Quantum work statistics in regular and classical-chaotic dynamical billiard systems

In the thermodynamics of nanoscopic systems the relation between classical and quantum mechanical description is of particular importance. To scrutinize this correspondence we have chosen two two-dimensional billiard systems. Both systems are studied in the classical and the quantum mechanical settings. The classical conditional probability density $p(E,L|E_0,L_0)$ as well as the quantum mechanical transition probability $P(n,l|n_0,l_0)$ are calculated, which build the basis for the statistical analysis. We calculate the work distribution for one particle. The results in the quantum case in particular are of special interest since a suitable definition of mechanical work in small quantum systems is already controversial. Furthermore, we analyze the probability of both zero work and zero angular momentum difference. Using connections to an exactly solvable system analytical formulas are given for both systems. In the quantum case we get numerical results with some interesting relations to the classical case.

Figure 1

The radially breathing circle (System 1) and the horizontally breathing stadium (System 2).

Figure 2

Typical trajectories of classical particles at $x_0=0.3R_0$, $y_0=0$, $v=20u$ (bold plus sign) and a shooting angle of ${45}^{\circ }$ (red) and ${46}^{\circ }$ (blue) in (a) System 1 and (b) System 2.

Figure 3

Trajectory end points of classical particles with initial conditions $x_0=0.3R_0$, $y_0=0$ and $v=20u$ (bold plus sign) at different shooting angles between 0 and $2\pi$ for System 1 (a, b) and (c, d). In (a) and (c) 360, and in (b) and (d) 3600 equidistant varying shooting angles are used. The color code shows the final energies.

Figure 4

Initial wave function for $n=6$ and $l=3$ (a) and its final state in System 1 (b) or in System 2 (c). The calculations were performed with parameters according to Eqs. (4) and (16).

Figure 5

Cumulative conditional probability for (a) System 1 and (b) System 2. Classical calculations were performed for ${10}^5$ particles with an initial kinetic energy of $E_{5,2}$ regardless of the angular momentum (blue) and with explicit consideration of $L=\hslash l_0$ (solid black). In the quantum case (red) we start with a wave function in the eigenstate $n_0=5$ and $l_0=2$. The calculations were performed with parameters according to Eqs. (4) and (16).

Figure 6

Classical (blue) and quantum mechanical (red) cumulative work distribution at different temperatures $1/\overline{\beta }$ for (a–c) System 1 and (d–f) System 2. The calculations were performed with parameters according to Eqs. (4) and (16).

Figure 7

Classical (blue) and quantum mechanical (red) probability of no energy change depending on temperature $1/\overline{\beta }$ for (a) System 1 and (b) System 2. The blue solid lines are given by Eq. (23), and the dashed line is given by Eq. (24). The calculations were performed with parameters according to Eqs. (4) and (16).

Figure 8

Classical (blue) and quantum mechanical (red) probability of no angular momentum change depending on temperature $1/\overline{\beta }$ for System 2. The blue solid line is given by Eq. (28). The calculations were performed with parameters according to Eqs. (4) and (16).