Entropy production in an energy balance Daisyworld model J. G. Dyke University of Sussex, Brighton, East Sussex, UK, BN1 9QH [email protected] Abstract that certain energetically open and driven systems, such as planetary atmospheres, are in states that maximise the rate of Daisyworld is a simple mathematical model of a planetary system that exhibits self-regulation due to the nature of feed- entropy production. It is straightforward to introduce ther- back between life and its environment. A two-box Daisy- modynamic constraints into the two-box Daisyworld such world is developed that shares a number of features with en- that maximum entropy production can be achieved. If the ergy balance climate models. Such climate models have been Earth’s atmosphere is in a non-equilibrium state that maxim- used to explore the hypothesis that non-equilibrium, dissipa- ses entropy production then it seems reasonable to ask what tive systems such as planetary atmospheres are in a state of maximum entropy production with respect to the latitudinal would the effects on Daisyworld regulation and stability be flux of heat. When values for heat diffusion in the two-box if it were in a MEPP state? Daisyworld are selected in order to maximize this rate of en- tropy production, the viability range of the daisies is maxi- mized. Consequently planetary temperature is regulated over Daisyworld the widest possible range of solar forcing. Daisyworld is an imaginary grey planet orbiting a star sim- ilar to the Sun. It is home to two daisy types: black and Introduction white. Albedo is a measure of the reflectivity of an object. Although not intended as such, Daisyworld can be regarded In Daisyworld the black daisies have a low albedo (0.25), as an example of an artificial life model. It develops a the grey bare earth intermediate albedo (0.5) and the white conceptual framework that explores life and environment as daisies a high albedo (0.75). The white daisies, having the they could be on a planet similar to the Earth. Daisyworld highest albedo in the model, reflect more of the short wave was originally proposed as a mathematical proof of con- energy from the star and so have a lower temperature than ei- cept for James Lovelock’s Gaia Hypothesis: that the Earth ther the grey planet or the black daisies. The same applies to and its biota are a self-regulating system that will reduce the black daisies but in reverse. The black and white daisies the effects of otherwise deleterious perturbations (Lovelock, share a viability range of temperature. They are only able to 1995). Since its creation some twenty five years ago (Love- grow when the local ambient temperature is within 5-40 de- lock, 1983) Daisyworld has been extended and modified in grees Celsius. Within this range growth rates of the daisies order to address a number of different research questions. vary, with optimum growth being achieved when the temper- See (Wood et al., 2008) for a recent review. The consensus ature is 22.5C. Simulations begin when the star is dim and is that it has matured to the extent that it can be considered the temperature of the planet is below 5C. As the star in- as a model in its own right rather than as an ancillary com- creases in brightness the planetary temperature reaches 5C ponent of Gaia theory. A central message from Daisyworld and black daisies begin to grow. The increase in the num- is that when life affects the environment as well as being ber of black daisies is further increased via a feedback loop: affected by the environment, self-regulation may emerge. more black daisies leads to a lower planetary albedo and so Modelling of the Earth’s climate can take place at various more energy is absorbed that warms the ground that leads to levels of complexity, from zero-dimensional models such as an increase of the growth rate. This feedback loop is reg- the original Daisyworld to three-dimensional general circu- ulated by the parabolic growth rate of the daisies. As the latory models that are very computationally expensive and temperature increases past 22.5C, the daisy growth rate de- defy exact analysis. A relatively recent development of a creases. At steady state the ambient temperature, growth two-box Daisyworld (Harvey, 2004) is conceptually simi- rate and death rate are at equilibrium. As the energy output lar to one-dimensional energy balance climate models that of the star continues to increase, coverage in black daisies have been used to explore the Maximum Entropy Produc- decreases and white daisies begin to grow. This initiates a tion Principle (MEPP) (Lorenz et al., 2001). It is proposed feedback loop that is the inverse of the effect of the black Artificial Life XI 2008 189 daisies and so decreases