Quantum-to-classical transition in the work distribution for chaotic systems

The work distribution is a fundamental quantity in nonequilibrium thermodynamics mainly due to its connection with fluctuation theorems. Here, we develop a semiclassical approximation to the work distribution for a quench process in chaotic systems that provides a link between the quantum and classical work distributions. The approach is based on the dephasing representation of the quantum Loschmidt echo and on the quantum ergodic conjecture, which states that the Wigner function of a typical eigenstate of a classically chaotic Hamiltonian is equidistributed on the energy shell. Using numerical simulations, we show that our semiclassical approximation accurately describes the quantum distribution as the temperature is increased.

Figure 1

Schematic depiction of the quench scheme: $H_0\to H_0+V\left(\xi \right)$, where $H_0=\left(p_x^2+p_y^2\right)$. The Gaussians in the quench potential are centered at $(x_1,y_1)=(0.2,0.4), (x_2,y_2)=(0.67,0.5), (x_3,y_3)=(0.5,0.15)$, and $(x_4,y_4)=(0.3,0.75)$.

Figure 2

Quantum (light blue/gray) and semiclassical (black) work distribution for different temperatures, from (a)–(d): $\beta =2^{-6}, 2^{-8}, 2^{-10}, 2^{-12}$. In the inset we show the characteristic function for each temperature (the $y$ axis is in ${log}_{10}$ scale).

Figure 3

Jarzinsky relation for the quenched described in the text. The squares correspond to the results obtained using the quantum work distribution, and the circles have been computed using the semiclassical approximation of the characteristic function for different temperatures. The scale of the $xy$ axis is ${log}_{10}-{log}_{10}$.

Figure 4

Classical and semiclassical work distributions for $\beta =2^{-12}$, and $\hslash =0.01$, 0.1, 0.5, 1. Semiclassical results were obtained by the numerical evaluation of Eq. (6). The scale of the $y$ axis is ${log}_{10}$.