= Solution
Let $q=(\mathbf q_1,\ldots,\mathbf q_N)$ denote the particle positions. In the <Ghirardi--Rimini--Weber model>, the wavefunction obeys the ordinary <Schrodinger equation>
$$
i\hbar\frac{\partial\psi}{\partial t}=H\psi
$$
between random collapses. Each particle $i$ has an independent <Poisson process> of collapse times with rate $\lambda$. At such a time the state jumps according to
$$
\psi(q)\longmapsto
\frac{L_i(\mathbf X)\psi(q)}{\|L_i(\mathbf X)\psi\|},
\qquad
L_i(\mathbf X)=\frac{1}{(\pi r_C^2)^{3/4}}
\exp\left[-\frac{(\mathbf q_i-\mathbf X)^2}{2r_C^2}\right],
$$
where the random centre has <probability density> $\|L_i(\mathbf X)\psi\|^2$. The normalization is arranged so that $\int d^3X\,L_i(\mathbf X)^2=I$.
The original GRW scales are approximately
$$
r_C\sim10^{-7}\ {\rm m},
\qquad
\lambda\sim10^{-16}\ {\rm s}^{-1}.
$$
An isolated microscopic particle is therefore exceedingly unlikely to collapse during a laboratory experiment. A macroscopic pointer containing about $N\sim10^{23}$ relevant particles has total collapse rate $N\lambda\sim10^7\ {\rm s}^{-1}$ and collapse time about $10^{-7}\ {\rm s}$. After a <measurement interaction> correlates different microscopic outcomes with pointer positions separated by much more than $r_C$, one constituent's localization suppresses all incompatible pointer branches. This <GRW amplification mechanism> produces one definite macroscopic outcome with probabilities given by the <Born rule>, while leaving ordinary microscopic <unitary time evolution> almost unchanged.
For one spatial coordinate, average over the random centre of one collapse. The resulting <density operator> has position-space kernel
$$
\rho'(x,x')=
\exp\left[-\frac{(x-x')^2}{4r_C^2}\right]\rho(x,x').
$$
Its diagonal is unchanged, so $\langle x\rangle$ and $\langle x^2\rangle$ are unchanged. The first derivative of the Gaussian factor vanishes at $x=x'$, so $\langle p\rangle$ is also unchanged. Its second derivative does not vanish, and the <position representation of the momentum operator> gives
$$
\boxed{\langle p^2\rangle'
=\langle p^2\rangle+\frac{\hbar^2}{2r_C^2}}.
$$
In three dimensions the increase in total $\mathbf p^2$ is $3\hbar^2/(2r_C^2)$, corresponding to kinetic-energy increase $3\hbar^2/(4mr_C^2)$ per collapse. These are ensemble statements: conditioning on one specified collapse centre can shift the position moments. Repetition at rate $\lambda$ predicts <GRW spontaneous heating>, so precision searches for anomalous bulk heating, spontaneous radiation, momentum diffusion, and loss of matter-wave interference test the model.
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