One of the biggest challenges in characterizing 2-D topographies is succinctly communicating the dominant nature of local configurations. In a 2-D grid composed of bistate units, this could be expressed as finding the characteristic configuration variables such as nearest-neighbor pairs and triplet combinations. The 2-D cluster variation method (CVM) provides a theoretical framework for associating a set of configuration variables with only two parameters, for a system that is at free energy equilibrium. This work presents a method for determining which of many possible two-parameter sets provides the ``most suitable'' match for a given 2-D topography, drawing from methods used for variational inference. This particular work focuses exclusively on topographies for which the activation enthalpy parameter (epsilon_0) is zero, so that the distribution between two states is equiprobable. This condition is used since, when the two states are equiprobable, there is an analytic solution giving the configuration variable values as functions of the h-value, where we define h in terms of the interaction enthalpy parameter (epsilon_1) as h = exp(2*epsilon_1). This allows the computationally-achieved configuration variable values to be compared with the analytically-predicted values for a given h-value. The method is illustrated using four patterns derived from three different naturally-occurring black-and-white topographies, where each pattern meets the equiprobability criterion. We achieve expected results, that is, as the patterns progress from having relatively low numbers of like-near-like nodes to increasing like-near-like masses, the h-values for each corresponding free energy-minimized model also increase. Further, the corresponding configuration variable values for the (free energy-minimized) model patterns are in approximate alignment with the analytically-predicted values.
翻译:2D 地形特征化的最大挑战之一是简洁地传达本地配置的主导性。 在由两州单位组成的 2D 网格中, 这可以表现为找到典型配置变量, 如近邻配对和三重组合。 2D 群集变异法( CVM) 提供了一个理论框架, 将一组配置变量与仅有两个参数挂钩, 用于一个处于自由能源平衡的系统。 这项工作提供了一个方法, 用以确定哪些可能的许多两平方的直径数据集提供给给给定的 2D 地形的“ 最合适的” 匹配值, 从用于变异性推断的规律模式中提取。 具体的工作只侧重于启动 enthalpy 参数( epsalon_0) 的地形变量, 这样两个州之间的分布是可配置的。 这个条件之所以被使用,是因为当两个州都具备了可配置的系统时, 有一种解析性解决方案, 将配置变量的值作为 h值的函数, 我们用三个交互值- 值值值值值值-, 将每平面值- 的直值值值值值值值值值值值值值值值值值值值值值比值值比值值值- 的直值值值值值值值值值值值值值值值值值值值值值值值值-, 将比值值- 直值- 的计算为直值- 的计算值- 直值- 。