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We study the thermodynamics of a crystalline solid by applying intermediate statistics manifested by q-deformation. We based part of our study on both Einstein and Debye models, exploring primarily deformed thermal and electrical conductivities as a function of the deformed Debye specific heat. The results revealed that the q-deformation acts in two different ways but not necessarily as independent mechanisms. It acts as a factor of disorder or impurity, modifying the characteristics of a crystalline structure, which are phenomena described by q-bosons, and also as a manifestation of intermediate statistics, the B-anyons (or B-type systems). For the latter case, we have identified the Schottky effect, normally associated with high-Tc superconductors in the presence of rare-earth-ion impurities, and also the increasing of the specific heat of the solids beyond the Dulong-Petit limit at high temperature, usually related to anharmonicity of interatomic interactions. Alternatively, since in the q-bosons the statistics are in principle maintained the effect of the deformation acts more slowly due to a small change in the crystal lattice. On the other hand, B-anyons that belong to modified statistics are more sensitive to the deformation.
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We propose a measure, which we call the dissipative spectral form factor (DSFF), to characterize the spectral statistics of non-Hermitian (and non-Unitary) matrices. We show that DSFF successfully diagnoses dissipative quantum chaos, and reveals correlations between real and imaginary parts of the complex eigenvalues up to arbitrary energy (and time) scale. Specifically, we provide the exact solution of DSFF for the complex Ginibre ensemble (GinUE) and for a Poissonian random spectrum (Poisson) as minimal models of dissipative quantum chaotic and integrable systems respectively. For dissipative quantum chaotic systems, we show that DSFF exhibits an exact rotational symmetry in its complex time argument $tau$. Analogous to the spectral form factor (SFF) behaviour for Gaussian unitary ensemble, DSFF for GinUE shows a dip-ramp-plateau behavior in $|tau|$: DSFF initially decreases, increases at intermediate time scales, and saturates after a generalized Heisenberg time which scales as the inverse mean level spacing. Remarkably, for large matrix size, the ramp of DSFF for GinUE increases quadratically in $|tau|$, in contrast to the linear ramp in SFF for Hermitian ensembles. For dissipative quantum integrable systems, we show that DSFF takes a constant value except for a region in complex time whose size and behavior depends on the eigenvalue density. Numerically, we verify the above claims and additionally compute DSFF for real and quaternion real Ginibre ensembles. As a physical example, we consider the quantum kicked top model with dissipation, and show that it falls under the universality class of GinUE and Poisson as the `kick is switched on or off. Lastly, we study spectral statistics of ensembles of random classical stochastic matrices or Markov chains, and show that these models fall under the class of Ginibre ensemble.
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In this paper, we give a general discussion on the calculation of the statistical distribution from a given operator relation of creation, annihilation, and number operators. Our result shows that as long as the relation between the number operator and the creation and annihilation operators can be expressed as $a^{dagger}b=Lambdaleft(Nright) $ or $N=Lambda^{-1} left( a^{dagger}bright)$, where $N$, $a^{dagger}$, and $b$ denote the number, creation, and annihilation operators, i.e., $N$ is a function of quadratic product of the creation and annihilation operators, the corresponding statistical distribution is the Gentile distribution, a statistical distribution in which the maximum occupation number is an arbitrary integer. As examples, we discuss the statistical distributions corresponding to various operator relations. In particular, besides Bose-Einstein and Fermi-Dirac cases, we discuss the statistical distributions for various schemes of intermediate statistics, especially various $q$-deformation schemes. Our result shows that the statistical distributions corresponding to various $q$-deformation schemes are various Gentile distributions with different maximum occupation numbers which are determined by the deformation parameter $q$. This result shows that the results given in much literature on the $q$-deformation distribution are inaccurate or incomplete.
We study the thermodynamics of metals by applying q-deformed algebras. We shall mainly focus our attention on q-deformed Sommerfeld parameter as a function of q-deformed electronic specific heat. The results revealed that q-deformation acts as a factor of disorder or impurity, modifying the characteristics of a crystalline structure and thereby controlling the number of electrons per unit volume.
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