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i have been confused by one statement on the spatial correlation funciton in the statistical physics textbook. They say for a spatial correlation function f(x1,x2), where x1 and x2 are the coordinate of particle 1 and 2, if the system has translational symmetry, then f depends only on the distance between 1 and 2, not on the coordinates of the particles. That is, f(x1-x2)=f(|x1-x2|) in this case.

i mean this sounds reasonable, but i just can't tell why. i am wondering if there is a solid demonstration on this. Could anybody give me some explanation on this? Thanks a lot for your help!

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# Correlation funciton of a system with translational symmetry

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