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We study non-relativistic particles and waves in N dimensions in a time-dependent potential V (r / I(r))/(l(f))', which describes a force tield that expands and weakens as the scale factor I increases. I t I: is a quadratic function of time, then, in a reference frame expanding with the system and employing clocks recalibrated to read a scaled time that depends on I (f) , the classical and quantum evolutions can be described byaconservative Hamiltonian differing from the original one by an 'inertial' term quadratic in the position variables. The quantal 'expanding modes' form a complete set whose energies decrease and which carry current outwards from the centre of expansion. Non-equilibrium statistical ensembles can be constructed, expanding with the force field.

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The theory of first-passage times of Brownian motion is developed in general, and it is shown that for certain special boundaries—the only ones of any importance—mean first-passage times can be derived very simply, avoiding the usual method involving series. Moreover, these formulae have a close analytical relationship to the better-known type of formulae for average 'displacements’ in given intervals; there exist certain pairs of reciprocal relations. Some new formulae, of mathematical interest, for translational Brownian motion are given. The main application of the general theory, however, lies in the derivation of experimentally particularly useful formulae for rotational Brownian motion. Special cases when external forces are present, and mean reciprocal first-passage times are discussed briefly, and finally it is shown how finite times of observation modify the mean first-passage time formulae of free Brownian motion.