Shortcuts to isothermality and nonequilibrium work relations

In conventional thermodynamics, it is widely acknowledged that the realization of an isothermal process for a system requires a quasistatic controlling protocol. Here we propose and design a strategy to realize a finite-rate isothermal transition from an equilibrium state to another one at the same temperature, which is named the “shortcut to isothermality.” By using shortcuts to isothermality, we derive three nonequilibrium work relations, including an identity between the free-energy difference and the mean work due to the potential of the original system, a Jarzynski-like equality, and the inverse relationship between the dissipated work and the total driving time. We numerically test these three relations by considering the motion of a Brownian particle trapped in a harmonic potential and dragged by a time-dependent force.

Figure 1

Comparison of estimates of $\mathrm{\Delta }F$ from Eqs. (28) (triangles), (29) (circles), (35) (squares), and the Jarzynski equality (36) (diamonds). The dashed line is a fitted curve for Eq. (29) with the consideration of Eq. (32), while the solid line represents the theoretical value for $\mathrm{\Delta }F=-32$.

Figure 2

Comparison of estimates of $\mathrm{\Delta }F$ from relation (35) with different initial fixed states.

Figure 3

Comparison of estimates of $\mathrm{\Delta }F$ from relation (35) with different initial fixed states. The process is conducted under a new driving protocol, $\tilde{\lambda }\left(s\right)=1-\cos \left(\pi s\right)$.