Investigation of the Finite Size Properties of the Ising Model Under Various Boundary Conditions

Abstract

The one-dimensional Ising model with various boundary conditions is considered. Exact expressions for the thermodynamic and magnetic properties of the model using different kinds of boundary conditions [Dirichlet (D), Neumann (N), and a combination of Neumann–Dirichlet (ND)] are presented in the absence (presence) of a magnetic field. The finite-size scaling functions for internal energy, heat capacity, entropy, magnetisation, and magnetic susceptibility are derived and analysed as function of the temperature and the field. We show that the properties of the one-dimensional Ising model is affected by the finite size of the system and the imposed boundary conditions. The thermodynamic limit in which the finite-size functions approach the bulk case is also discussed.

The qualitative properties of 1D mixed-type Potts-SOS model with 1-spin and its dynamical behavior

“…So, we get We define σ N+1 to be equal to σ 1 in order to establish periodic boundary conditions (see [30] for details). Therefore, from (A2) we obtain the partition function as Now, it is feasible to define the entire thermodynamic properties of the model by utilizing either the partition function and the finite-size free energy or one of its derived forms [16,29,30,35]. From (A4), we obtain the free energy associated with the model as Many researchers have examined other thermodynamic properties as a result of derivatives of the free energy function provided in (A5) with regard to certain parameters [16,29,30,35].…”

“…Ostilli and Mukhamedov [31] determined the free energy by deriving the transfer matrix for a 1D three-state Potts model in the presence of a q-component external field. However, under certain special circumstances, calculations were performed in the absence of the q-component external field (see [29,35] for more information). It is worth noting that in section 3, for non-zero external field, obtaining the transfer matrix, we will specifically develop the transfer matrix by assuming the external magnetic field variable to be zero as applications.…”

“…Using the Kramers-Wannier transfer technique, which requires building a transfer matrix and determining its eigenvalues, we can solve the problem [16,29,35]. The matrix transfer approach is widely employed in statistical mechanics due to its broad range of applications, making it one of the most frequently utilized methods [29][30][31].…”

On the critical behavior of the spin-s Ising model

“…In the thermodynamic limit N → ∞, we obtain the wellknown bulk magnetization and susceptibility for the system of spin-1/2 [40]:…”