The generalized density of states in a one-dimensional Ising model with ferrromagnetic and antiferromagnetic interactions

Expressions for the density of states $D(E)$, where $D(E)$ is the number of states of energy $E$, are well known. The present paper offers the expressions for generalized density of states $D_N(E,m)$, where $D_N(E,m)$ is the number of states with energy $E$ and magnetization $m$ in a one-dimensional $N$-spin chain. The expressions obtained here can be considered as reference ones, since all the main characteristics were obtained without them: using the transfer matrix technique or using well-known expressions for the density of states $D(E)=\sum_m{D_N(E,m)}$. Nevertheless, the knowledge of quantity $D_N(E,m)$ helps to understand the model properties and allows the analysis of the temporal behavior of magnetization $m=m(\tau)$. In particular, we demonstrate that in a one-dimensional model spontaneous magnetization can be observed at a non-zero temperature. However, the spontaneous magnetization can randomly change its sign, which results in the magnetization averaged over a very long observation period becoming zero $\langle m(\tau)\rangle$.