How multiplicity determines entropy and the derivation of the maximum entropy principle for complex systems

The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems, by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and Markovian systems in statistical mechanics, information theory, and statistics. For several decades there exists an ongoing controversy whether the notion of the maximum entropy principle can be extended in a meaningful way to non-extensive, non-ergodic, and complex statistical systems and processes. In this paper we start by reviewing how Boltzmann-Gibbs-Shannon entropy is related to multiplicities of independent random processes. We then show how the relaxation of independence naturally leads to the most general entropies that are compatible with the first three Shannon-Khinchin axioms, the (c,d)-entropies. We demonstrate that the MEP is a perfectly consistent concept for non-ergodic and complex statistical systems if their relative entropy can be factored into a generalized multiplicity and a constraint term. The problem of finding such a factorization reduces to finding an appropriate representation of relative entropy in a linear basis. In a particular example we show that path-dependent random processes with memory naturally require specific generalized entropies. The example is the first exact derivation of a generalized entropy from the microscopic properties of a path-dependent random process.Comment: 6 pages, 1 figure. To appear in PNA

Decoherent Histories Quantum Mechanics with One 'Real' Fine-Grained History

Decoherent histories quantum theory is reformulated with the assumption that there is one "real" fine-grained history, specified in a preferred complete set of sum-over-histories variables. This real history is described by embedding it in an ensemble of comparable imagined fine-grained histories, not unlike the familiar ensemble of statistical mechanics. These histories are assigned extended probabilities, which can sometimes be negative or greater than one. As we will show, this construction implies that the real history is not completely accessible to experimental or other observational discovery. However, sufficiently and appropriately coarse-grained sets of alternative histories have standard probabilities providing information about the real fine-grained history that can be compared with observation. We recover the probabilities of decoherent histories quantum mechanics for sets of histories that are recorded and therefore decohere. Quantum mechanics can be viewed as a classical stochastic theory of histories with extended probabilities and a well-defined notion of reality common to all decoherent sets of alternative coarse-grained histories.Comment: 11 pages, one figure, expanded discussion and acknowledgment

Neutrino Models of Dark Energy

I consider a scenario proposed by Fardon, Nelson and Weiner where dark energy and neutrinos are connected. As a result, neutrino masses are not constant but depend on the neutrino number density. By examining the full equation of state for the dark sector, I show that in this scenario the dark energy is equivalent to having a cosmological constant, but one that "runs" as the neutrino mass changes with temperature. Two examples are examined that illustrate the principal feautures of the dark sector of this scenario. In particular, the cosmological constant is seen to be negligible for most of the evolution of the Universe, becoming inportant only when neutrinos become non-relativistic. Some speculations on features of this scenario which might be present in a more realistic theory are also presented.Comment: 12 pages, 6 figures. Added comments on why FNW scenario always leads to a running cosmological constant and a few references. To be published in Phys. Rev.

Possible evidence of extended objects inside the proton

Recent experimental determinations of the Nachtmann moments of the inelastic structure function of the proton F2p(x, Q**2), obtained at Jefferson Lab, are analyzed for values of the squared four-momentum transfer Q**2 ranging from ~ 0.1 to ~ 2 (GeV/c)**2. It is shown that such inelastic proton data exhibit a new type of scaling behavior and that the resulting scaling function can be interpreted as a constituent form factor consistent with the elastic nucleon data. These findings suggest that at low momentum transfer the inclusive proton structure function originates mainly from the elastic coupling with extended objects inside the proton. We obtain a constituent size of ~ 0.2 - 0.3 fm.Comment: 1 reference adde

Decoherence Functional and Probability Interpretation

We confirm that the diagonal elements of the Gell-Mann and Hartle's decoherence decoherence functional are equal to the relative frequencies of the results of many identical experiments, when a set of alternative histories decoheres. We consider both cases of the pure and mixed initial states.Comment: 9 pages, UCSBTH-92-40 and MMC-M-

