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Calculating the Configurational Entropy of a Landscape Mosaic

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Calculating the Configurational Entropy of a Landscape Mosaic Landscape Ecol (2016) 31:481–489 DOI 10.1007/s10980-015-0305-2 PERSPECTIVE Calculating the configurational entropy of a landscape mosaic Samuel A. Cushman Received: 15 August 2014 / Accepted: 29 October 2015 / Published online: 7 November 2015 Ó Springer Science+Business Media Dordrecht (outside the USA) 2015 Abstract of classes and proportionality can be arranged (mi- Background Applications of entropy and the second crostates) that produce the observed amount of total law of thermodynamics in landscape ecology are rare edge (macrostate). and poorly developed. This is a fundamental limitation given the centrally important role the second law plays Keywords Entropy Á Landscape Á Configuration Á in all physical and biological processes. A critical first Composition Á Thermodynamics step to exploring the utility of thermodynamics in landscape ecology is to define the configurational entropy of a landscape mosaic. In this paper I attempt to link landscape ecology to the second law of Introduction thermodynamics and the entropy concept by showing how the configurational entropy of a landscape mosaic Entropy and the second law of thermodynamics are may be calculated. central organizing principles of nature, but are poorly Result I begin by drawing parallels between the developed and integrated in the landscape ecology configuration of a categorical landscape mosaic and literature (but see Li 2000, 2002; Vranken et al. 2014). the mixing of ideal gases. I propose that the idea of the Descriptions of landscape patterns, processes of thermodynamic microstate can be expressed as unique landscape change, and propagation of pattern-process configurations of a landscape mosaic, and posit that relationships across scale and through time are all the landscape metric Total Edge length is an effective governed and constrained by the second law of measure of configuration for purposes of calculating thermodynamics (Cushman 2015). This direct linkage configurational entropy. to thermodynamics and entropy was noted in several Conclusions I propose that the entropy of a given of the pioneering works in the field of landscape landscape configuration can be calculated using the ecology (e.g. Forman and Godron 1986; O’neill et al. Boltzmann equation. Specifically, the configurational 1989). Yet in the subsequent decades our field has entropy can be defined as the logarithm of the number largely failed to embrace and utilize these relation- of ways a landscape of a given dimensionality, number ships and concepts. Vranken et al. (2014) presented an overview of the use of entropy in landscape ecology and identified & S. A. Cushman ( ) three main uses of the entropy concept in past USDA Forest Service, Rocky Mountain Research Station, 2500 S. Pine Knoll Dr., Flagstaff, AZ 86001, USA landscape ecology research, including spatial hetero- e-mail: [email protected] geneity, unpredictability of pattern dynamics, and 123 482 Landscape Ecol (2016) 31:481–489 pattern dependence on scale. They conclude from their thermodynamic microstate can be expressed as unique review that thermodynamic interpretations of spatial configurations of a landscape mosaic, and propose that heterogeneity in the literature are not relevant, that the landscape metric Total Edge length is an appro- thermodynamic interpretations related to scale depen- priate measurement with which to define thermody- dence are highly questionable, and that, of all appli- namic microstates of a landscape mosaic. I posit that cations of entropy in landscape ecology, only the entropy of a given landscape configuration can be unpredictability could be thermodynamically relevant calculated using Boltzmann’s equation for entropy if appropriate measurements were performed to test it. once the number of ways a landscape of that dimen- In an editorial that was published concurrently to sionality, number of classes and proportionality can be the Vranken et al. (2014) paper, I noted that the near arranged (microstates) that produce the observed absence of thermodynamic focus from the landscape amount of total edge (macrostate). ecology literature is particularly strange given the focus of landscape ecology on understanding pattern process relationships across scale and through time Landscape configurations and entropy and the fundamental importance of entropy and the second law of thermodynamics to understanding in There are n! unique ways to arrange the cells of a natural sciences (e.g. Atkins 1992; Craig 1992). Every landscape represented as a lattice of n cells. This does interaction between entities leads to irreversible not mean, however, that there will be n! unique change which increases the entropy and decreases landscape configurations. For example, if the land- the free energy of the closed system in which they scape is all of one class there is only 1 possible reside and descriptions of landscape patterns, pro- configuration, because every one of the n! arrange- cesses of landscape change, propagation of pattern- ments of the cells produce the same landscape pattern process relationships across scale and through time are of a single uniform patch of class 1. In a two class all governed, constrained, and in large part directed by landscape with class 1 amounting to proportion p and thermodynamics. This direct linkage to thermody- class 2 proportion q of the landscape, there will be n!