Effective Hamiltonian approach to kinetic Ising models: Application to an infinitely long-range Husimi-Temperley model

The probability distribution (PD) of spin configurations in kinetic Ising models has been cast in the form of the canonical Boltzmann PD with a time-dependent effective Hamiltonian (EH). It has been argued that in systems with extensive energy EH depends linearly on the number of spins $N$ leading to the exponential dependence of PD on the system size. In macroscopic systems the argument of the exponential function may reach values of the order of the Avogadro number which is impossible to deal with computationally, thus making unusable the linear master equation (ME) governing the PD evolution. To overcome the difficulty, it has been suggested to use instead the nonlinear ME (NLME) for the EH density per spin. It has been shown that in spatially homogeneous systems NLME contains only terms of order unity even in the thermodynamic limit. The approach has been illustrated with the kinetic Husimi-Temperley model (HTM) evolving under the Glauber dynamics. At finite $N$ the known numerical results has been reproduced and extended to broader parameter ranges. In the thermodynamic limit an exact nonlinear partial differential equation of the Hamilton-Jacobi type for EH has been derived. It has been shown that the average magnetization in HTM evolves according to the conventional kinetic mean field equation.

Figure 1

Equilibrium FFE density Eq. (22) for $T=0.8T_c$ and $h=0.015$. A, B, and C denote, respectively, the two local minima and a local maximum appearing in $f^0\left(m\right)$ when $T<T_c$ and $\left|h\right|<|h_{\text{SP}}|$ where $h_{\text{SP}}$ is $h$ at the spinodal point (further explanations are given in the text).

Figure 2

Probability of survival of the metastable states Eq. (38) calculated with the use of Eq. (27) for HTM with $N={10}^3, T=0.8T_c$, and $h=0.06$. The points are spaced with time step $100{\tau }_0$ and their size exceeds the accuracy of calculations. The exponential law has been fulfilled better than the accuracy of the drawing (see the text).

Figure 3

Solid line—FFE $f$ corresponding to the solution of Eq. (27) at $t=500{\tau }_0$ when $n_A\approx n_B$ with the model parameters $\beta =1.25(T=0.8T_c)$ and $h=0.06$ with the initial condition Eqs. (35) and (36); dashed lines correspond to local fits of $f^0$ in Eq. (41).

Figure 4

Comparison of Eq. (62) [10] with lifetimes found from the solution of Eqs. (27) (open symbols) and (56) (filled symbols and crosses). Circles correspond to $T=0.5T_c$, squares to $T=0.8T_c$; smaller (larger) symbols correspond to $N={10}^3$ ($N=2\times {10}^3$). The cross at larger (smaller) $\tau$ is for $N={10}^5$ ($N={10}^6$); the triangle marks symmetric case $h=0$ with $N={10}^3$ and $T=0.8T_c$; the straight line corresponds to $\tilde{\tau }=\tau$. The symbol sizes do not reflect the data accuracy which is higher than the drawing resolution.

Figure 5

Time derivative of FFE found numerically from Eq. (27) for HTM with the same parameters as in Fig. 2 and for the same five time values of (from bottom to top) as in the text following Eq. (40).

Figure 6

FFE Eq. (59) calculated with the use of Eqs. (56) and (55) for values of $\lambda$ given in Table 1.