Fluctuation relations for the entropy production in non equilibrium stationary states of Ising models are investigated by Monte Carlo simulations. Systems in contact with heat baths at two different temperatures or subject to external driving will be studied. In the first case, by considering different kinetic rules and couplings with the baths, the behavior of the probability distributions of the heat exchanged in a time $τ$ with the thermostats, both in the disordered and in the low temperature phase, are discussed. The fluctuation relation is always verified in the large $τ$ limit and deviations from linear response theory are observed. Finite-$τ$ corrections are shown to obey a scaling behavior. In the other case the system is in contact with a single heat bath but work is done by shearing it. Also for this system the statistics collected for the mechanical work shows the validity of the fluctuation relation and preasymptotic corrections behave analogously to the case with two baths.
The statistical mechanics of a one-dimensional Ising model in thermal equilibrium is well-established, textbook material. Yet, when driven far from equilibrium by coupling two sectors to two baths at different temperatures, it exhibits remarkable phenomena, including an unexpected 'freezing by heating.' These phenomena are explored through systematic numerical simulations. Our study reveals complicated relaxation processes as well as a crossover between two very different steady-state regimes.
Nonequilibrium phase transitions and stationary state solutions of a three-dimensional random-field Ising model under a time dependent periodic external field
Many-body non-equilibrium steady states can be described by a Landau-Ginzburg theory if one allows non-analytic terms in the potential. We substantiate this claim by working out the case of the Ising magnet in contact with a thermal bath and undergoing stochastic reheating: It is reset to a paramagnet at random times. By a combination of stochastic field theory and Monte Carlo simulations, we unveil how the usual $φ^4$ potential is deformed by non-analytic operators of intrinsic non-equilibrium nature. We demonstrate their infrared relevance at low temperatures by a renormalization-group analy…
While the Ising model belongs to the realm of equilibrium statistical mechanics, the voter model is an example of a nonequilibrium system. We examine an opinion formation model, which is a mixture of Ising and voter agents with concentrations $p$ and $1-p$, respectively. Although in our model for $p<1$ a detailed balance is violated, on a complete graph the average magnetization in the stationary state for any $p>0$ is shown to satisfy the same equation as for the pure Ising model ($p=1$). Numerical simulations confirm such a behavior, but the equivalence with the pure Ising model apparently h…
Notwithstanding great strides that statistical mechanics has made in recent decades, an analytic solution of arguably the simplest model of relaxation dynamics, the Ising model in an applied external field remains elusive even in $1d$. Extant studies are based on numerics using single-spin-flip Glauber dynamics. There is no reason why this algorithm should lead to the global minimum energy state of the system. With this in mind, we explore multi-spin-flip parallel and sequential Glauber dynamics of Ising spins in $1d$ and also on a regular random graph of coordination number $z=3$. We view our…
We show that, contrary to previous suggestions based on computer simulations or erroneous theoretical treatments, the critical points of the random-field Ising model out of equilibrium, when quasi-statically changing the applied source at zero temperature, and in equilibrium are not in the same universality class below some critical dimension $d_{DR}\approx 5.1$. We demonstrate this by implementing a non-perturbative functional renormalization group for the associated dynamical field theory. Above $d_{DR}$, the avalanches, which characterize the evolution of the system at zero temperature, bec…
We investigate the dynamics of a 2D Ising model on a square lattice with conservative Kawasaki dynamics in the bulk, coupled with two external reservoirs that pull the dynamics out of equilibrium. Two different mechanisms for the action of the reservoirs are considered. In the first, called ISF, the condition of local equilibrium between reservoir and the lattice is not satisfied. The second mechanism, called DB, implements a detailed balance condition, thus satisfying the local equilibrium property. We provide numerical evidence that, for a suitable choice of the temperature (i.e. below the c…
Usually, in a non-equilibrium setting, a current brings mass from the highest density regions to the lowest density ones. Although rare, the opposite phenomenon (known as "uphill diffusion") has also been observed in multicomponent systems, where it appears as an artificial effect of the interaction among components. We show here that uphill diffusion can be a substantial effect, i.e. it may occur even in single component systems as a consequence of some external work. To this aim we consider the 2D ferromagnetic Ising model in contact with two reservoirs that fix, at the left and the right bo…
Magnetic properties of an Ising bilayer system defined on a honeycomb lattice with non-magnetic interlayers which interact via an indirect exchange coupling have been investigated by Monte Carlo simulation technique. Equilibrium properties of the system exhibit ferrimagnetism with $P$-, $N$- and $Q$- type behaviors. Compensation phenomenon suddenly disappears with decreasing strength of indirect ferrimagnetic interlayer exchange coupling. Qualitative properties are in a good agreement with those obtained by effective field theory. In order to investigate the stochastic dynamics of kinetic Isin…
We investigate the maximum caliber variational principle as an inference algorithm used to predict dynamical properties of complex nonequilibrium, stationary, statistical systems in the presence of incomplete information. Specifically, we maximize the path entropy over discrete time step trajectories subject to normalization, stationarity, and detailed balance constraints together with a path-dependent dynamical information constraint reflecting a given average global behavior of the complex system. A general expression for the transition probability values associated with the stationary rando…
We study the approach towards equilibrium in a dynamic Ising model, the Q2R cellular automaton, with microscopic reversibility and conserved energy for an infinite one-dimensional system. Starting from a low-entropy state with positive magnetisation, we investigate how the system approaches equilibrium characteristics given by statistical mechanics. We show that the magnetisation converges to zero exponentially. The reversibility of the dynamics implies that the entropy density of the microstates is conserved in the time evolution. Still, it appears as if equilibrium, with a higher entropy den…
We consider the spatial correlation function of the two-dimensional Ising spin glass under out-equilibrium conditions. We pay special attention to the scaling limit reached upon approaching zero temperature. The field-theory of a non-interacting field makes a surprisingly good job at describing the spatial shape of the correlation function of the out-equilibrium Edwards-Anderson Ising model in two dimensions.
