Fig. 4: Verification of the exact mean-field solutions by simulating the kinetic Ising systems with synchronous and asynchronous updates.
From: Nonequilibrium thermodynamics of the asymmetric Sherrington-Kirkpatrick model
We repeated 400, 000 simulations of systems under synchronous (top) and asynchronous (bottom) updates with Θi,u = 0 and ΔJ = 0.5: a, d Sample estimates of the mean activation rate \(\frac{1}{N}{\sum }_{i}{\left[\left\langle {s}_{i,u}\right\rangle \right]}_{{{{{{{{\bf{J}}}}}}}},{{{{{{{\boldsymbol{\tau }}}}}}}}}\) compared with the theoretical order parameter m. b, e Sample estimates of the average delayed self-covariances \(\frac{1}{N}{\sum }_{i}{\left[\left\langle {s}_{i,u}{s}_{i,u-d}\right\rangle \right]}_{{{{{{{{\bf{J}}}}}}}},{{{{{{{\boldsymbol{\tau }}}}}}}}}- \frac{1}{N}{\sum }_{i}{\left[\left\langle {s}_{i,u}\right\rangle \right]}_{{{{{{{{\bf{J}}}}}}}},{{{{{{{\boldsymbol{\tau }}}}}}}}} \frac{1}{N}{\sum }_{i}{\left[\left\langle {s}_{i,u-d}\right\rangle \right]}_{{{{{{{{\bf{J}}}}}}}},{{{{{{{\boldsymbol{\tau }}}}}}}}} \) (d = 1 for the synchronous system and d = 10N for the asynchronous one) computed from samples compared with the theoretical order parameter q-m2. c, f Sample estimates of the entropy production and entropy production rate (Supplementary Eqs. (S6.6) and (S6.7)) compared with its mean-field value at the thermodynamic limit \(\frac{1}{N}{\left[\sigma \right]}_{{{{{{{{\bf{J}}}}}}}}},\frac{1}{N}{\left[\frac{d\sigma }{dt}\right]}_{{{{{{{{\bf{J}}}}}}}},{{{{{{{\boldsymbol{\tau }}}}}}}}}\) (Eqs. (54) and (58)).
