Saturation of entropy production in quantum many-body systems

Bridging the second law of thermodynamics and microscopic reversible dynamics has been a longstanding problem in statistical physics. Here, we address this problem on the basis of quantum many-body physics, and discuss how the entropy production saturates in isolated quantum systems under unitary dynamics. First, we rigorously prove that the entropy production does indeed saturate in the long time regime, even when the total system is in a pure state. Second, we discuss the non-negativity of the entropy production at saturation, implying the second law of thermodynamics. This is based on the eigenstate thermalization hypothesis, which states that even a single energy eigenstate is thermal. We also numerically demonstrate that the entropy production saturates at a non-negative value even when the initial state of a heat bath is a single energy eigenstate. Our results reveal fundamental properties of the entropy production in isolated quantum systems at late times.

Figure 1

Schematic of a time evolution of the entropy production. Starting from a nonequilibrium initial state, the entropy production increases, and then saturates at some value $\langle {\sigma }_{\infty }\rangle$. The long time behavior is the main focus of the present work, while the short time behavior can be explained by our previous work [38] based on the Lieb-Robinson bound.

Figure 2

Schematic of our setup. The total system consists of system $\mathrm{S}$ and bath $\mathrm{B}$.

Figure 3

Initial state dependence of the effective dimensions $D_{\mathrm{eff}}$ for the one-dimensional nonintegrable model of hard-core bosons with the bath size $L=8,12,16$. We consider the case that bath $\mathrm{B}$ is $1/4$ filling ($N_{\mathrm{b}}/L=1/4$) and the initial state is a pure state $|1\rangle |E_j^{\mathrm{B}}\rangle$. The parameters are given by $\epsilon /\gamma =1,{\gamma }_{\mathrm{I}}/\gamma =1,g/\gamma =0.1,{\gamma }^{\prime }/\gamma =0.2,\text{and}{g}^{\prime }/\gamma =0.1$. The effective dimension increases with $L$.

Figure 4

Time dependence of the entropy production with $L=16$ and $N_{\mathrm{b}}=5$. The entropy production is plotted against dimensionless time $\gamma t$, for three interaction strengths between system $\mathrm{S}$ and bath $\mathrm{B}$: ${\gamma }_{\mathrm{I}}/\gamma =0.5,1,2$. We set other parameters as $\epsilon /\gamma =1,g/\gamma =0.1,{\gamma }^{\prime }/\gamma =0.2,\text{and}{g}^{\prime }/\gamma =0.1$. The initial state is the product state $|1\rangle |E_j^{\mathrm{B}}\rangle$ and the inverse temperature of the initial energy eigenstate is $\beta =0.1$. The entropy production is non-negative in all time scales, and saturates in the long time $\gamma t\gtrsim {10}^2$. We note that the saturation values are almost the same for all the interaction strengths ${\gamma }_{\mathrm{I}}$.