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arXiv 0809.1166cond-mat.stat-mechhep-thquant-ph
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汽液平衡中的信息几何

Information geometry in vapour-liquid equilibrium

Dorje C. Brody, Daniel W. Hook

摘要:

使用平方根映射p-> \sqrt{p},概率密度函数p可以表示为平方可积函数的希尔伯特空间中的单位球面S的点。如果密度函数光滑地依赖于一组参数,则映射的像形成S中的黎曼子流形M。由S的周围球面几何导出的M上的度量是Fisher信息矩阵。由参数密度函数p建模的系统的统计特性可以用信息几何来表示。信息几何的基本介绍,其次是一个精确的几何特征的家庭高斯密度函数。当参数密度函数描述物理系统的平衡态时,某些物理特征可以用相关信息流形M的几何特征来识别。应用这一思想,汽-液相变的性质在几何方面阐明。对于理想气体,没有相变,M的几何形状是平坦的。在这种情况下,测地线方程的解产生绝热状态方程。对于范德瓦尔斯气体,M的相关几何是高度非平凡的。M的标量曲率沿着包围相图中非物理区域的旋节边界发散。因此,曲率与系统的稳定性密切相关。

英文摘要:

Using the square-root map p-->\sqrt{p} a probability density function p can be represented as a point of the unit sphere S in the Hilbert space of square-integrable functions. If the density function depends smoothly on a set of parameters, the image of the map forms a Riemannian submanifold M in S. The metric on M induced by the ambient spherical geometry of S is the Fisher information matrix. Statistical properties of the system modelled by a parametric density function p can then be expressed in terms of information geometry. An elementary introduction to information geometry is presented, followed by a precise geometric characterisation of the family of Gaussian density functions. When the parametric density function describes the equilibrium state of a physical system, certain physical characteristics can be identified with geometric features of the associated information manifold M. Applying this idea, the properties of vapour-liquid phase transitions are elucidated in geometrical terms. For an ideal gas, phase transitions are absent and the geometry of M is flat. In this case, the solutions to the geodesic equations yield the adiabatic equations of state. For a van der Waals gas, the associated geometry of M is highly nontrivial. The scalar curvature of M diverges along the spinodal boundary which envelopes the unphysical region in the phase diagram. The curvature is thus closely related to the stability of the system.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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