  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 $\rho(T,\alpha) =
4T\chi(T,\alpha)$ 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) $\Delta(\alpha)$.
