New Class of Classical Uncertainty Relations Giving Uncertainty for Long and Certainty for Short Times

Abstract

Bohr's concept of complementarity enters any statistical description of physical phenomena. In quantum theory the complementary quantities are dynamical variables. In classical theory complementarity exists between dynamical and statistical variables. Slowing down of a "large" particle of mass $m$ by multiple collisions with a gas of "small" molecules leads to certainty for "short times." For "long times," the multiple collisions introduce statistics leading to uncertainty. An uncertainty relation has been derived for the coordinate $q$ as the dynamical and the drift momentum $p_d$ as the statistical complementary variable: $\Delta p_d\Delta q\ge mD(1-e^{-\frac{t}{\tau }})$, where $D$ is a diffusion coefficient and $\tau$ a relaxation time. This relation gives uncertainty for "long" and certainty for "short" times.