Generalized entropies and the transformation group of superstatistics

Superstatistics describes statistical systems that behave like superpositions of different inverse temperatures $\beta$, so that the probability distribution is $p(\epsilon_i) \propto \int_{0}^{\infty} f(\beta) e^{-\beta \epsilon_i}d\beta$, where the `kernel' $f(\beta)$ is nonnegative and normalized ($\int f(\beta)d \beta =1$). We discuss the relation between this distribution and the generalized entropic form $S=\sum_i s(p_i)$. The first three Shannon-Khinchin axioms are assumed to hold. It then turns out that for a given distribution there are two different ways to construct the entropy. One approach uses escort probabilities and the other does not; the question of which to use must be decided empirically. The two approaches are related by a duality. The thermodynamic properties of the system can be quite different for the two approaches. In that connection we present the transformation laws for the superstatistical distributions under macroscopic state changes. The transformation group is the Euclidean group in one dimension.Comment: 5 pages, no figur

Numerical indications of a q-generalised central limit theorem

We provide numerical indications of the $q$-generalised central limit theorem that has been conjectured (Tsallis 2004) in nonextensive statistical mechanics. We focus on $N$ binary random variables correlated in a {\it scale-invariant} way. The correlations are introduced by imposing the Leibnitz rule on a probability set based on the so-called $q$-product with $q \le 1$. We show that, in the large $N$ limit (and after appropriate centering, rescaling, and symmetrisation), the emerging distributions are $q_e$-Gaussians, i.e., $p(x) \propto [1-(1-q_e) \beta(N) x^2]^{1/(1-q_e)}$, with $q_e=2-\frac{1}{q}$, and with coefficients $\beta(N)$ approaching finite values $\beta(\infty)$. The particular case $q=q_e=1$ recovers the celebrated de Moivre-Laplace theorem.Comment: Minor improvements and corrections have been introduced in the new version. 7 pages including 4 figure

Neutrino Mass Matrices with a Texture Zero and a Vanishing Minor

We study the implications of the simultaneous existence of a texture zero and a vanishing minor in the neutrino mass matrix. There are thirty six possible texture structures of this type, twenty one of which reduce to two texture zero cases which have, already, been extensively studied. Of the remaining fifteen textures only six are allowed by the current data. We examine the phenomenological implications of the allowed texture structures for Majorana type CP-violating phases, 1-3 mixing angle and Dirac type CP-violating phase. All these possible textures can be generated through the seesaw mechanism and realized in the framework of discrete abelian flavor symmetry. We present the symmetry realization of these texture structures.Comment: To appear in Phys. Rev.

Neutrino Mixing Predictions of a Minimal SO(10) Model with Suppressed Proton Decay

During the past year, a minimal renormalizable supersymmetric SO(10) model has been proposed with the following properties: it predicts a naturally stable dark matter and neutrino mixing angles theta_atm and theta_13 while at the same time accommodating CKM CP violation among quarks with no SUSY CP problem. Suppression of proton decay for all allowed values of tan beta strongly restricts the flavor structure of the model making it predictive for other processes as well. We discuss the following predictions of the model in this paper, e.g. down-type quark masses, and neutrino oscillation parameters, U_e3, delta_MNSP, which will be tested by long baseline experiments such as T2K and subsequent experiments using the neutrino beam from JPARC. We also calculate lepton flavor violation and the lepton asymmetry of the Universe in this model.Comment: 22 pages, 11 figure

Constraints on leptogenesis from a symmetry viewpoint

It is shown that type I seesaw models based on the standard model Lagrangian extended with three heavy Majorana right-handed fields do not have leptogenesis in leading order, if the symmetries of mass matrices are also the residual symmetry of the Lagrangian. In particular, flavor models that lead to a mass-independent leptonic mixing have a vanishing leptogenesis CP asymmetry. Based on symmetry arguments, we prove that in these models the Dirac-neutrino Yukawa coupling combinations relevant for leptogenesis are diagonal in the physical basis where the charged leptons and heavy Majorana neutrinos are diagonal.Comment: 5 pages; a few comments added; final version to appear in Phys. Rev.