Geometric Flow Appearing in Conservation Law in Classical and Quantum Mechanics

N. Ogawa
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Abstract

The appearance of a geometric flow in the conservation law of particle number in classical particle diffusion and in the conservation law of probability in quantum mechanics is discussed in the geometrical environment of a two-dimensional curved surface with thickness $\epsilon$ embedded in $R_3$. In such a system with a small thickness $\epsilon$, the usual two-dimensional conservation law does not hold and we find an anomaly. The anomalous term is obtained by the expansion of $\epsilon$. We find that this term has a Gaussian and mean curvature dependence and can be written as the total divergence of some geometric flow. We then have a new conservation law by adding the geometric flow to the original one. This fact holds in both classical diffusion and quantum mechanics when we confine particles to a curved surface with a small thickness.
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几何流动出现在经典力学和量子力学的守恒定律中
在厚度$\epsilon$嵌入$R_3$的二维曲面的几何环境中,讨论了经典粒子扩散中的粒子数守恒定律和量子力学中的概率守恒定律中几何流的出现。在这样一个厚度很小的系统中,通常的二维守恒定律不成立,我们发现了一个异常。反常项是通过展开$\epsilon$得到的。我们发现这一项与高斯曲率和平均曲率相关,可以写成某些几何流的总散度。然后我们有了一个新的守恒定律通过在原来的基础上加上几何流。这一事实在经典扩散和量子力学中都成立,当我们把粒子限制在一个小厚度的曲面上时。
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来源期刊
Geometry, Integrability and Quantization
Geometry, Integrability and Quantization Mathematics-Mathematical Physics
CiteScore
0.70
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0.00%
发文量
4
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