Untitled21.ipynb

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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For individual point differences, we have"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$s = (v_i - v_j)^2$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "so "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\sigma_s^2 = \\left(\\frac{\\partial{s}}{\\partial{v_i}}\\right)^2\\sigma_{v_i}^2 + \\left(\\frac{\\partial{s}}{\\partial{v_j}}\\right)^2\\sigma_{v_j}^2$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "and "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\left(\\frac{\\partial{s}}{\\partial{v_i}}\\right) = 2(v_i - v_j)$\n",
      "\n",
      "$\\left(\\frac{\\partial{s}}{\\partial{v_j}}\\right) = -2(v_i - v_j)$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "therefore "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\sigma_s^2 = 4(v_i-v_j)^2\\sigma_{v_i}^2 + 4(v_i-v_j)^2\\sigma_{v_j}^2$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "but"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\sigma_{v_i} = \\sigma_{v_j} = \\sigma_v$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "so "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\sigma_s^2 = 8(v_i-v_j)^2\\sigma_v^2 = 8s\\sigma_v^2$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "but the structure function is not $s$, but the *average* of $s$:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$SF = \\langle(v_i-v_j)^2\\rangle$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So I'm not sure how to take it past this point, e.g. how to compute the error on the bin from $\\sigma_s$."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}