Driven Brownian particle as a paradigm for a nonequilibrium heat bath: Effective temperature and cyclic work extraction

We apply the concept of a frequency-dependent effective temperature based on the fluctuation-dissipation ratio to a driven Brownian particle in a nonequilibrium steady state. Using this system as a thermostat for a weakly coupled harmonic oscillator, the oscillator thermalizes according to a canonical distribution at the respective effective temperature across the entire frequency spectrum. By turning the oscillator from a passive thermometer into a heat engine, we realize the cyclic extraction of work from a single thermal reservoir, which is feasible only due to its nonequilibrium nature.

Figure 1

Scheme of the model. The coupling is realized by a linear velocity-dependent potential. The oscillator is used as a thermometer of the driven Brownian system.

Figure 2

(a) Probability distribution of the internal energy $E_{\mathrm{h}}$ of an oscillator with eigenfrequency ${\omega }_{\mathrm{h}}=6.0$ sampled from simulations (symbols) of the equilibrium case ($f=0$) and two NESSs ($f=5,7$). The data are fitted (lines) with respect to the oscillator temperature $T_{\mathrm{h}}$ using Eq. (7). (b) Marginal distributions of rescaled oscillator position ${\omega }_{\mathrm{h}}{q}_{\mathrm{h}}$ (lines) and velocity ${\dot{q}}_{\mathrm{h}}$ (symbols) for driving forces $f=5$ (green line and crosses) and $f=7$ (red line and circles). The Gaussian distributions match in both cases, suggesting that an equipartition principle holds with respect to quadratic terms in the internal oscillator energy.

Figure 3

Correlation and response functions for the equilibrium case ($f=0$) and three NESSs ($f=3,5,9$). In equilibrium at a bath temperature $T=1$, the FDT requires that $C\left(\tau \right)/R\left(\tau \right)=1$ for all $\tau$. Here the oscillations in $C\left(\tau \right)$ and $R\left(\tau \right)$ reflect the underdamped motion in the potential wells. They are damped out by the driving for intermediate forces ($f=3$). In the running-state regime ($f=5,9$), oscillations reappear but are now caused by the impact of the periodic potential on the drifting particle.

Figure 4

Frequency-dependent effective temperature $T_{\mathrm{eff}}\left(\omega \right)$ (lines) for various driving forces $f$ in direct comparison with the kinetic and potential oscillator temperatures $T_{\mathrm{h}}^{\mathrm{kin}}$ (circles) and $T_{\mathrm{h}}^{\mathrm{pot}}$ (crosses). Standard deviations of the oscillator temperatures are always smaller than the symbol size. For $f=5$, the frequencies ${\omega }_1$ and ${\omega }_2$ and the respective values of the effective temperature $T_1$ and $T_2$ will be used in the cyclic protocol described at the end.

Figure 5

Effective temperature $T_{\mathrm{eff}}\left(\omega \right)$ for intermediate driving forces $f$ against frequency $\omega$ rescaled by the characteristic jump frequency ${\omega }_c\left(f\right)$ [Eq. (11)].

Figure 6

Scheme of the cyclic process as explained in the text.