Work relations for a system governed by Tsallis statistics

We derive analogs of the Jarzynski equality and Crooks relation to characterize the nonequilibrium work associated with changes in the spring constant of an overdamped oscillator in a quadratically varying spatial temperature profile. The stationary state of such an oscillator is described by Tsallis statistics, and the work relations for certain processes may be expressed in terms of $q$-exponentials. We suggest that these identities might be a feature of nonequilibrium processes in circumstances where Tsallis distributions are found.

Figure 1

Illustration of a particle trajectory $x\left(t\right)$ in a nonisothermal environment (indicated by background color) as the spring constant changes in a sequence of steps. The initial distribution of positions takes the stationary Tsallis form $P_{\mathrm{st}}^{\kappa \left(0\right)}$. At time $\tau$ the distribution need not take the stationary form $P_{\mathrm{st}}^{\kappa \left(\tau \right)}$, shown for comparison.

Figure 2

Distribution of values of $Y={\prod }_{i=1}^4{e}_{q_i}\left(-\mathrm{\Delta }{W}_i/k_B{T}_0\right)$ for a cyclic process with ${\kappa }_0=1, {\kappa }_1=1.5, {\kappa }_2=2, {\kappa }_3=0.5$, and ${\kappa }_4=1$, with $t_2-t_1=t_3-t_2=t_4-t_3=0.1$, and $m=\gamma =k_B=T_0={\kappa }_T=1$, obtained from ${10}^5$ realizations with a time step of ${10}^{-4}$. The mean of the distribution is 0.999 with statistical uncertainty of 0.001, verifying the analog Jarzynski equality (25) for this process.

Figure 3

Probability distributions of work for process $F$ consisting of a step up from ${\kappa }_0={\kappa }_T$ to ${\kappa }_1=2{\kappa }_T$ (positive values of $W$), and process $R$ specified by a step down from ${\kappa }_1$ to ${\kappa }_0$ (negative values of $W$), both starting from a stationary state. The mean work for the $F$ process is $k_B{T}_0$ and for the $R$ process it is $-k_B{T}_0/3$.

Figure 4

Verification of the analog Crooks relation Eq. (30) for a step up from ${\kappa }_0={\kappa }_T$ to ${\kappa }_1=2{\kappa }_T$ and a step down from ${\kappa }_1$ to ${\kappa }_0$, based on ${10}^6$ particle trajectories with a time step of ${10}^{-4}$, with $m=\gamma =k_B=T_0={\kappa }_T=1$.