The Einstein specific heat model for finite systems

•Extension of the Einstein specific heat model for finite solids.•Obtainment of the specific heat (χχ) solving transcendental equations.•Obtainment of the chemical potential (mm) using a discrete Leibniz integral rule.•Discontinuities of χ(τ)χ(τ) and βm(τ)βm(τ) derivatives appear only for finite NN.•Achievement of BEC temperature has an inverse of harmonic series dependence on NN.

The theoretical model proposed by Einstein to describe the phononic specific heat of solids as a function of temperature consists of the very first application of the concept of energy quantization to describe the physical properties of a real system. Its central assumption lies in the consideration of a total energy distribution among NN (in the thermodynamic limit N→∞N→∞) non-interacting oscillators vibrating at the same frequency (ωω). Nowadays, it is well-known that most materials behave differently at the nanoscale, having thus some cases physical properties with potential technological applications. Here, a version of the Einstein model composed of a finite number of particles/oscillators is proposed. The main findings obtained in the frame of the present work are: (i) a qualitative description of the specific heat in the limit of low-temperatures for systems with nano-metric dimensions; (ii) the observation that the corresponding chemical potential function for finite solids becomes null at finite temperatures as observed in the Bose–Einstein condensation and; (iii) emergence of a first-order like phase transition driven by varying NN.