Equivalence condition for the canonical and microcanonical ensembles in coupled spin systems

It is typically assumed, without justification, that a weak coupling between a system and a bath is a necessary condition for the equivalence of a canonical ensemble and a microcanonical ensemble. For instance, in a canonical ensemble, temperature emerges if the system and the bath are uncoupled or weakly coupled. We investigate the validity region of this weak-coupling approximation, using a coupled composite-spin system. Our results show that the spin coupling strength can be as large as the level spacing of the system, indicating that the weak-coupling approximation has a much wider region of validity than usually expected.

Figure 1

(Color online) Schematic of a spin system (red arrows with circles) coupled to a spin bath (blue arrows without circles). The dashed lines denote the coupling between them.

Figure 2

(Color online) Canonical relation between $\text{ln}\left(P_i\right)$ and the system energy $E_i^S$ for various system-bath coupling strengths: $n=0$ (no coupling, blue dashed line with circles), $n=3$ (weak coupling, green solid line with crosses), $n=10$ (medium coupling, red solid line with upper triangles), and $n=30$ (strong coupling, cyan solid line with down triangles). Here $P_i$ is the probability that the system is in the $i\text{th}$ eigenstate with energy $E_i^S$. For weak and medium couplings $\left(n=0,3,10\right)$, the curves are almost linear. See also Table .

Figure 3

(Color online) Normalized density of states for system-bath coupling strength $n=0$ (a), 3 (b), 10 (c), 30 (d). The red lines show the energy range $\left[E,E+\delta \right]$ for small $\delta ⪡E$. Here, $E=-1.722$.

Figure 4

(Color online) Same as Fig. 2 except that the bath sizes are $K=21$ (left) and $K=25$ (right). The energy shell lies at $E=-1.722$ for both cases. Comparing with the case of $K=18$ (Fig. 2), we find that larger bath sizes do not qualitatively affect the equivalence condition between canonical and microcanonical ensembles.

Figure 5

(Color online) Same as Fig. 2 except that the bath size is $K=25$ for two other energy shells: $E=-2.179$ (left); $E=-2.637$ (right). Comparing with the bottom panel of Fig. 4, we do not observe a significant effect of the position of the energy shell on the equivalence condition between canonical and microcanonical ensembles.

Figure 6

(Color online) Same as Fig. 2 except that the system level distribution is now uneven. Note that this implies that an uneven level spacing distribution is equally sensitive to the system-bath coupling strength.

Figure 7

(Color online) Same as Fig. 3 except for a system with uneven level distribution. The total energy (the red vertical dashed line) is $E=-2.214$. Note that the DOS is not significantly affected by the uneven distribution of the system energy levels.