Noncanonical statistics of a finite quantum system with non-negligible system-bath coupling

The canonical statistics describes the statistical properties of an open system by assuming its coupling with the heat bath is infinitesimal in comparison with the total energy in thermodynamic limit. In this paper, we generally derive a noncanonical density matrix for the open system with a finite coupling to the heat bath, which deforms the energy shell to effectively modify the conventional canonical way. The obtained noncanonical distribution reflects the back action of system on the bath and thus depicts the statistical correlations between two subsystems by the mutual information as a result of energy conservation.

Figure 1

The energy spectra of the system and the heat bath. The spectral density of the heat bath should be much larger than the system.

Figure 2

The subenergy shell without and with the consideration of the system-bath coupling are schematically shown in a manner analogous to classical phase space.

Figure 3

The noncanonical (blue solid line) and canonical (pink dashed line) distribution functions for coupled oscillators system. We choose $\omega =1$, $E=100$, $\beta =0.5$, and $\kappa =0.00235$. The highest energy level is $n_{\max }=160$. The noncanonical and canonical distributions are similar for low energy levels while they differ greatly for high energy levels.

Figure 4

The internal energy $U$ of the system with respect to $\beta$ for noncanonical and canonical statistics. We plot the cases of $\kappa =0.002$, $n_{\max }=138$ (blue solid line); $\kappa =0.0015$, $n_{\max }=122$ (orange dotted-dash line); $\kappa =0.001$, $n_{\max }=112$ (pink dashed line); and the canonical case for $\kappa =0$, $n_{\max }=112$. For small $\beta$, the internal energy $U$ is explicitly larger than that given by canonical statistics.

Figure 5

The internal energy relative fluctuation ${\left(\Delta U\right)}^2$ of the system with respect to $\beta$ for noncanonical and canonical statistics. The parameters are chosen the same as Fig. 4. A local maximum appears in the energy fluctuation for small $\beta$ which results of the noncanonical statistics.

Figure 6

The mutual entropy of two identical harmonic oscillators (HOs) by noncanonical and canonical statistics. The eigenfrequency of two identical HOs is $\omega =1$, and $\kappa =0.002$, $n_{\max }=138$ (blue solid line); $\kappa =0.0015$, $n_{\max }=122$ (orange dotted-dash line); $\kappa =0.001$, $n_{\max }=112$ (pink dashed line); and the canonical case for $\kappa =0$, $n_{\max }=112$.