Ising ferromagnet and nonadditive entropies: equilibrium and nonequilibrium properties

Abstract

Some equilibrium and nonequilibrium properties of the one-dimensional nearest-neighbor Ising ferromagnet that are grounded on nonadditive entropies are reviewed. First, we focus on the known fact that the nonadditive entropy \(S_q\) for a special value of q, namely \(q^\star =\sqrt{37}-6 \simeq 0.0828\), yields entropic extensivity, thus satisfying the Legendre structure of classical thermodynamics, at the quantum critical point of the spin-1/2 ferromagnetic Ising chain in the presence of an external transverse field. Then we address a well-known issue in phase transitions, namely that, at the critical point, quantities such as the susceptibility and the Grüneisen ratio diverge within Boltzmann–Gibbs statistical mechanics (\(q=1\)). Moreover, we show that such thermostatistical quantities diverge for \(q>q^\star \), vanish for \(q<q^\star \), and are finite for \(q=q^\star \). Second, we focus on a transport phenomenon, specifically, the heat transport in a one-dimensional Ising model. We do so by analyzing two classical inertial anisotropic XY models that recover the equilibrium and nonequilibrium properties of the Ising model under the extreme anisotropy limit. In other words, we determine, from first principles, the thermal conductivity behavior of the Ising chain with regard to temperature and lattice size, concluding that the model exhibits normal heat conduction, i.e., it satisfies Fourier’s law.