The random field Ising model in an asymmetric, anisotropic and linearly field dependent trimodal probability distribution

The Ising model, under the influence of an asymmetric and anisotropic external random magnetic field, is investigated for any temperature T and any other parameters; the random field is drawn from the trimodal probability density function P(hi)=phiδ(hi−h0)+qhiδ(hi+λh0)+rhiδ(hi)=11+λ[1h0+λ(1−r)]hiδ(hi−h0)+11+λ(1−r−1h0)hiδ(hi+λh0)+rhiδ(hi), dependent linearly on the field hi, as well. The partial probabilities p,q,r obey the constraint p+q+r=1, asymmetric distribution; hi is the random field with absolute value h0 (strength); λ is a positive competition parameter making the random fields competitive, anisotropic distribution; the presence of the multiplicative factor hi is to enhance the influence of the magnetic fields. The trimodal probability distribution is an extension of the bimodal one allowing for the existence in the lattice of vacant sites or non magnetic particles, of fraction r. The current random field Ising system undergoes, mainly, second order phase transitions, which, for some values of λ and h0, are followed by first order phase transitions, both joined smoothly at a tricritical point. Using the variational principle, the equilibrium equation for the magnetization is written down and solve it for both phase transitions as well as at the tricritical point, in order to determine the magnetization profile with respect to h0 and temperature; the stability for each phase transition and at the tricritical point is examined, as well.