121 research outputs found
Validity and Failure of the Boltzmann Weight
The dynamics and thermostatistics of a classical inertial XY model,
characterized by long-range interactions, are investigated on -dimensional
lattices ( and 3), through molecular dynamics. The interactions between
rotators decay with the distance like~ (), where and respectively correspond to the
nearest-neighbor and infinite-range interactions. We verify that the momenta
probability distributions are Maxwellians in the short-range regime, whereas
-Gaussians emerge in the long-range regime. Moreover, in this latter regime,
the individual energy probability distributions are characterized by long
tails, corresponding to -exponential functions. The present investigation
strongly indicates that, in the long-range regime, central properties fall out
of the scope of Boltzmann-Gibbs statistical mechanics, depending on and
through the ratio .Comment: 10 pages, 6 figures. To appear in EP
q-Gaussians in the porous-medium equation: stability and time evolution
The stability of -Gaussian distributions as particular solutions of the
linear diffusion equation and its generalized nonlinear form,
\pderiv{P(x,t)}{t} = D \pderiv{^2 [P(x,t)]^{2-q}}{x^2}, the
\emph{porous-medium equation}, is investigated through both numerical and
analytical approaches. It is shown that an \emph{initial} -Gaussian,
characterized by an index , approaches the \emph{final}, asymptotic
solution, characterized by an index , in such a way that the relaxation rule
for the kurtosis evolves in time according to a -exponential, with a
\emph{relaxation} index . In some cases,
particularly when one attempts to transform an infinite-variance distribution
() into a finite-variance one (), the relaxation towards
the asymptotic solution may occur very slowly in time. This fact might shed
some light on the slow relaxation, for some long-range-interacting many-body
Hamiltonian systems, from long-standing quasi-stationary states to the ultimate
thermal equilibrium state.Comment: 20 pages, 6 figure
Computation of energy exchanges by combining information theory and a key thermodynamic relation: Physical applications
Abstract By considering a simple thermodynamic system, in thermal equilibrium at a temperature T and in the presence of an external parameter A , we focus our attention on the particular thermodynamic (macroscopic) relation d U = T d S + ÎŽ W . Using standard axioms from information theory and the fact that the microscopic energy levels depend upon the external parameter A , we show that all usual results of statistical mechanics for reversible processes follow straightforwardly, without invoking the Maximum Entropy principle. For the simple system considered herein, two distinct forms of heat contributions appear naturally in the Clausius definition of entropy, T d S = ÎŽ Q ( T ) + ÎŽ Q ( A ) = C A ( T ) d T + C T ( A ) d A . We give a special attention to the amount of heat ÎŽ Q ( A ) = C T ( A ) d A , associated with an infinitesimal variation d A at fixed temperature, for which a "generalized heat capacity", C T ( A ) = T ( â S / â A ) T , may be defined. The usefulness of these results is illustrated by considering some simple thermodynamic cycles
Spin-Glass Attractor on Tridimensional Hierarchical Lattices in the Presence of an External Magnetic Field
A nearest-neighbor-interaction Ising spin glass, in the presence of an
external magnetic field, is studied on different hierarchical lattices that
approach the cubic lattice. The magnetic field is considered as uniform, or
random (following either a bimodal or a Gaussian probability distribution). In
all cases, a spin-glass attractor is found, in the plane magnetic field versus
temperature, associated with a low-temperature phase. The physical consequences
of this attractor are discussed, in view of the present scenario of the
spin-glass problem.Comment: Accepted for publication in Physical Review
Associating an Entropy with Power-Law Frequency of Events
Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy Sq. The relevant parameters become: (i) The entropic index q, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy #0, in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature Tq with kTq ” #0 (i.e., isothermal processes, for a given q), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index q and value of Tq are estimated.
The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures.Facultad de Ciencias ExactasInstituto de FĂsica La Plat
Associating an Entropy with Power-Law Frequency of Events
Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy Sq. The relevant parameters become: (i) The entropic index q, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy #0, in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature Tq with kTq ” #0 (i.e., isothermal processes, for a given q), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index q and value of Tq are estimated.
The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures.Facultad de Ciencias ExactasInstituto de FĂsica La Plat
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