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Register a new userThe Einstein specific heat model for finite systems
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Added by
Mariano de Souza Prof. Dr.
Publication date
2016
fields
Physics
and research's language is
English
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The theoretical model proposed by Einstein to describe the phononic specific heat of solids as a function of temperature consists 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 N (in the thermodynamic limit $N rightarrow infty$) non-interacting oscillators vibrating at the same frequency ($omega$). 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 Einsteins 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 $N$.
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The theory of small-system thermodynamics was originally developed to extend the laws of thermodynamics to length scales of nanometers. Here we review this nanothermodynamics, and stress how it also applies to large systems that subdivide into a heterogeneous distribution of internal subsystems that we call regions. We emphasize that the true thermal equilibrium of most systems often requires that these regions are in the fully-open generalized ensemble, with a distribution of region sizes that is not externally constrained, which we call the nanocanonical ensemble. We focus on how nanothermodynamics impacts the statistical mechanics of specific models. One example is an ideal gas of indistinguishable atoms in a large volume that subdivides into an ensemble of small regions of variable volume, with separate regions containing atoms that are distinguishable from those in other regions. Combining such subdivided regions yields the correct entropy of mixing, avoiding Gibbs paradox without resorting to macroscopic quantum symmetry for semi-classical particles. Other models are based on Ising-like spins (binary degrees of freedom), which are solved analytically in one-dimension, making them suitable examples for introductory courses in statistical physics. A key result is to quantify the net increase in entropy when large systems subdivide into small regions of variable size. Another result is to show similarity in the equilibrium properties of a two-state model in the nanocanonical ensemble and a three-state model in the canonical ensemble. Thus, emergent phenomena may alter the thermal behavior of microscopic models, and the correct ensemble is necessary for accurate predictions.
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