Fluctuation theorem with two independent field parameters: The one-dimensional compressible Ising model

In this work we consider the nonequilibrium mechanical and magnetic work performed on a one-dimensional compressible Ising model. In the harmonic approximation we easily integrate the mechanical degrees of freedom of the model, and the resulting effective Hamiltonian depends on two external parameters, the magnetic field and the force applied along the chain. As the model is exactly soluble in one dimension we can determine the free energy difference between two arbitrary thermodynamic states of the system. We show the validity of the Jarzynski equality, which relates the free energy difference between two thermodynamic states of the system and the average work performed by external agents in a finite time, through nonequilibrium paths between the same thermodynamic states. We have found that the Jarzynski theorem remains valid for all the values of the rate of variation of the magnetic field and the mechanical force applied to the system.

Figure 1

Schematic representation of the one-dimensional compressible Ising model. Black points represent the chain particles, and the arrow the compression force along the chain.

Figure 2

Monte Carlo simulations for the one-dimensional compressible ferromagnetic Ising model. (A) Magnetization per particle $m$ versus magnetic field $H$, in the interval $H=0.0\to 1.0$, where the rate of change of the magnetic field is given by $\mathrm{\Delta }H=(H_{\max }-H_{\min })/n$. Note that (A1) corresponds to $\lambda =0.0$ and (A2) $\lambda =1.0$. Lattice size $N=128$, and temperature $T=1.0$. We show some selected values of $n$, number of steps taken for the variation of the field and “Eq” means the equilibrium, which is reached for $n=500$. The arrows indicate the direction of variation of the corresponding field parameter.

Figure 3

Monte Carlo simulations for the one-dimensional compressible ferromagnetic Ising model. (B) Exchange energy per particle $e$ versus external force $\lambda$, in the interval $\lambda =0.0\to 1.0$, where the rate of change of $\lambda$ is given by $\mathrm{\Delta }\lambda =({\lambda }_{\max }-{\lambda }_{\min })/n$ and $\tilde{J}=(1.0+\lambda )$ is the effective exchange coupling. In (B1) $H=0.0$ and in (B2) $H=1.0$. Lattice size $N=128$, and temperature $T=1.0$. We show some selected values of $n$, number of steps taken for the variation of the force and “Eq” means the equilibrium, which is reached for $n=500$. The arrows indicate the direction of variation of the corresponding field parameter.

Figure 4

(Straight lines) Evolution between equilibrium states, where one field parameter is fixed. (Broken line) Evolution along a trajectory where both external parameters change. “Eq” means equilibrium states.

Figure 5

Monte Carlo simulations for the determination of the free energy difference ($\mathrm{\Delta }F$) and the average work performed on the system ($\langle W\rangle$) as a function of the number of steps of variation of the field. We take $N=128$ and the temperature is fixed at $T=1.0$. Full lines correspond to the exact value of the free energy difference through the use of Eq. (13). (A1) changes in the magnetic field $H$ for $\lambda =0.0$. (B2) changes in $\lambda$ with $H=1.0$. (a) and (b) Forward and reverse processes, respectively. Error bars not shown are smaller than the size of the points.

Figure 6

Monte Carlo simulations for the determination of the free energy difference ($\mathrm{\Delta }F$) and the average work performed on the system ($\langle W\rangle$), as a function of the number of steps of variation of the field, for the lattice size $N=128$ and temperature $T=1.0$. Full lines correspond to the exact value of the free energy difference through the use of Eq. (13). (B1) changes in the force for $H=0.0$. (A2) changes in the magnetic field with $\lambda =1.0$. (a) and (b) Forward and reverse processes, respectively. Error bars not shown are smaller than the size of the points.

Figure 7

Free energy difference, $\mathrm{\Delta }F$, and the average total work, $\langle W\rangle$, as a function of the number of steps of variation of the field, when both magnetic field and force change. Monte Carlo simulations for the lattice size $N=128$ and temperature $T=1.0$. (C) represent changes in $H$ and $\lambda$. (a) and (b) Forward and reverse processes, respectively. Full lines represent the free energy differences calculated through Eq. (13).