their temperature. Again, this feed- leznev, 2006) to place the MEPP on firm information theo- back loop is regulated by the parabolic growth rate of the retic foundations. See (Ozawa et al., 2003) for a review. daisies. Increasing the amount of energy from the star results in a progressive increase in white daisies (and decrease in Two-Box Daisyworld black daisies) until the maximum coverage of white daisies The two-box Daisyworld was motivated by a desire to pro- is reached. Any further increase in energy takes the ambi- duce the simplest implementation of the Daisyworld control ent temperature past the point where growth rates balance system. Whilst the original Daisyworld was itself a simple death rates and so the coverage of white daisies decreases. model of complex real-world phenomena, analysis was not This leads to a rapid collapse of white daisies similar in na- trivial. The two-box model simplifies Daisyworld further by ture to the population explosion of the black daisies. The removing space competition dynamics by having each daisy differential coverage of white and black daisies results in a occupy separate ‘beds’ or boxes and dispensing with a sep- system that effectively regulates ambient planetary temper- arate birth and death rate and finding steady state daisy cov- ature to within the viability range. Whereas the tempera- erage with a single linear function. Each grey box is seeded ture of a bare lifeless planet would increase in an approx- with either black or white daisies. When the seeds are dor- imately linear fashion with increases in luminosity, when mant, the boxes have the same temperature. As the seeds black and white daisies are present, ambient temperature re- germinate and daisies begin to cover each box, a temper- mains within the viability range over a wide range of solar ature gradient is established. The two-box Daisyworld is forcing. represented schematically in Figure 1. The Maximum Entropy Production Principle The Earth’s atmosphere can be viewed as a heat engine: mo- tions are driven by the flow of heat from the hot equator to Star the cold polar regions. The amount of work that can be done by this flow of heat depends on the temperatures of the reser- Eb Ew I voirs: the greater the drop in temperature for a given amount b Iw of heat flow, the greater the thermodynamic efficiency of the system and so a greater amount of work output and entropy produced. Such heat processes do not produce entropy at an arbitrary rate. Two extreme principles have been formu- Black Daisy Box F White Daisy Box lated to describe their characteristic behaviour. For systems near thermodynamic equilibrium with fixed boundary con- ditions, (Prigogine, 1962) formulated the principle of min- I - E - F = 0 I - E + F = 0 imum entropy production (MinEP) stating that the steady b b w w state of the process is associated with a MinEP state. How- ever, many processes do not have fixed boundary conditions Figure 1: Schematic of Two-Box Daisyworld and are far from equilibrium. For those processes it has been The two-box Daisyworld which is conceptually similar to a proposed that they maintain steady states in which the pro- two-box energy balance climate model (North et al., 1981). duction of entropy is maximized if there are sufficient de- Incoming energy in the form of short-wave energy Ii warms grees of freedom associated with the processes - the Maxi- each box and is radiated back into space as long-wave emis- mum Entropy Production Principle (MEPP). It may not be sions Ei with an amount of heat flux F proportional to the necessary to understand the detailed internal dynamics of temperature gradient between the boxes. Only black daisies such systems in order to make accurate models and predic- are seeded in the black daisy box and only white daisies in tions. An impressive demonstration of this was with (Pal- the white daisy box. White daisies, being lighter than the tridge, 1975) who successfully reproduced the Earth’s latitu- black daisies will reflect more energy from the star. Black dinal temperature profile in a simple energy balance climate daisies, being darker, absorb more energy. Hence the black model by assuming that the rate of diffusion was that which daisies are warmer and the white daisies are cooler than the maximized the rate of entropy production via latitudinal heat grey bare earth. The two boxes are coupled via a heat con- transport. However, in the absence of any proposed mech- ducting medium which when there is non-zero coverage of anism or other system that maximized entropy production daisies allows heat to flow from the black to white daisy box. in this manner, Paltridge’s results failed to gain significant traction within the scientific community.