Construction of a non-standard quantum field theory through a generalized Heisenberg algebra

We construct a Heisenberg-like algebra for the one dimensional quantum free Klein-Gordon equation defined on the interval of the real line of length $L$. Using the realization of the ladder operators of this type Heisenberg algebra in terms of physical operators we build a 3+1 dimensional free quantum field theory based on this algebra. We introduce fields written in terms of the ladder operators of this type Heisenberg algebra and a free quantum Hamiltonian in terms of these fields. The mass spectrum of the physical excitations of this quantum field theory are given by $\sqrt{n^2 \pi^2/L^2+m_q^2}$, where $n= 1,2,...$ denotes the level of the particle with mass $m_q$ in an infinite square-well potential of width $L$.Comment: Latex, 16 page

Generalized quantum field theory: perturbative computation and perspectives

We analyze some consequences of two possible interpretations of the action of the ladder operators emerging from generalized Heisenberg algebras in the framework of the second quantized formalism. Within the first interpretation we construct a quantum field theory that creates at any space-time point particles described by a q-deformed Heisenberg algebra and we compute the propagator and a specific first order scattering process. Concerning the second one, we draw attention to the possibility of constructing this theory where each state of a generalized Heisenberg algebra is interpreted as a particle with different mass.Comment: 19 page

Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations

A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The H-theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard -- linear Fokker-Planck equation -- may be also related to a family of nonlinear Fokker-Planck equations.Comment: 19 pages, no figure

Generalized entropy arising from a distribution of q-indices

It is by now well known that the Boltzmann-Gibbs (BG) entropy $S_{BG}=-k\sum_{i=1}^W p_i \ln p_i$ can be usefully generalized into the entropy $S_q=k (1-\sum_{i=1}^Wp_i^{q}) / (q-1)$ ($q\in \mathcal{R}; S_1=S_{BG}$). Microscopic dynamics determines, given classes of initial conditions, the occupation of the accessible phase space (or of a symmetry-determined nonzero-measure part of it), which in turn appears to determine the entropic form to be used. This occupation might be a uniform one (the usual {\it equal probability hypothesis} of BG statistical mechanics), which corresponds to $q=1$; it might be a free-scale occupancy, which appears to correspond to $q \ne 1$. Since occupancies of phase space more complex than these are surely possible in both natural and artificial systems, the task of further generalizing the entropy appears as a desirable one, and has in fact been already undertaken in the literature. To illustrate the approach, we introduce here a quite general entropy based on a distribution of $q$-indices thus generalizing $S_q$. We establish some general mathematical properties for the new entropic functional and explore some examples. We also exhibit a procedure for finding, given any entropic functional, the $q$-indices distribution that produces it. Finally, on the road to establishing a quite general statistical mechanics, we briefly address possible generalized constraints under which the present entropy could be extremized, in order to produce canonical-ensemble-like stationary-state distributions for Hamiltonian systems.Comment: 14 pages including 3 figure

Distribution of Eigenvalues of Ensembles of Asymmetrically Diluted Hopfield Matrices

Using Grassmann variables and an analogy with two dimensional electrostatics, we obtain the average eigenvalue distribution $\rho(\omega)$ of ensembles of $N \times N$ asymmetrically diluted Hopfield matrices in the limit $N \rightarrow \infty$. We found that in the limit of strong dilution the distribution is uniform in a circle in the complex plane.Comment: 9 pages, latex, 4 figure

Influence of Refractory Periods in the Hopfield model

We study both analytically and numerically the effects of including refractory periods in the Hopfield model for associative memory. These periods are introduced in the dynamics of the network as thresholds that depend on the state of the neuron at the previous time. Both the retrieval properties and the dynamical behaviour are analyzed.Comment: Revtex, 7 pages, 7 figure

Generalized Simulated Annealing

We propose a new stochastic algorithm (generalized simulated annealing) for computationally finding the global minimum of a given (not necessarily convex) energy/cost function defined in a continuous D-dimensional space. This algorithm recovers, as particular cases, the so called classical ("Boltzmann machine") and fast ("Cauchy machine") simulated annealings, and can be quicker than both. Key-words: simulated annealing; nonconvex optimization; gradient descent; generalized statistical mechanics.Comment: 13 pages, latex, 4 figures available upon request with the authors