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Register a new userThe minimal length uncertainty and the nonextensive thermodynamics
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In this paper, we study the thermodynamics of quantum harmonic oscillator in the Tsallis framework and in the presence of a minimal length uncertainty. The existence of the minimal length is motivated by various theories such as string theory, loop quantum gravity, and black-hole physics. We analytically obtain the partition function, probability function, internal energy, and the specific heat capacity of the vibrational quantum system for $1<q<frac{3}{2}$ and compare the results with those of Tsallis and Boltzmann-Gibbs statistics without the minimal length scale.
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We study the thermodynamics of various physical systems in the framework of the Generalized Uncertainty Principle that implies a minimal length uncertainty proportional to the Planck length. We present a general scheme to analytically calculate the quantum partition function of the physical systems to first order of the deformation parameter based on the behavior of the modified energy spectrum and compare our results with the classical approach. Also, we find the modified internal energy and heat capacity of the systems for the anti-Snyder framework.
Studies in string theory and quantum gravity suggest the existence of a finite lower limit $Delta x_0$ to the possible resolution of distances, at the latest on the scale of the Planck length of $10^{-35}m$. Within the framework of the euclidean path integral we explicitly show ultraviolet regularisation in field theory through this short distance structure. Both rotation and translation invariance can be preserved. An example geometry is studied in detail.
We have shown that the weak-coupling limit superconductors are well described by $ q sim 1 $, where $ q $ is a real parameter which characterizes the degree of nonextensivity of Tsallis entropy. Nevertheless, small deviations with respect to q=1 provide better agreement when compared with experimental results. We have also shown that the generalized BCS theory with $ q eq 1 $ exhibit power-law behavior of several measurable macroscopic functions in the low-temperature regime. These power-law properties are found in many high-Tc oxides superconductors and motivated us to extend Tsallis entropy calculations to these systems. Therefore, we have calculated the phase diagram and the specific heat and we compare our results with the experimental data for the YBCO compound.
Modifications of Heisenbergs uncertainty relations have been proposed in the literature which imply a minimum position uncertainty. We study the low energy effects of the new physics responsible for this by examining the consequent change in the quantum mechanical commutation relations involving position and momenta. In particular, the modifications to the spectrum of the hydrogen atom can be naturally interpreted as a varying (with energy) fine structure constant. From the data on the energy levels we attempt to constrain the scale of the new physics and find that it must be close to or larger than the weak scale. Experiments in the near future are expected to change this bound by at least an additional order of magnitude.
In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relation $[X,P]=ihbarleft(1+beta P^2right)$ where $beta$ is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycieslki (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, i.e., $X=x$ and $P=tanleft(sqrt{beta}pright)/sqrt{beta}$ where $[x,p]=ihbar$, the BBM inequality is still valid in the form $S_x+S_pgeq1+lnpi$ as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.
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