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B. H. Lavenda

Publications and source records attributed to B. H. Lavenda.

29 records · Page 2Linked to original sources
arXiv2004

Mean Entropies

Entropies must correspond to mean values for them to be measurable. The Shannon entropy corresponds to the weighted arithmetic mean, whereas the Renyi entropy corresponds to the exponential mean. These means refer to code lengths, which are converted into entropies by replacing the length of a sequence by the negative logarithm of the probability of its occurrence. Only affine and exponential generating functions of means preserve the property of additivity and invariance under translations, and hence are Kolmogorov-Nagumo functions, resulting in the Shannon and Renyi entropies, respectively. Pseudo-additive entropies are generating functions of means of order 0\leτ<1, which is the exponential Renyi entropy, or, in the τ=0 limit, the Shannon entropy. Means of any order cannot be expressed as escort averages because such averages contradict the fact that the means are monotonically increasing functions of their order. Exponential mean error functions of Renyi, in general, and Shannon, in particular, are shown to be measures of the extent of a distribution.

cond-mat.stat-mech↗
arXiv2004

Information and coding discrimination of pseudo-additive entropies (PAE)

PAE cannot be made a basis for either a generalized statistical mechanics or a generalized information theory. Either statistical independence must be waived, or the expression of the averaged conditional probability as the difference between the marginal and joint entropies must be relinquished. The same inequality, relating the PAE to the Rényi entropy, when applied to the mean code length produces an expression that it is without bound as the order of the code length approaches infinity. Since the mean code length associated with the Rényi entropy is finite and can be made to come as close to the Hartley entropy as desired in the same limit, the PAE have a more limited range of validity than the Rényi entropy which they approximate.

cond-mat.stat-mech↗
arXiv2004

The linear Ising model and its analytic continuation, random walk

A generalization of Gauss's principle is used to derive the error laws corresponding to Types II and VII distributions in Pearson's classification scheme. Student's $r$-pdf (Type II) governs the distribution of the internal energy of a uniform, linear chain, Ising model, while analytic continuation of the uniform exchange energy converts it into a Student $t$-density (Type VII) for the position of a random walk in a single spatial dimension. Higher dimensional spaces, corresponding to larger degrees of freedom and generalizations to multidimensional Student $r$- and $t$-densities, are obtained by considering independent and identically distributed random variables, having rotationally invariant densities, whose entropies are additive and generating functions are multiplicative.

cond-mat.stat-mech↗
arXiv2004

On the definition of fluctuating temperature

The Maxwell distribution is derived from the $F$-distribution in the limit where one of the degrees of freedom of the $χ^2$ variates tends to infinity. The estimator of the temperature is consistent, and, hence coincides with the temperature of the heat reservoir in the asymptotic limit; it is also unbiased and efficient. Consequently, there is a contradiction between indentifying the Lagrange multiplier in the variational formalism that Tsallis and co-workers use to maximize his nonadditive entropy with respect to escort expectation values in order to derive the Student $t$- and $r$-distributions and the physical meaning of these variables. Only in the asymptotic limit when these distributions become the $χ^2$-distributions of MBG statistics can the Lagrange multiplier be interpreted as the inverse temperature. Hence, there is no generalization of the $χ^2$-distributions that can be made which involves interpreting the Lagrange multiplier as the inverse temperature. The frequency interpretation of the fluctuating temperature is contrasted with the Bayesian approach that treats a parameter to be estimated as a random variable which is equipped with a probability distribution.

cond-mat.stat-mech↗
arXiv2003

Qualms regarding "Dynamical Foundations of Nonextensive Statistical Mechanics" by C. Beck (cond-mat/0105374)

The derivation of Student's pdf from superstatistics is a mere coincidence due to the choice of the $χ^2$ distribution for the inverse temperature which is actually the Maxwell distribution for the speed. The difference between the estimator and the variance introduces a fluctuating temperature that is generally different than the temperature of the variance. Fluctuations in the energy of a composite system are transferred to fluctuations in the temperature via Lévy's transform. The randomization of the operational parameter leading to a subordinated process requires the same number of degrees of freedom in the directing as in the original process. In place of Student's pdf we find the subordinated process to Maxwell's speed distribution has a Pareto distribution which is interpreted as the density of the length of the random velocity vector.

cond-mat.stat-mech↗
arXiv2003

Qualms concerning Tsallis's condition of pseudo-additivity as a definition of non-extensivity

The pseudo-additive relation that the Tsallis entropy satisfies has nothing whatsoever to do with the super- and sub- additivity properties of the entropy. The latter properties, like concavity and convexity, are couched in geometric inequalities and cannot be reduced to equalities. Rather, the pseudo-additivity relation is a functional equation that determines the functional forms of the random entropies. The Arimoto entropy satisfies a similar pseudo-additive relation and yet it is a first order homogeneous form. Hence no conclusions can be drawn on the extensive nature of the system from either the Tsallis or Arimoto entropy based on the pseudo-additive equation.

cond-mat.stat-mech↗
arXiv2003

Three tests of general relativity via Fermat's principle and the phase of Bessel functions

Fermat's principle applied to a flat metric in the plane yields the phase of a Bessel function in the periodic domain for a constant index of refraction. Gravitational forces cause the index of refraction to vary and lead to a modified phase of the Bessel function. A distinction is made between the forces that cause acceleration: the gravitational force affects the optical properties of the medium whereas the centrifugal force does not, the latter being built into the phase of oscillations of the Bessel function. The time delay in radar echoes from planets is determined from Fermat's principle where the velocity of light is the phase velocity and the index of refraction varies on account of the gravitational potential. The deflection of light by a massive body is shown to be produced by a quadrupole interaction, and the perihelion shift requires both the gravitational potential, producing a closed orbit, and the quadrupole, causing the perihelion to rotate.

math-ph↗
arXiv2003

Additive Entropies of degree-q and the Tsallis Entropy

The Tsallis entropy is shown to be an additive entropy of degree-q that information scientists have been using for almost forty years. Neither is it a unique solution to the nonadditive functional equation from which random entropies are derived. Notions of additivity, extensivity and homogeneity are clarified. The relation between mean code lengths in coding theory and various expressions for average entropies is discussed.

physics.class-ph↗