Large Deviations for Nonlocal Stochastic Neural Fields

We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a $Q$-Wiener process and focus on noise-induced transitions and escape. We argue that proving a sharp Kramers' law for neural fields poses substanial difficulties but that one may transfer techniques from stochastic partial differential equations to establish a large deviation principle (LDP). Then we demonstrate that an efficient finite-dimensional approximation of the stochastic neural field equation can be achieved using a Galerkin method and that the resulting finite-dimensional rate function for the LDP can have a multi-scale structure in certain cases. These results form the starting point for an efficient practical computation of the LDP. Our approach also provides the technical basis for further rigorous study of noise-induced transitions in neural fields based on Galerkin approximations.Comment: 29 page

Computation of the p6p^6 order low-energy constants with tensor sources

We present the results of calculations of the $p^4$ and $p^6$ order low-energy constants for the chiral Lagrangian with tensor sources for both two and three flavors of pseudoscalar mesons. This is a generalization of our previous work on similar calculations without tensor sources in terms of the quark self-energy $\Sigma(p^2)$, based on the first principle derivation of the low-energy effective Lagrangian and computation of the low-energy constants with some rough approximations. With the help of partial integration and some epsilon relations, we find that some $p^6$ order operators with tensor sources appearing in the literature are related to each other. That leaves 98 independent terms for $n$-flavor, 92 terms for three-flavor, and 65 terms for two-flavor cases. We also find that the odd-intrinsic-parity chiral Lagrangian with tensor sources cannot independently exist in any order of low-energy expansion.Comment: 29 page

On the Thermodynamics of Particles Obeying Monotone Statistics

The aim of the present paper is to provide a preliminary investigation of the thermodynamics of particles obeying monotone statistics. To render the potential physical applications realistic, we propose a modified scheme called block-monotone, based on a partial order arising from the natural one on the spectrum of a positive Hamiltonian with compact resolvent. The block-monotone scheme is never comparable with the weak monotone one and is reduced to the usual monotone scheme whenever all the eigenvalues of the involved Hamiltonian are non-degenerate. Through a detailed analysis of a model based on the quantum harmonic oscillator, we can see that: (a) the computation of the grand-partition function does not require the Gibbs correction factor $n!$ (connected with the indistinguishability of particles) in the various terms of its expansion with respect to the activity; and (b) the decimation of terms contributing to the grand-partition function leads to a kind of "exclusion principle" analogous to the Pauli exclusion principle enjoined by Fermi particles, which is more relevant in the high-density regime and becomes negligible in the low-density regime, as expected.Comment: Published in Entropy, 2 Figure