We show that the equilibrium interfaces in the disordered phase have critical percolation fractal dimension over a wide range of length scales. We confirm that the system falls out of equilibrium at a temperature that depends on the cooling rate as predicted by the Kibble-Zurek argument and we prove that the dynamic growing length once the cooling reaches the critical point satisfies the same scaling. We determine the dynamic scaling properties of the interface winding angle variance and we show that the crossover between critical Ising and critical percolation properties is determined by the …
Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function
对于伊辛自旋，我们导出了涨落耗散定理的非平衡推广，它在形式上与以前用朗之万动力学[L. F. Cugliandolo，J.Kurchan和G.Parisi，J.Phys.I France \textbf{4}，1641（1994）]对软自旋得到的结果相同。该结果具有一般性，对具有守恒和非守恒序参量的动力学都成立。基于这种涨落耗散关系，我们构造了一种有效的计算线性响应函数的数值算法，而不施加扰动场，这是替代Chatelain [J.Phys. A \textbf{36}，10739（2003）]和Ricci-Tersenghi [Phys.Rev.E { \bf 68}，065104（R）（2003）]的算法。作为新算法的应用，我们提出了非常准确的数据的线性响应函数的伊辛链，保守和非保守的序参量动力学，发现在这两种情况下的结构是相同的一个非常简单的物理解释。我们还计算了二维伊辛模型的集成响应函数，证实它服从缩放$χ(t,t_w)\simeq t_w^{-a}f(t/t_w)$，与$a =0.26\pm 0.01$，如以前发现的不同的方法。
We introduce a model of interacting lattices at different resolutions driven by the two-dimensional Ising dynamics with a nearest-neighbor interaction. We study this model both with tools borrowed from equilibrium statistical mechanics as well as non-equilibrium thermodynamics. Our findings show that this model keeps the signature of the equilibrium phase transition. Moreover the critical temperature of the equilibrium models correspond to the state maximizing the entropy and delimits two out-of-equilibrium regimes, one satisfying the Onsager relations for systems close to equilibrium and one …
A general understanding of optimal control in non-equilibrium systems would illuminate the operational principles of biological and artificial nanoscale machines. Recent work has shown that a system driven out of equilibrium by a linear response protocol is endowed with a Riemannian metric related to generalized susceptibilities, and that geodesics on this manifold are the non-equilibrium control protocols with the lowest achievable dissipation. While this elegant mathematical framework has inspired numerous studies of exactly solvable systems, no description of the thermodynamic geometry yet …
We present here the non-equilibrium dynamics of the recently studied quasiperiodic Ising model. The zero temperature phase diagram of this model mainly consists of three phases, where each of these three phases can have extended, localized or critically delocalized low energy excited states. We explore the nature of excitations in these different phases by studying the evolution of entanglement entropy after performing quenches of different strengths to different phases. Our results on non-equilibrium dynamics of entanglement entropy are concurrent with the nature of excitations discussed in R…
Long-range interacting Hamiltonian systems are believed to relax generically towards non-equilibrium states called "quasi-stationary" because they evolve towards thermodynamic equilibrium very slowly, on a time-scale diverging with particle number. We show here that, by applying a suitable perturbation operator for a finite time interval, we obtain, in a family of long-range systems, non-equilibrium states which appear to be strictly stationary. They exist even in the case of a harmonic potential, and are characterised by an ordered microscopic phase space structure. We give some simple heuris…
In this paper, we study perturbations of the $d$-dimensional Ising model for $d\geq 2$, including long range ones to which the Pirogov-Sinai theory is not applicable. We show that the uniqueness of the equilibrium state of the Ising model at high temperature and the coexistence of equilibrium states at low temperature are preserved by spin-flip symmetric perturbations.
We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like models. The stability analysis of the fixed points in the mean field calculation shows the existence of an exponent that depends on the coordination number $z$ in the Ising model. For the generalised voter model, a phase diagram is obtained based on this study. Numerical results for the Ising model for both the mean field case and short ranged models on lattices with different values of $z$ are also obtained. A related study is the behaviour of the exit probability $E(x_0)$, defined as the probabil…
Quasi-stationary criticality of the Order-Parameter of the d=3 Random-Field Ising Antiferromagnet Fe(0.85)Zn(0.15)F2: A Synchrotron X-ray Scattering Study
