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自然迭代法
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     Under the restriction of unitarity the same conclusion can also be obtained by adding a set of "intermediate" variables. The monotonous decrease of grand potential and the convergence of nature iteration method are proven by these two steps of analysis. This method can be used for proving convergence of any kind of approximation in CVM.
     加上归一性条件的限制之后,通过引入“中间”变量,也证明了函数△Ф的非负性,即证明了迭代过程中的巨势函数的值总是单调减小的 最终依据热力学中稳定结构对应巨势函数极小值的结论证明了自然迭代法的收敛性.对于集团变分法中各种近似计算的自然迭代过程的收敛性问题,这种证明方法不同于 Kikuchi提出的直接比较的方法,具有一般性
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     On Natural Appreciation of the Beautiful
     自然审美
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     The Natural Meaning
     自然的心迹
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     Convergence Analysis of a Family Iteration methods
     一族迭代法的收敛性
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     Fractal in Iteration Method
     迭代法中的分形
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  natural iteration method
The equilibrium cluster distribution is then found by minimizing the free energy by utilizing the natural iteration method developed by Kikuchi.
      
On the Convergence of Kikuchi's Natural Iteration Method
      
In this article we investigate on the convergence of the natural iteration method, a numerical procedure widely employed in the statistical mechanics of lattice systems, to minimize Kikuchi's cluster variational free energies.
      
Finally, we consider a recently proposed class of methods for the minimization of Kikuchi functionals, showing that the natural iteration method turns out as a particular instance of that class.
      


Based on the selection of the bcc alloy as an example, irregular tetrahedron is used as the basic cluster in the cluster variation method (CVM) calculation. The difference of grand potentials between two consecutive iteration steps △Ф=Ф -Ф is obtained by grouping variables connected by the superposition expression into "input" and "output". The function △Ф is proven not less than zero when it is seen as function of "output" variables which are independent one another, and "input" variables and other parameters...

Based on the selection of the bcc alloy as an example, irregular tetrahedron is used as the basic cluster in the cluster variation method (CVM) calculation. The difference of grand potentials between two consecutive iteration steps △Ф=Ф -Ф is obtained by grouping variables connected by the superposition expression into "input" and "output". The function △Ф is proven not less than zero when it is seen as function of "output" variables which are independent one another, and "input" variables and other parameters are constants. Under the restriction of unitarity the same conclusion can also be obtained by adding a set of "intermediate" variables. The monotonous decrease of grand potential and the convergence of nature iteration method are proven by these two steps of analysis. This method can be used for proving convergence of any kind of approximation in CVM.

以bcc合金为例,选择四面体顶点作为集团变分法的基本原子团,将通过迭代方程联系起来的两组变量区分为“输入”和“输出”变量组,得到前后两步巨势函数差△Ф=Ф-Ф的表达式 首先将函数△Ф视为相互独立的“输出”变量的函数,而将“输入”变量以及其它参数均看成是常量,然后采用极值定理和极值的充分条件,证明了函数△Ф的非负性;加上归一性条件的限制之后,通过引入“中间”变量,也证明了函数△Ф的非负性,即证明了迭代过程中的巨势函数的值总是单调减小的 最终依据热力学中稳定结构对应巨势函数极小值的结论证明了自然迭代法的收敛性.对于集团变分法中各种近似计算的自然迭代过程的收敛性问题,这种证明方法不同于 Kikuchi提出的直接比较的方法,具有一般性

 
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