/ namics and entropy was noted in several of the ((np)!(nq)!) unique configurations of the landscape pioneering works in the field of landscape ecology pattern. Each of these unique configurations will occur (e.g. Forman and Godron 1986; O’neill et al. 1989), (np)!(nq)! times. This can be generalized to a but in the subsequent decades research has largely landscape of any number of classes by adding (nxi)! abandoned this framework and focus, as demonstrated terms for each additional class where xi is the by the Vranken et al. (2014) review (but see Li 2000, proportion of the landscape in the ith class. 2002; Zaccarelli et al. 2013). The most basic measure of the entropy of a Given how central entropy and the second law of landscape could be called the ‘‘compositional thermodynamics are to understanding nature, land- entropy’’ which is equivalent to the entropy of mixing scape ecologists should strenuously explore the con- ideal gases (Craig 1992). Boltzmann entropy defines nections between landscape ecological concepts, entropy of a system as the natural logarithm of the theories and methods and entropy (Cushman 2015). number of microstates in that macrostate multiplied by In particular, there is a critical need to define the the Boltzmann constant, which scales the relationship configurational entropy of landscape mosaics to define to the ideal gas laws, giving units of Joules per Kelvin. a benchmark with which to compare changes in A classic example is temperature gradients in a fluid landscape patterns and to associate these patterns with causing heat to flow from hotter regions to colder ones, processes across space and time. by the random movement of particles. In the modern In this paper I attempt small step in linking literature the concept embodied in Boltzmann entropy landscape ecology to the second law of thermody- is often used in a more general sense and refers to any namics and the entropy concept by showing how the kinetic equation that describes the change of a configurational entropy of a landscape mosaic may be macroscopic quantity in a thermodynamic system, calculated. I begin by drawing parallels between the such as energy, charge or particle number. In configuration of a categorical landscape mosaic and landscape ecology this macroscopic could be defined the mixing of ideal gases. I propose that the idea of the as the arrangement of the cells of a landscape mosaic. 123 Landscape Ecol (2016) 31:481–489 483 In calculating the entropy of a landscape no scaling entropy of mixtures of ideal gasses is calculated (Craig constant is needed, since the system is not an ideal gas, 1992). In the case of ideal gases one can know the and entropy may be calculated as simply the natural proportions of each molecular species in the mixture, logarithm of the number of microstates in a given but it is infeasible to measure the unique states of macrostate, producing a unitless measure of entropy arrangement themselves (positions and velocities of (Eq. 1). Thus the entropy of a landscape is defined as: the individual gas molecules). As such, the entropy equation measures the disorder of the system given a S ¼ lnðÞ W ; ð1Þ land particular number and proportion of different molec- where W is the number of arrangements of a landscape ular species. (microstates) with the dimensionality, number of classes and proportionality of the focal landscape that produce the same value for the macrostate. Number of Calculating the configurational entropy arrangements in this case refers to the number of ways of a landscape mosaic the cells of a given dimensionality can be arranged. Dimensionality of a landscape refers to the number of In contrast to gaseous mixtures, it is readily possible to cells in that landscape. Number of classes refers to the measure the characteristics of particular landscape number of different types, or classes of cells, such as configurations. That is the purpose of landscape pattern cover types in a land cover mosaic. analysis and there are an array of landscape metrics Given this definition of compositional entropy of a available to quantify the configurational characteristics landscape we can readily calculate the entropy of of landscape mosaics (e.g. Neel et al. 2004; Cushman landscapes of different dimensions, numbers of et al. 2008). Quantitative measurement of landscape classes and different proportions of each class configuration enables us to define additional macro- (Eq. 1). We can imagine the compositional entropy states corresponding to unique landscape configura- as a function of the number of microstates in a tions. Indeed, distinguishing the patterns of different macrostate defined by the dimensionality of a land- landscapes and understanding the implications of these scape, the number of classes of a landscape and the patterns for particular ecological processes is the proportion of e ach class.
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