53,334 research outputs found
Renormalization in the Henon family, I: universality but non-rigidity
In this paper geometric properties of infinitely renormalizable real
H\'enon-like maps in are studied. It is shown that the appropriately
defined renormalizations converge exponentially to the one-dimensional
renormalization fixed point. The convergence to one-dimensional systems is at a
super-exponential rate controlled by the average Jacobian and a universal
function . It is also shown that the attracting Cantor set of such a map
has Hausdorff dimension less than 1, but contrary to the one-dimensional
intuition, it is not rigid, does not lie on a smooth curve, and generically has
unbounded geometry.Comment: 42 pages, 5 picture
Semiclassical Series from Path Integrals
We derive the semiclassical series for the partition function in Quantum
Statistical Mechanics (QSM) from its path integral representation. Each term of
the series is obtained explicitly from the (real) minima of the classical
action. The method yields a simple derivation of the exact result for the
harmonic oscillator, and an accurate estimate of ground-state energy and
specific heat for a single-well quartic anharmonic oscillator. As QSM can be
regarded as finite temperature field theory at a point, we make use of Feynman
diagrams to illustrate the non-perturbative character of the series: it
contains all powers of and graphs with any number of loops; the usual
perturbative series corresponds to a subset of the diagrams of the
semiclassical series. We comment on the application of our results to other
potentials, to correlation functions and to field theories in higher
dimensions.Comment: 18 pages, 4 figures. References update
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