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It is shown that in systems with time-dependent and/or spatially nonuniformtemperature $T(t,x)$, (i) most of the transport processes is weaklynon-ergodic, and (ii) the diffusion (Brownian motion, BM) is anomalous. A fewexamples of simple arrangements, easy for experimental realization, arediscussed in detail. Proposed measurements will enable also the observation oftransitions from ergodic to weakly non-ergodic and from normal to anomalousdiffusion. New effects are predicted: (i) zero-mean oscillations of $T(t)$accelerate BM (pumping effect), (ii) the combination of temporal and spatialvariations of temperature may lead to superballistic BM, (iii) linear gradientsof $T(x)$ result in an exponential acceleration of BM. One can expect similareffects in inflationary systems with time-dependent metrics.

How to generate and measure anomalous weakly non-ergodic Brownian motion in simple systems