Destructive effect of fluctuations on the performance of a Brownian gyrator

The Brownian gyrator (BG) is often called a minimal model of a nano-engine performing a rotational motion, judging solely upon the fact that in non-equilibrium conditions its torque, specific angular momentum 0000000111101100 0000011000010010 0000010000100010 0000100000100100 0000100001001000 0000000111110000 0000001110000000 0000000100000000 0011011100010000 0100010000100000 0100011001000000 0011100110000000 and specific angular velocity 001100001100000110 010010000100001010 000010000100001000 000001000100001000 000001000010010000 000001000110010000 000000101010010000 000000101001010000 000000101001010000 000000010000100000 000000100000010000 have non-zero mean values. For a time-discretised (with time-step δt) model we calculate here the previously unknown probability density functions (PDFs) of and . We show that for finite δt, the PDF of has exponential tails and all moments are therefore well-defined. At the same time, this PDF appears to be effectively broad - the noise-to-signal ratio is generically bigger than unity meaning that is strongly not self-averaging. Concurrently, the PDF of exhibits heavy power-law tails and its mean is the only existing moment. The BG is therefore not an engine in the common sense: it does not exhibit regular rotations on each run and its fluctuations are not only a minor nuisance - on contrary, their effect is completely destructive for the performance. Our theoretical predictions are confirmed by numerical simulations and experimental data. We discuss some plausible improvements of the model which may result in a more systematic rotational motion.