Campbell Baker Hausdorff Research Articles

Diffusion in a laminar flow: Shear rate dependence of correlation functions and of effective transport coefficients

The diffusion equation for independent Brownian particles suspended in a fluid undergoing plane Couette shear flow is solved in Fourier space by means of the Campbell–Baker–Hausdorff expansion for the product of exponentials of noncommuting operators. Explicit solutions are derived and numerically evaluated for an initial Gaussian distribution with no source and for a continuous stationary source with a Gaussian spatial distribution. For the latter problem, the tensor describing the curvature of the steady-state distribution at the origin is analyzed in some detail and is shown to possess a dependence on shear rate very similar to that of the pressure tensor obtained in computer simulations of simple liquids under shear by Hanley and Evans.

Lie Algebras of Vector Fields and Local Approximation of Attainable Sets

Consider an analytic, n-dimensional control system described by ${{dx} / {dt}} = X(x) + u(t)Y(x)$, $x(0) = p$ and let $\mathcal{A}(t,p)$ denote the set of states attainable at time t by use of all admissible control functions u, which we take as measurable functions with values $|u(t)| \leqq 1$. Our goal is to derive second order conditions to determine if the reference solution, $(\exp tX)(p)$, corresponding to $u(t) \equiv 0$, lies on the boundary or interior of $\mathcal{A}(t,p)$ for small $t > 0$. If $t_1 > 0$ and $p^1 = (\exp t_1 X)(p)$ the approach is to use the Campbell–Baker–Hausdorff formula to obtain second order tangent vectors to $\mathcal{A}(t_1 ,p)$ at $p^1 $. These involve elements of the Lie algebra generated by X and Y having an arbitrary number of X factors and two Y factors. With certain hypotheses, for $n = 2,3$ relatively complete results are obtained.

Lie series and invariant functions for analytic symplectic maps

Symplectic maps (canonical transformations) are treated from the Lie algebraic point of view using Lie series and Lie algebraic techniques. It is shown that under very general conditions an analytic symplectic map can be written as a product of Lie transformations. Under certain conditions this product of Lie transformations can be combined to form a single Lie transformation by means of the Campbell–Baker–Hausdorff theorem. This result leads to invariant functions and generalizes to several variables a classic result of Birkhoff for the case of two variables. It also provides a new approach since the connection between symplectic maps, Lie algebras, invariant functions, and Birkhoff’s work has not been previously recognized and exploited. It is expected that the results obtained will be applicable to the normal form problem in Hamiltonian mechanics, the use of the Poincaré section map in stability analysis, and the behavior of magnetic field lines in a toroidal plasma device.