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Appendix 3: Derivation of the Sphere Density Function

In this appendix we determine the probability density distribution of a set of Dindependent normally distributed random variables as a function of radial distance in the space defined by those variables. By definition, the probability density along the j^th axis in the space is:

$$f(y_{j}) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-y_{j}^{2}/2\sigma^{2}} .$$ (46)

To determine the overall probability density function in the space, we integrate over spherical shells. The probability of the machine being in a small shell of volume dVat radius r is

 

$$f(y_{1} , \ldots , y_{j}, \ldots, y_{D} ) dV$$ p(r) = (47)

$$\Pi_{j=1}^{D} f(y_{j}) dV$$ =

$$\frac{1}{\sigma^{D}\sqrt{2 \pi}^{D}} e^{-r^{2}/2\sigma^{2}} dV .$$ =

If f_D(r)is the probability density of the sphere as a function of radius, then the probability of the machine being in a small interval of radius dr is also p(r) = f_D(r) dr. Combining the two equations for p(r) with equation (27) we obtain

$$f_{D}(r) = \frac { r^{D-1} e^{-r^{2}/2\sigma^{2}} } {\Gamma \left( \frac{D}{2} \right) \: \sigma^{D} \: 2^{\frac{D}{2}-1} } ,$$ (48)

which has a maximum at $r_{max} = \sigma \sqrt{D-1}$. If D is sufficiently high, then the f_D(r) function can be approximated by a Gaussian distribution. Since any Gaussian with mean $\mu$ and standard deviation ${\sigma}{'}$ has the property that $f(\mu + {\sigma}{'})/f(\mu)=e^{-1/2}$, we may estimate the fuzziness or thickness of the shell from the two intercepts with e^-1/2 in Fig. 4: ${{\sigma}{'}}^{-}$and ${{\sigma}{'}}^{+}$. This can also be calculated by noting that $r^{D-1} = e^{(D-1)\ln r}$, expanding the log by $\ln(x+1) \approx x- x^{2}/2$[Thomas, 1968] and setting r_max = 1. This leads to ${\sigma}{'}\approx \frac{1}{\sqrt{2 (D - 1)}}$when r_max = 1. For the curves in Fig. 4, ${{\sigma}{'}}^{-}< {\sigma}{'}< {{\sigma}{'}}^{+}$.

The f_D(r) function is the probability density function of a $\chi^2$ distribution for the variable $x = r^2 / \sigma^2$ and D degrees of freedom [Weast et al., 1988,International Telephone and Telegraph Corporation, 1956]. Fig. 7 is essentially a series of $\chi^2$ tests. The curves for the lower dimensions are named after well known physicists: D = 1 is a Gaussian distribution [Wannier, 1966,Waldram, 1985]; D = 2 is a Rayleigh distribution [International Telephone and Telegraph Corporation, 1956]; and D = 3 is a Maxwellian or Maxwell-Boltzmann speed distribution [Wannier, 1966,Castellan, 1971,Waldram, 1985].