Flux fluctuations in the one dimensional nearest neighbors symmetric simple exclusion process

Let $J(t)$ be the the integrated flux of particles in the symmetric simple exclusion process starting with the product invariant measure $\nu_\rho$ with density $\rho$. We compute its rescaled asymptotic variance: $$\lim_{t\to\infty} t^{-1/2} \V J(t) = \sqrt{2/\pi} (1-\rho)\rho$$ Furthermore we show that $t^{-1/4}J(t)$ converges weakly to a centered normal random variable with this variance. From these results we compute the asymptotic variance of a tagged particle in the nearest neighbor case and show the corresponding central limit theorem, results previously proven by Arratia.