Cushman Samuel A
USDA Forest Service, Rocky Mountain Research Station, Flagstaff, AZ 86004, USA.
Entropy (Basel). 2021 Oct 28;23(11):1420. doi: 10.3390/e23111420.
There has been a recent surge of interest in theory and methods for calculating the entropy of landscape patterns, but relatively little is known about the thermodynamic consistency of these approaches. I posit that for any of these methods to be fully thermodynamically consistent, they must meet three conditions. First, the computed entropies must lie along the theoretical distribution of entropies as a function of total edge length, which Cushman showed was a parabolic function following from the fact that there is a normal distribution of permuted edge lengths, the entropy is the logarithm of the number of microstates in a macrostate, and the logarithm of a normal distribution is a parabolic function. Second, the entropy must increase over time through the period of the random mixing simulation, following the expectation that entropy increases in a closed system. Third, at full mixing, the entropy will fluctuate randomly around the maximum theoretical value, associated with a perfectly random arrangement of the lattice. I evaluated these criteria in a test condition involving a binary, two-class landscape using the Cushman method of directly applying the Boltzmann relation (s = klogW) to permuted landscape configurations and measuring the distribution of total edge length. The results show that the Cushman method directly applying the classical Boltzmann relation is fully consistent with these criteria and therefore fully thermodynamically consistent. I suggest that this method, which is a direct application of the classical and iconic formulation of Boltzmann, has advantages given its direct interpretability, theoretical elegance, and thermodynamic consistency.
最近，人们对景观格局熵的计算理论和方法兴趣激增，但对这些方法的热力学一致性却知之甚少。我认为，要使这些方法中的任何一种在热力学上完全一致，它们必须满足三个条件。首先，计算出的熵必须符合熵随总边长变化的理论分布，库什曼表明这是一个抛物线函数，原因在于排列后的边长呈正态分布，熵是宏观状态下微观状态数的对数，而正态分布的对数是抛物线函数。其次，在随机混合模拟期间，熵必须随时间增加，这符合封闭系统中熵增加的预期。第三，在完全混合时，熵将围绕最大理论值随机波动，这与晶格的完美随机排列相关。我在一个涉及二元、两类景观的测试条件下，使用库什曼方法评估了这些标准，该方法是将玻尔兹曼关系（s = klogW）直接应用于排列后的景观构型并测量总边长的分布。结果表明，直接应用经典玻尔兹曼关系的库什曼方法完全符合这些标准，因此在热力学上完全一致。我认为，这种直接应用玻尔兹曼经典且具有代表性公式的方法，因其直接的可解释性、理论的简洁性和热力学一致性而具有优势。
