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The antiferromagnetic $J_1-J_2$ model is a spin-1/2 chain with isotropicexchange $J_1 > 0$ between first neighbors and $J_2 = \alpha J_1$ betweensecond neighbors. The model supports both gapless quantum phases withnondegenerate ground states and gapped phases with $\Delta(\alpha) > 0$ anddoubly degenerate ground states. Exact thermodynamics is limited to $\alpha =0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of smallsystems at frustration $\alpha$ followed by density matrix renormalizationgroup (DMRG) calculations returns the entropy density $S(T,\alpha,N)$ andmagnetic susceptibility $\chi(T,\alpha,N)$ of progressively larger systems upto $N = 96$ or 152 spins. Convergence to the thermodynamics limit,$S(T,\alpha)$ or $\chi(T,\alpha)$, is demonstrated down to $T/J \sim 0.01$ inthe sectors $\alpha < 1$ and $\alpha > 1$. $S(T,\alpha)$ yields the criticalpoints between gapless phases with $S^\prime(0,\alpha) > 0$ and gapped phaseswith $S^\prime(0,\alpha) = 0$. The $S^\prime(T,\alpha)$ maximum at$T^*(\alpha)$ is obtained directly in chains with large $\Delta(\alpha)$ and byextrapolation 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 $\chi(T,\alpha)$ analysisalso yields power-law deviations, but with exponent $\eta(\alpha)$ thatdecreases with $\alpha$. $S(T,\alpha)$ and the spin density $\rho(T,\alpha) =4T\chi(T,\alpha)$ probe the thermal and magnetic fluctuations, respectively, ofstrongly 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) $\Delta(\alpha)$.

Low temperature thermodynamics of the antiferromagnetic $J_1-J_2$ model: Entropy, critical points and spin gap