Increase of Boltzmann entropy in a quantum forced harmonic oscillator

Recently, a quantum-mechanical proof of the increase of Boltzmann entropy in quantum systems that are coupled to an external classical source of work has been given. Here we illustrate this result by applying it to a forced quantum harmonic oscillator. We show plots of the actual temporal evolution of work and entropy for various forcing protocols. We note that entropy and work can be partially or even fully returned to the source, although both work and entropy balances are non-negative at all times in accordance with the minimal work principle and the Clausius principle, respectively. A necessary condition for the increase of entropy is that the initial distribution is decreasing (e.g., canonical). We show evidence that for a nondecreasing distribution (e.g., microcanonical), the quantum expectation of entropy may decrease slightly. Interestingly, the classical expectation of entropy cannot decrease, irrespective of the initial distribution, in the forced harmonic oscillator.

Figure 1

Classical transition probability, Eq. (31), for $\Phi =2.5$ and $W=1.5$. The initial microcanonical distribution evolves to a U-shaped distribution.

Figure 2

(Color online) Quantum entropy as a function of the initial quantum number $n$ for $W=10$. Solid line: classical case, Eq. (29). Dots: quantum case, Eq. (41); the sum has been truncated at $m=1000$. The quantum phenomena smooth out the sharp angle in the classical graph.

Figure 3

Quantum entropy change for an initial microcanonical condition with $n=1,2,3,4,5$, as a function of work $W$.

Figure 4

Work as a function of time for the exponential driving protocol in Eq. (45) with $L=8$, $\tau =3$. This is the same in the quantum and classical cases.

Figure 5

(Color online) Entropy change as a function of time for the exponential driving protocol in Eq. (45) with $L=8$, $\tau =3$, and the initial microcanonical distribution of energy $E=n+1∕2$ with $n=2$. Solid green line, classical case, Eq. (29). Blue dashed line, quantum case, Eq. (41); the summation is truncated at $m=100$.

Figure 6

(Color online) Entropy change as a function of time for the exponential driving protocol in Eq. (45) with $L=8$, $\tau =3$, and the initial canonical distribution of inverse temperature $\beta =2$. The canonical distribution is truncated at $n=20$. Solid green line, classical case, Eq. (29). Blue dashed line, quantum case, Eq. (41); the summation is truncated at $m=100$.

Figure 7

Work as a function of time for the sinusoidal driving protocol in Eq. (47) with $L=3∕2$, $\Omega =2$.

Figure 8

(Color online) Entropy change as a function of time for the sinusoidal driving protocol in Eq. (47) with $L=3∕2$, $\Omega =2$, and the initial microcanonical distribution of energy $E=n+1∕2$ with $n=2$. Solid green line, classical case, Eq. (29). Blue dashed line, quantum case, Eq. (41); the summation is truncated at $m=100$.

Figure 9

(Color online) Entropy change as a function of time for the sinusoidal driving protocol in Eq. (47) with $L=3∕2$, $\Omega =2$, and the initial canonical distribution of inverse temperature $\beta =2$. The canonical distribution is truncated at $n=20$. Solid green line, classical case, Eq. (29). Blue dashed line, quantum case, Eq. (41); the summation is truncated at $m=100$.

Figure 10

(Color online) Entropy change for the force in Eq. (47) with $L=6$, $n=2$, as a function of switching time $T$, for the initial microcanonical condition. Solid line: classical case, Eq. (29), with $\Phi =n+1∕2=5∕2$. Dashed line: quantum case, Eq. (41); the summation is truncated at $m=1000$.

Figure 11

(Color online) Entropy change for the force in Eq. (47) with $L=6$, $\beta =2$, as a function of switching time $T$, for the initial canonical distribution. The canonical distribution is truncated at $n=100$. Solid line: classical case, Eq. (29), with $\Phi =n+1∕2=5∕2$. Dashed line: quantum case, Eq. (41); the summation is truncated at $m=1000$.