Discrete density of states

Abstract

By considering the quantum-mechanically minimum allowable energy interval, we exactly count number of states (NOS) and introduce discrete density of states (DOS) concept for a particle in a box for various dimensions. Expressions for bounded and unbounded continua are analytically recovered from discrete ones. Even though substantial fluctuations prevail in discrete DOS, they're almost completely flatten out after summation or integration operation. It's seen that relative errors of analytical expressions of bounded/unbounded continua rapidly decrease for high NOS values (weak confinement or high energy conditions), while the proposed analytical expressions based on Weyl's conjecture always preserve their lower error characteristic. arXiv:1602.05219v1 [cond-mat.stat-mech]

Quantum shape effects and novel thermodynamic behaviors at nanoscale

“…Therefore, instead of solving the Schrödinger equation for the 3D domain, we solve it for its 2D cross section and use the eigenvalues obtained from this numerical solution in the transverse part of partition function (see Methods section for the details of numerical calculations). For longitudinal part, we can easily obtain a precise analytical expression containing QSE corrections, by using the first two terms of Poisson summation formula (PSF) or equivalently by Weyl conjecture 17,20 . By multiplying longitudinal and transverse parts obtained from analytical and numerical approaches respectively, the partition function reads…”

“…At nanoscale, changing the size of a domain alters the thermodynamic properties of particles confined within 17,18 . Lower dimensional geometric elements of a domain such as surface area (A), peripheral length (P ) and even the number of vertices (NV ) enter as control parameters on thermodynamic state functions at nanoscale in addition to volume (V ) [17][18][19][20] . By controlling these size variables it is possible to manipulate and tailor the certain features of confined systems (e.g.…”

Discrete and Weyl density of states for photonic dispersion relation

“…1(a, b, c), Eqs. (3), (10) and ( 8) are used to draw DDOS, WDOS and CDOS functions represented by black dots, blue and red curves, respectively. The extremely fluctuating feature of DDOS is evident in Fig.…”

“…As DOS functions are used along with distribution functions in statistical mechanics to calculate physical properties of a system, throughout the article, dimensionless energy (ε = ε/k B T ) is adopted rather than the energy itself for the compactness of the expressions. DDOS function converts multiple summations over discrete quantum states into a single summation over discrete energy states as follows [10]…”

“…Analytical expressions of WDOS functions are obtained by considering bounded continuum approximation, which corresponds to the first two terms of Poisson Summation Formula (PSF) [20]. To find the analytical expressions of WNOS and WDOS functions for linear dispersion relation, we first apply the same methodology, based on the first two terms of PSF, presented in [10] to DNOS function, Eq. ( 4).…”