arXiv subjects
Publications and source records attributed to Marcelo R. Ubriaco.
With use of the Second Inverse Maximum Entropy Principle we find entropy functions for systems with fractal distribution functions with order parameter $q$. We compare these entropy functions with those given by the Bose-Einstein and Fermi-Dirac cases.
Starting with the average particle distribution function for bosons and fermions for non-extensive thermodynamics , as proposed in \cite{CMP}, we obtain the corresponding density matrix operators and hamiltonians. In particular, for the bosonic case the corresponding operators satisfy a deformed bosonic algebra and the hamiltonian involves interacting terms in powers of $a^{\dagger}_ja_j$ standard creation and annihilation operators. For the unnormalized density matrix we obtain a nonlinear equation that leads to a two-parameter solution relevant to anomalous diffusion phenomena.
In this manuscript, we calculate the scalar curvature of a two-dimensional thermodynamic space to study the properties of two thermodynamic systems. In particular, we study the stability and possible anyonic behavior of quantum group invariant systems and systems with fractal distribution functions.
We study the thermodynamics of systems based on a Fock space representation inspired by the differential-difference operators proposed in Ref. \cite{Dunkl}. We calculate thermodynamic functions as the entropy and heat capacity and compare them with the standard boson case. A calculation of the second virial coefficient and the scalar curvature in two and three dimensions show that these systems becomes repulsive within an interval of negative values of the reflection operator parameter $μ_0$. In addition, the stability of this system is examined as a function of $μ_0$
Starting with the partition function $Z$ for systems with $M$-statistics, as proposed in \cite{WND}, we calculate from the metric $g_{αη}=\frac{\partial \ln Z}{\partialβ^αβ^η}$ the scalar curvature $R$ in two and three dimensions. Our results exhibit the details of the anyonic behavior as a function of the fugacity $z$ and the identical particle maximum occupancy number $M$. We also compare the stability of systems for $M>1$with the fermionic (M=1), and bosonic ($M\rightarrow \infty$), cases.
Starting with the partition functions for quantum group invariant systems we calculate the metric in the two-dimensional space defined by the parameters $β$ and $γ=-βμ$ and the corresponding scalar curvature for these systems in two and three spatial dimensions. Our results exhibit the details of the anyonic behavior of quantum group boson and fermion systems as a function of the fugacity $z$ and the quantum group parameter $q$. For the case of the quantum group $SU_q(2)$, we compare the stability of these systems with the stability of SU(2) invariant boson and fermion systems.
Starting with the relative entropy for two close statistical states we define the metric and calculate the scalar curvature $R$ for systems with classical, boson and fermion fractal distribution functions with moment order parameter $q$. In particular, we find that for $q\neq 1$ the scalar curvature is closer to zero implying that the fractal bosonic and fermionic systems are more stable than the standard ones.
Starting with the relative entropy based on a previously proposed entropy function $S_q[p]=\int dx p(x)(-\ln p(x))^q$, we find the corresponding Fisher's information measure. After function redefinition we then maximize the Fisher information measure with respect to the new function and obtain a differential operator that reduces to a space coordinate second derivative in the $q\to 1$ limit. We then propose a simple differential equation for anomalous diffusion and show that its solutions are a generalization of the functions in the Barenblatt-Pattle solution. We find that the mean squared displacement, up to a $q$-dependent constant, has a time dependence according to $ \sim K^{1/q}t^{1/q}$, where the parameter $q$ takes values $q=\frac{2n-1}{2n+1}$ (superdiffusion) and $q=\frac{2n+1}{2n-1}$ (subdiffusion), $\forall n\geq 1$.
We propose entropy functions based on fractional calculus. We show that this new entropy has the same properties than the Shannon entropy except additivity, therefore making this entropy non-extensive. We show that this entropy function satisfies the Lesche and thermodynamic stability criteria.
We propose a model for describing the growth on an untreated tumor, which is characterized in a simple way by a minimal number of parameters with a well-defined physical interpretation. The model is motivated by invoking the Master Equation and the Principle of Detailed Balance in the present context, and it is easily generalizable to include the effects of various types of therapies. In the simplest version that we consider here, it leads to a linear equation that describes the population growth in a dynamic environment, for which a complete solution can be given in terms of the integral of the growth rate. The essential features of the general solution for this case are illustrated with a few examples.
We study the consequences of introducing quantum group invariance in the formalism of nonextensive quantum statistical mechanics. We find that the corresponding thermodynamical system is equivalent to a Bose-Einstein gas in the Boltzmann-Gibbs formalism with a higher critical temperature than the standard Bose-Einstein case.
We study the long range behavior of a gas whose partition function depends on a parameter q and it has been claimed to be a good approximation to the partition function proposed in the formulation of nonextensive statistical mechanics. We compare our results, at large temperatures and at the critical point, with the case of Boltzmann-Gibbs thermodynamics for the case of a Bose-Einstein gas. In particular, we find that for all temperatures the long range correlations in a Bose gas decrease when the value of q departs from the standard value q=1.
Starting with the fractal inspired distribution functions for Maxwell-Boltzmann, Bose-Einstein and Fermi systems, as reported by F. Büyükkiliç and D. Demirhan, we obtain the corresponding probability distributions and study their thermodynamic behavior. We compare our results with those corresponding to ideal gases (q=1), and Bose-Einstein and Fermi systems with quantum group symmetry. In particular, we show that the hamiltonian that gives the Bose-Einstein generalized distribution function can be interpreted as a q-deformation of the ideal gas hamiltonian.
We study the properties of a thermodynamic system having the symmetry of a quantum group and interacting with a harmonic potential. We calculate the dependence of the chemical potential, heat capacity and spatial distribution of the gas on the quantum group parameter $q$ and the number of spatial dimensions $D$. In addition, we consider a fourth-order interaction in the quantum group fields $Ψ$, and calculate the ground state energy up to first order.
We show that $SU_q(2)$ invariant systems in one and two dimensions exhibit Bose-Einstein condensation for $q >1$. For these systems there is a $λ$-transition at the critical temperature. The critical temperature and the gap in the heat capacity increase more rapidly for small deviations from the standard value $q=1$, and they become approximately constant for large values of $q$. For low temperatures and $q>1$ the entropy is lower than the entropy of an ideal Bose gas.
We study the Bose-Einstein condensation of a gas with $SU_q(2)$ symmetry. We show, in the thermodynamic limit, that the boson interactions introduced by the quantum group symmetries enhance Bose-Einstein condensation giving a discontinuity in the heat capacity $C_v$ at the critical temperature $T_c$. The critical temperature and the gap in $C_v$ increase with the value of the parameter $q$ and become approximately constant for $q>3$.
We discuss the role that quantum group symmetries, in particular $SU_q(2)$, play in a thermodynamic system at high temperatures. We show that the interactions introduced by the quantum group symmetries, are such that a quantum group gas describe repulsive and attractive behavior in two and three spatial dimensions.
We first introduce and discuss the formalism of $SU_q(N)$-bosons and fermions and consider the simplest Hamiltonian involving these operators. We then calculate the grand partition function for these models and study the high temperature (low density) case of the corresponding gases for $N=2$. We show that quantum group gases exhibit anyonic behavior in $D=2$ and $D=3$ spatial dimensions. In particular, for a $SU_q(2)$ boson gas at $D=2$ the parameter $q$ interpolates within a wider range of attractive and repulsive systems than the anyon statistical parameter.