Consistent thermodynamics for spin echoes

Spin-echo experiments are often said to constitute an instant of antithermodynamic behavior in a concrete physical system that violates the second law of thermodynamics. We argue that a proper thermodynamic treatment of the effect should take into account the correlations between the spin and the translational degrees of freedom of the molecules. To this end, we construct an entropy functional using Boltzmann macrostates that incorporate both spin and translational degrees of freedom. With this definition there is nothing special in the thermodynamics of spin echoes: dephasing corresponds to Hamiltonian evolution and leaves the entropy unchanged; dissipation increases the entropy. In particular, there is no phase of entropy decrease in the echo. We also discuss the definition of macrostates from the underlying quantum theory and we show that the decay of net magnetization provides a faithful measure of entropy change.

Figure 1

Minimum value of total entropy $S_{\mathrm{tot}}=S_B-\beta \mathrm{H̄}$ plotted as an increasing function of time $t$, for ${\Gamma }_T/(g\mu B)=0.05$ and different values of temperature $T$. Curve a corresponds to $T=0.3g\mu B$; curve b, to $T=0.5g\mu B$; and curve c, to $T=g\mu B$.

Figure 2

The increase in total entropy $\Delta S_{\mathrm{tot}}$ during an echo is a monotonously decreasing function of the magnetization decay $D(\tau )=\frac{M_x(2\tau )}{M_x(0)}$. Here, curve a corresponds to $T_1/T_2=0.75$; curve b, to $T_1/T_2=1.5$; and curve c, to $T_1/T_2=3$.