The antiferromagnetic $J_1-J_2$ model is a spin-1/2 chain with isotropic exchange $J_1 > 0$ between first neighbors and $J_2 = alpha J_1$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $Delta(alpha) > 0$ and doubly degenerate ground states. Exact thermodynamics is limited to $alpha = 0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $alpha$ followed by density matrix renormalization group (DMRG) calculations returns the entropy density $S(T,alpha,N)$ and magnetic susceptibility $chi(T,alpha,N)$ of progressively larger systems up to $N = 96$ or 152 spins. Convergence to the thermodynamics limit, $S(T,alpha)$ or $chi(T,alpha)$, is demonstrated down to $T/J sim 0.01$ in the sectors $alpha < 1$ and $alpha > 1$. $S(T,alpha)$ yields the critical points between gapless phases with $S^prime(0,alpha) > 0$ and gapped phases with $S^prime(0,alpha) = 0$. The $S^prime(T,alpha)$ maximum at $T^*(alpha)$ is obtained directly in chains with large $Delta(alpha)$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,alpha)$ down to $T = 0$ indicates power-law deviations $T^{-gamma(alpha)}$ from $exp(-Delta(alpha)/T)$ with exponent $gamma(alpha)$ that increases with $alpha$. The $chi(T,alpha)$ analysis also yields power-law deviations, but with exponent $eta(alpha)$ that decreases with $alpha$. $S(T,alpha)$ and the spin density $rho(T,alpha) = 4Tchi(T,alpha)$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,alpha)/rho(T,alpha)$ for $T < 0.10$. Remarkably, the ratio decreases (increases) with $T$ in chains with large (small) $Delta(alpha)$.
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