Prof. Predrag Cvitanović
Professor, School of Physics, Georgia Tech
Noise is your friend, or: How well can we resolve state space?
Monday, Nov 16, 2015, 3:00pm to 4:00pm | Room 5-314
All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. What is the best resolution possible for a given physical system?
It turns out that for nonlinear dynamical systems the noise itself is highly nonlinear, with the effective noise different for different regions of system's state space. The best obtainable resolution thus depends on the observed state, the interplay of local stretching/contraction with the smearing due to noise, as well as the memory of its previous states. We show how that is computed, orbit by orbit. But noise also associates to each a finite state space volume, thus helping us by both smoothing out what is deterministically a fractal strange attractor, and restricting the computation to a set of unstable periodic orbits of finite period. By computing the local eigenfunctions of the Fokker-Planck evolution operator, forward operator along stable linearized directions and the adjoint operator along the unstable directions, we determine the 'finest attainable' partition for a given hyperbolic dynamical system and a given weak additive noise. The space of all chaotic spatiotemporal states is infinite, but noise kindly coarse-grains it into a finite set of resolvable states.
Predrag Cvitanović is theoretical physicist at the Georgia Institute of Technology. He believes passionately in the unity of exact sciences, and in his ChaosBook.org he is developing theoretical tools that apply equally to classical chaos (turbulence, cardiac dynamics), stochastic dynamics and quantum theories. He has contributed to quantum field theory, the theory of exceptional Lie groups, and his nonlinear dynamics researches span a broad range of physical problems, from renormalization in transitions to chaos to periodic orbit theory of quantum systems to dynamical theory of hydrodynamical turbulence. Cvitanović is also known for his very reasonably priced webbooks on Field Theory, Group Theory, and Chaos.