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版本 2023/09/08 Ver0.1

References

1. Kardar, Mehran. Statistical physics of particles. Cambridge University Press, 2007.

Essential Math

Probability - Multi-var

In Kardar (particle) 2.4, Many random variables, everything is so clear that I don’t want to elaborate. For the joint Gaussian distribution, the Wick’s theorem is important in the course.

Recall what we’ve learnt from 2.2 One random variable (also in 概率论), and Symmetric matrices diagonalization by unitary matrices in 线性代数.

d-Dimensional Space Gaussian Integrals

In Kardar (particle) 4.4. In Calculus (微积分), we’ve learnt \(I_d≡(\displaystyle\int_{-\infty}^{\infty}dxe^{-x^2})^d=\pi^{d/2}\) by changing coordinates. We can also “drag a surface out”, as \(I_d=\displaystyle\int_0^{\infty}dRS_dR^{d-1}e^{-R^2}=\frac{S_d}{2}(d/2-1)!\). Then we get \(S_d=\frac{2\pi^{d/2}}{(d/2-1)!}\).