Single-particle stochastic heat engine

We have performed an extensive analysis of a single-particle stochastic heat engine constructed by manipulating a Brownian particle in a time-dependent harmonic potential. The cycle consists of two isothermal steps at different temperatures and two adiabatic steps similar to that of a Carnot engine. The engine shows qualitative differences in inertial and overdamped regimes. All the thermodynamic quantities, including efficiency, exhibit strong fluctuations in a time periodic steady state. The fluctuations of stochastic efficiency dominate over the mean values even in the quasistatic regime. Interestingly, our system acts as an engine provided the temperature difference between the two reservoirs is greater than a finite critical value which in turn depends on the cycle time and other system parameters. This is supported by our analytical results carried out in the quasistatic regime. Our system works more reliably as an engine for large cycle times. By studying various model systems, we observe that the operational characteristics are model dependent. Our results clearly rule out any universal relation between efficiency at maximum power and temperature of the baths. We have also verified fluctuation relations for heat engines in time periodic steady state.

Figure 1

Stochastic heat engine in a harmonic potential: the time dependence of our periodically driven stiffness constant (protocol) $k\left(t\right)$ for the full cycle $\left(0\le t\le \tau \right)$.

Figure 2

Schematic representation for a cyclic process of stochastic heat engine operating between two reservoirs kept at temperatures $T_h$ and $T_l$. The cycle consists of two isothermal steps and two adiabatic steps according to the time varying protocol $k\left(t\right)$. The blue line denotes a one dimensional potential $V(x,t)$ and the filled region denotes the corresponding steady state distribution.

Figure 3

Phase diagram for different $T_h$ and $\tau$ but for fixed $T_l=0.1$.

Figure 4

Variation of $\langle w\rangle ,\langle q_1\rangle$, and $\langle q_2\rangle$ with cycle time $\tau$.

Figure 5

Variation of $\langle \eta \rangle$ and $\overline{\eta }$ with cycle time $\tau$. The dotted blue straight line denotes the quasistatic limit for $\overline{\eta }$.

Figure 6

Variation of average power $\langle p\rangle$ with cycle time $\tau$.

Figure 7

Distribution of $w$ for different cycle times $(\tau =0.7$, 7.0, 70.0).

Figure 8

Distribution of $q_1$ for different cycle times.

Figure 9

Distribution of $\eta$ for different cycle times.

Figure 10

Joint distribution of $w$ and $q_1$ for different $\tau$. In (a) $\tau =0.7$, in (b) $\tau =7.0$, and in (c) $\tau =70$.

Figure 11

Initial phase space distribution at different cycle times $\tau$. In (a) $\tau =0.7$, in (b) $\tau =7.0$, in (c) $\tau =70$. The asymmetric position of the red broken line along the major axis of the elliptical Gaussian distribution for lower values of $\tau$ $(=0.7$ and 7.0) indicates nonzero $\langle xv\rangle$. This correlation becomes zero for larger $\tau$ $\left(=70\right)$, where the position of the major axis also becomes symmetric.

Figure 12

Distribution of the internal energy change in one cycle for underdamped steady state for $\tau =7.0$.

Figure 13

Distribution of system entropy change and total entropy production in one cycle for underdamped steady state for $\tau =7.0$

Figure 14

Variation of $\langle w\rangle$ vs $\tau$ for symmetric as well as asymmetrical cycles. Here, ${\tau }_h$ and ${\tau }_l$ are contact times of the particle with hot and cold baths, respectively. Thus, $\tau ={\tau }_h+{\tau }_l=7.0$ for our case.

Figure 15

Variation of $\langle \eta \rangle$ vs $\tau$ for symmetric as well as asymmetrical cycles. Inset: comparison of $\langle \eta \rangle$ with $\overline{\eta }$ for ${\tau }_h:{\tau }_l=3:1$.

Figure 16

Variation of $\langle \eta \rangle$ with $\tau$ for different types of potentials.

Figure 17

Variation of power $\langle p\rangle$ with $\tau$ for different types of potentials.

Figure 18

Phase diagram for different $T_h$ and $\tau$ but for fixed $T_l=0.1$.

Figure 19

Variation of $\langle w\rangle ,\langle q_1\rangle$, and $\langle q_2\rangle$ with cycle time $\tau$.

Figure 20

Variation of $\langle \eta \rangle$ and $\overline{\eta }$ with cycle time $\tau$. The dotted blue straight line denotes the quasistatic limit for $\overline{\eta }$.

Figure 21

Variation of $\langle p\rangle$ with cycle time $\tau$.

Figure 22

Distribution of $w$ for different cycle times in the overdamped case.

Figure 23

Distribution of $q_1$ for different cycle times in the overdamped case.

Figure 24

Distribution of $\eta$ for different cycle times in the overdamped case.