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.