Jarzynski equality: Connections to thermodynamics and the second law

The one-dimensional expanding ideal gas model is used to compute the exact nonequilibrium distribution function. The state of the system during the expansion is defined in terms of local thermodynamics quantities. The final equilibrium free energy, obtained a long time after the expansion, is compared against the free energy that appears in the Jarzynski equality. Within this model, where the Jarzynski equality holds rigorously, the free energy change that appears in the equality does not equal the actual free energy change of the system at any time of the process. More generally, the work bound that is obtained from the Jarzynski equality is an upper bound to the upper bound that is obtained from the first and second laws of thermodynamics. The cancellation of the dissipative (nonequilibrium) terms that result in the Jarzynski equality is shown in the framework of response theory. This is used to show that the intuitive assumption that the Jarzynski work bound becomes equal to the average work done when the system evolves quasistatically is incorrect under some conditions.

Figure 1

The initial equilibrium distribution function at $t=0$ is shown on the left and the nonequilibrium distribution function at $t=4$ is shown on the right. Here, $L_i=10$, $V=0.5$, and $T_i=1$ and the moving piston is on the right. The contours are drawn when $f(x,u)=0.005$, 0.01, 0.015, 0.02, 0.025, 0.03, and 0.035.

Figure 2

The local thermodynamics quantities defined by Eqs. (2.7, 2.8, 2.9, 2.10) are shown as a function of time from $t=0$ to $t=4$ for an expansion where $L_i=10$ and $V=0.5$. At $t=0$, all quantities are uniform for $0<x<L_i$ and zero elsewhere. The curves are equally spaced at time intervals of $4∕9$. Larger values of $t$ have larger nonuniform regions.

Figure 3

The fine-grained distribution function (on the left) is compared with the time-averaged distribution function that is later used to calculate the coarse-grained entropy according to Eq. (2.16b). Here, $t=1000$ and $\tau =100$. The contours are drawn when $f(x,u)=0.005$, 0.01, 0.015, 0.02, 0.025, 0.03, and 0.035.

Figure 4

The average work $W$ is obtained from Eq. (3.13) with $T=1$, $L_i=10$, and $Vt=10$. This is compared with the Jarzynski bound, which is independent of the piston velocity.

Figure 5

The real free energy difference $\Delta A$ between the two equilibrium states is compared with $\Delta A_J$. The conditions of the expansion are the same as in Fig. 4.