. Astrobiol.
**2002**, 1, 3–13. [Google Scholar] [CrossRef] - Dewar, R. Maximum entropy production and the fluctuation theorem. J. Phys. A
**2005**, 38, 371–381. [Google Scholar] [CrossRef] - Bumstead, H.A.; Van Name, R.G. (Eds.) The Scientific Papers of J. Willard Gibbs; Dover Publications: New York, NY, USA, 1961.
- Jellinek, A.M.; Lenardic, A.M. Effects of spatially varying roof cooling on Rayleigh-Benard convection in a fluid with temperature-dependent viscosity. J. Fluid Mech.
**2008**. in revision. [Google Scholar] - Dewar, R. Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states. J. Phys. A
**2003**, 36, 631. [Google Scholar] [CrossRef] - Dewar, R.C. Maximum Entropy Production and Non-Equilibrium Statistical Mechanics. In Non-equilibrium Thermodynamics and the Produciton of Entropy; Kleidon, A., Lorenz, R.D., Eds.; Springer: Berlin, Germany, 2005; pp. 41–53. [Google Scholar]
- Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics, II. Phys. Rev.
**1957**, 108, 171–190. [Google Scholar] [CrossRef] - Essex, C. Radiation and the irreversible thermodynamics of climate. J. Atmos. Sci.
**1984**, 41, 1985–1991. [Google Scholar] [CrossRef] - Popper, K.R. Conjectures and Refutations: The Growth of Scientific Knowledge; Routledge: Florence, KY, USA, 1963. [Google Scholar]
- Jaynes, E.T. Probability Theory: The Logic of Science; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- 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; p. 15. [Google Scholar]
- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Caticha, A. Information and entropy. In Bayesian Inference and Maximum Entropy Methods in Science and Engineering; Knuth, K.H., Ed.; AIP: New York, NY, USA, 2007; Volume 30. [Google Scholar]
- Caticha, A. From Information Geometry to Newtonian Dynamics. In Bayesian Inference and Maximum Entropy Methods in Science and Engineering; Knuth, K.H., Ed.; AIP: New York, NY, USA, 2007; Volume 30. [Google Scholar]
- Schymanski, S.J.; Kleidon, A.; Stieglitz, M.; Narula, J. Maximum Entropy Production allows simple representation of heterogeneity in arid ecosystems. Phil. Trans. Royal Soc. B: Biol. Sci.
**2009**, in press. [Google Scholar]

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license http://creativecommons.org/licenses/by/3.0/.

## Share and Cite

**MDPI and ACS Style**

Dyke, J.; Kleidon, A. The Maximum Entropy Production Principle: Its Theoretical Foundations and Applications to the Earth System. *Entropy* **2010**, *12*, 613-630.
https://doi.org/10.3390/e12030613

**AMA Style**

Dyke J, Kleidon A. The Maximum Entropy Production Principle: Its Theoretical Foundations and Applications to the Earth System. *Entropy*. 2010; 12(3):613-630.
https://doi.org/10.3390/e12030613

**Chicago/Turabian Style**

Dyke, James, and Axel Kleidon. 2010. "The Maximum Entropy Production Principle: Its Theoretical Foundations and Applications to the Earth System" *Entropy* 12, no. 3: 613-630.
https://doi.org/10.3390/e12030613