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We study through the time evolution of magnetization the low temperature (T)dynamics of the metastable coexisting phases created by traversing differentpaths in magnetic field (H) and T space in a shape memory alloy system,Ni$_{45}$Co$_{5}$Mn$_{38}$Sn$_{12}$. It is shown that these coexisting phasesconsisting of a fraction of kinetically arrested austenite phase and remainingfraction of low-T equilibrium martensitic phase undergo a slow relaxation tolow magnetization (martensitic) state but with very different thermomagnetichistory-dependent rates at the same T and H. We discovered that, when thenucleation of martensitic phase is initiated at much lower T through thede-arrest of the glass like arrested state contrasted with the respective firstorder transformation (through supercooling at much higher T), then the longtime relaxation rate scales with the non-equilibrium phase fraction but has avery weak dependence on T. This is explained on the basis of the H-T pathdependent size of the critical radii of the nuclei and the subsequent growth ofthe equilibrium phase through the motion of the interface.

History dependent nucleation and growth of the martensitic phase in a magnetic shape memory alloy Ni$_{45}$Co$_{5}$Mn$_{38}$Sn$_{12}$