Abstract
Entropy and the second law of thermodynamics are fundamental concepts that underlie all natural processes and patterns. Recent research has shown how the entropy of a landscape mosaic can be calculated using the Boltzmann equation, with the entropy of a lattice mosaic equal to the logarithm of the number of ways a lattice with a given dimensionality and number of classes can be arranged to produce the same total amount of edge between cells of different classes. However, that work seemed to also suggest that the feasibility of applying this method to real landscapes was limited due to intractably large numbers of possible arrangements of raster cells in large landscapes. Here I extend that work by showing that: (1) the proportion of arrangements rather than the number with a given amount of edge length provides a means to calculate unbiased relative configurational entropy, obviating the need to compute all possible configurations of a landscape lattice; (2) the edge lengths of randomized landscape mosaics are normally distributed, following the central limit theorem; and (3) given this normal distribution it is possible to fit parametric probability density functions to estimate the expected proportion of randomized configurations that have any given edge length, enabling the calculation of configurational entropy on any landscape regardless of size or number of classes. I evaluate the boundary limits (4) for this normal approximation for small landscapes with a small proportion of a minority class and show it holds under all realistic landscape conditions. I further (5) demonstrate that this relationship holds for a sample of real landscapes that vary in size, patch richness, and evenness of area in each cover type, and (6) I show that the mean and standard deviation of the normally distributed edge lengths can be predicted nearly perfectly as a function of the size, patch richness and diversity of a landscape. Finally, (7) I show that the configurational entropy of a landscape is highly related to the dimensionality of the landscape, the number of cover classes, the evenness of landscape composition across classes, and landscape heterogeneity. These advances provide a means for researchers to directly estimate the frequency distribution of all possible macrostates of any observed landscape, and then directly calculate the relative configurational entropy of the observed macrostate, and to understand the ecological meaning of different amounts of configurational entropy. These advances enable scientists to take configurational entropy from a concept to an applied tool to measure and compare the disorder of real landscapes with an objective and unbiased measure based on entropy and the second law.
Highlights
The second law of thermodynamics is the governing principle of nature, and entropy is the central concept of the second law
This disjunction is the primary motivation for this special issue on the linkage between entropy and landscape ecology, and the purpose of this paper is to advance a recent line of work that formalizes the entropy concept in landscapes [2], and develops the theory of how to measure it [11,12]
The goal of this paper is to of this paper evaluate each of these four assertions, demonstrate they of hold a sample of of evaluate eachisoftothese four assertions, demonstrate that they hold for athat sample realfor landscapes real landscapes of varying dimensionality, patch richness and evenness of cover type proportionality, varying dimensionality, patch richness and evenness of cover type proportionality, and provide and provide equations to that allow the researchers to calculateof the normal associated distribution of equations that allow researchers calculate normal distribution microstates with microstates associated with any landscape directlyoftolandscape enable rapid calculation landscape entropy any landscape directly to enable rapid calculation entropy withoutofthe need to conduct without randomization the need to conduct onerous randomization analyses
Summary
The second law of thermodynamics is the governing principle of nature, and entropy is the central concept of the second law. A recent review of the usage and application of the entropy concept in landscape ecology [1] unequivocally showed that landscape ecology has largely ignored thermodynamic theory, concepts or methods. This disjunction is the primary motivation for this special issue on the linkage between entropy and landscape ecology, and the purpose of this paper is to advance a recent line of work that formalizes the entropy concept in landscapes [2], and develops the theory of how to measure it [11,12]. The primary objective of this paper is to provide a practical analytical approach to measure and compare the entropies of real landscapes
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