Work distribution functions for hysteresis loops in a single-spin system

We compute the distribution of the work done in driving a single Ising spin with a time-dependent magnetic field. Using Glauber dynamics we perform Monte Carlo simulations to find the work distributions at different driving rates. We find that in general the work distributions are broad with a significant probability for processes with negative dissipated work. The special cases of slow and fast driving rates are studied analytically. We verify that various work fluctuation theorems corresponding to equilibrium initial states are satisfied while a steady state version is not.

Figure 1

(Color online) Distributions of the work done in driving a magnet at different rates when magnetic field is changed linearly.

Figure 2

(Color online) Average magnetization $m\left(\tau \right)$ for different rates, when magnetic field is changed linearly.

Figure 3

(Color online) Plot shows that the fluctuation theorem is valid for slow processes with Gaussian work distributions. The inset shows that for a fast rate $({\tau }_m=5.0)$ the probability of negative work is much larger than that predicted by the FT.

Figure 4

(Color online) Magnetic field changed over a cycle. In the symmetric case the total cycle time is $4{\tau }_m$ while in the asymmetric case it is $2({\tau }_m+{\tau }_n)$.

Figure 5

(Color online) Plot of work distributions for different driving rates when the magnetic field is changed in a symmetric cycle.

Figure 6

(Color online) Plot of work distributions obtained for different asymmetric cycles of the magnetic field.

Figure 7

(Color online) Hysteresis curves in driving a magnet at different rates when magnetic field is changed in a symmetric cycle and with the spin initially in equilibrium.

Figure 8

(Color online) Plot showing the work distribution for an asymmetric half cycle and the validity of Crook’s fluctuation theorem. Note that the probability of negative work processes is much higher than that predicted by usual TFT.

Figure 9

(Color online) Work distributions in the nonequilibrium steady state.

Figure 10

(Color online) Hysteresis curves in the nonequilibrium steady state.

Figure 11

(Color online) Violation of the fluctuation theorem for the steady state work distribution corresponding to ${\tau }_m=1.0$. The inset shows the same plots for the transient case where FT is clearly satisfied.

Figure 12

(Color online) Comparison of work distributions for slow rates obtained from simulations and from the analytic form. Solid lines show the analytic results.

Figure 13

(Color online) Comparison of work distribution for a fast rate obtained from simulation and from the analytic form.

Figure 14

(Color online) Comparison of work distributions for slow rates obtained from simulations of the cyclic case and from the analytic form. Solid lines show the analytic results.

Figure 15

(Color online) Comparison of work distribution, for the symmetric cycle with a fast rate, obtained from simulations and from the analytic results.