Stochastic cooling variance model ($V_{n+1}=(1-g{)}^2{V}_n+g^{2}N$)

Let $x_n$ be a centered scalar beam-deviation coordinate in a deliberately controlled toy model. Let the readout be $s_n=x_n+{\eta }_n$, where the centered measurement noise ${\eta }_n$ has variance $N$ and is independent of $x_n$ and of earlier noise. Stipulate a correctly timed correction of strength $g$: $x_{n+1}=x_n-g{s}_n$. This scalar observation is a teaching surrogate, not a claim that real cooling measures each particle. From variance additivity for independent random variables,For fixed current $V>0$ and $N\ge 0$, differentiate the next-step variance with respect to feedback gain:The unique one-step minimizing feedback gain and resulting variance areThe noisier the readout relative to the current spread, the smaller the useful correction. The exact one-step cooling condition follows by subtraction:Thus excessive positive feedback gain heats this model. The threshold depends on the current variance: a gain that initially cools can later heat if the beam has become narrower or the noise has grown.

For chosen dimensionless $V_0=1$ and $N=1/4$, $g_{\ast }=0.8$ gives $V_1=0.20$. Taking $g=1.8$ instead gives $V_1=1.45$, larger than the initial spread. Holding $g=0.8$ fixed givesso the variance approaches the nonzero noise floor $1/6$. More generally, for $0<g<2$ the fixed-gain limit is $V_{\infty }=gN/(2-g)$. This long-run stability condition differs from the condition for cooling the current beam on the very next step.

If the toy feedback gain is recomputed after every step and $N>0$, the reciprocal recurrence is even simpler:With the chosen inputs, $V_5=1/21\simeq 0.047619$. This ideal model has no additional diffusion, finite observation time or lost correlation; its indefinitely falling spread is not a prediction for an operating storage ring.

To expose the missing physics mathematically, replace $s=x+\eta$ by an arbitrary centered sample signal $s$. Defining $C=\mathrm{Cov}(x,s)$ and $S=\mathrm{Var}(s)>0$ givesUnwanted contributions from other particles or electronics can enlarge $S$ without increasing useful $C$; poor timing or lost correlation can reduce $C$ or even reverse its sign. This is why real stochastic cooling needs measured beam fluctuations, appropriate beam mixing and correctly phased feedback control, rather than an imagined catalogue of individual particle positions.