Abstract

Obtaining time-averaged quantities with sufficient accuracy can be challenging computationally for systems with a chaotic behaviour. We replace the quantity being averaged with another quantity having the same average but such that it is easier to average. If W(t) is a bounded differentiable function, then the infinite time average of its derivative is zero. Hence, rather than numerically averaging the quantity of interest, which we will denote F, one can average F+dW/dt. We explore first the simplest way of choosing W(t), which is to ensure that the fluctuation amplitude of F+dW/dt is smaller than the fluctuation amplitude of F. For this, F and dW/dt should be correlated. This can often be achieved by taking W(t)=V(x(t)), where x is the state of the dynamical system. The talk will discuss our tests of this idea. (A spoiler: the acceleration is only moderate but is worth doing because it is easy. Further improvement requires progress on interesting and challenging problems.)