黎曼曲面上的联合概率密度是对称张量密度

Manouchehr Amiri
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引用次数: 0

摘要

本文介绍了黎曼流形上联合概率密度的张量特性。首先,我们建立了一个二元数据矩阵,用于记录封闭系统中大量粒子在某一时刻的数值,从而获取相关变量的联合概率密度。通过引入面向粒子的坐标和以该坐标为基础的广义内积作为多线性运算,我们提取了联合概率集,并证明它们符合一般黎曼变量空间上的协变张量特性。根据标量场在黎曼流形中的泰勒展开,已经证明在黎曼流形上定义的累积概率函数的对称迭代协变导数也给出了与上述多线性方法等价的相关联合概率密度集。我们证明了这些协变张量可以简化为普通欧几里得空间与笛卡尔坐标中的经典普通偏导数,并给出了累积分布函数偏导数的联合概率的正式定义。对称协变导数与广义内积之间的等价性已经得出结论。一些著名物理张量的例子说明,许多确定性物理变量是作为张量密度呈现的,其解释类似于概率密度。
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Joint Probability Densities on Riemannian Manifolds are Symmetric Tensor Densities
This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially, we develop a binary data matrix to record the values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle-oriented coordinate and the generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar fields in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of the cumulative probability function defined on Riemannian manifolds also give the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and give the formal definition of joint probabilities by partial derivatives of the cumulative distribution function. The equivalence between the symmetrized covariant derivative and the generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.
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