The phenomenon of aging is an interesting property of non-equilibrium statistical mechanical systems whose relaxation slows down while time increases, meaning that time-translation invariance is broken. This phenomenon is observed in various systems, such as glasses and spin systems. In this work we review the aging properties of two related long-range spin models in one dimension: the voter model, where an agent copies the state of another at distance r with a probability P(r)proportional to r-alpha, and the Ising model with the interaction between spins decaying with an analogous power law J(r)proportional to r-alpha. In particular we focus on the autocorrelation function A(t,t ') and the Fisher-Huse exponent lambda, showing how the two models, which are equivalent in the case of nearest-neighbor interactions, behave differently as the interaction range increases (decreasing alpha).

In this paper, we start reviewing the main features of the one-dimensional Ising model with long-range interactions, where the spin-spin coupling decays as a power law, J(r) ∝ r^-α. We then discuss the key properties of the one-dimensional voter model, in which two agents (spins) at distance r interact with a power-law probability with the same form of J(r). The two models are compared, and the so-called p-voter model is presented, which provides a framework to interpolate between them. Specifically, the p-voter model reduces to the voter model for p = 1 and p = 2, while for p ≥ 3 it falls into the universality class of the Ising model.

Phase-separation plays a critical role within cell nuclei. Indeed, extensive experimental work has been done showing the importance of phase-separated condensates in the cell nucleus at different scales, ranging from liquid droplets of proteins to the bigger nucleoli, and there is growing evidence that this physical mechanism is crucial to orchestrate organization of genome architecture and to control gene regulation. On the other hand, computational and theoretical studies have been done to quantitatively explore the physics of this mechanism as well as its influence on biological processes relevant for genome activity, as formation of contacts between gene-enhancer and chromatin re-modelling. Methods commonly employed to study those systems include models based on classical statistical mechanics and numerical strategies such as Monte Carlo or molecular dynamics simulations. This short review focuses on recent theoretical and computational advances in this research field.

Recent advances have highlighted the rich low-temperature kinetics of the long-range Ising model (LRIM). This study investigates domain growth in an LRIM with quenched disorder, following a deep low-temperature quench. Specifically, we consider an Ising model with interactions that decay as J(r) ∼ r^-(D+σ), where D is the spatial dimension and σ> 0 is the power-law exponent. The quenched disorder is introduced via random pinning fields at each lattice site. For nearest-neighbor models, we expect that domain growth during activated dynamics is logarithmic in nature: R(t) ∼ (ln t)^α, with growth exponent α>0. Here, we examine how long-range interactions influence domain growth with disorder in dimensions D = 1 and D = 2. In D = 1, logarithmic growth is found to persist for various σ> 0. However, in D = 2, the dynamics is more complex due to the non-trivial interplay between extended interactions, disorder, and thermal fluctuations.

We discuss the relation between a system with quenched disorder in contact with a thermal bath at inverse temperature B and a companion one where the originally frozen disorder is annealed by thermalisation with a second thermal reservoir at a different temperature beta. In the limit in which the disorder variables become extremely slow and $\beta \to 0$beta -> 0 the two systems are expected to share the same static properties. Focusing on dynamical aspects, we discuss the non-equilibrium fluctuation-dissipation (FD) relation in the annealed system and the associated concept of effective temperature in four different models, which are studied numerically and/or analytically. We show that such effective temperature not only may bear a signature of the temperatures of the two reservoirs, but it may also reveal, in some cases, the FD relation of the corresponding quenched model in the ageing state after a temperature quench.

We study analytically the ordering kinetics of the two-dimensional long-range voter model on a two-dimensional lattice, where agents on each vertex take the opinion of others at distance $r$ with probability $P(r) \propto r^{-\al}$. The model is characterized by different regimes, as $\al$ is varied. For $\al > 4$ the behaviour is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as $L(t) \propto \sqrt{t}$, until consensus is reached in a time or order $N\ln N$, $N$ being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slow as $\rho(t) \propto 1/\ln t$. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbors interactions. For $0<\al \leq 4$ standard scaling is reinstated, and the correlation length increases algebraically as $L(t)\propto t^{1/z}$, with $1/z=2/\al$ for $3<\al<4$ and $1/z=2/3$ for $0<\al<3$. In addition, for $\al \le 3$, $L(t)$ depends on $N$ at any time $t>0$. Such coarsening, however, only leads the system to a partially ordered metastable state where correlations decay algebraically with distance, and whose lifetime diverges in the $N\to \infty$ limit. In finite systems consensus is reached in a time of order $N$ for any $\al <4$.

Due to its inherent intertwinement with irreversibility, entropy production is a prime observable to monitor in systems of active particles. In this numerical study, entropy production in the liquid, hexatic and solid phases of a two-dimensional system of active Brownian particles is examined at both average and fluctuation level. The trends of averages as functions of density show no singularity and marked changes in their derivatives at the hexatic-solid transition. Distributions show instead peculiar tail structures interpreted by looking at microscopic configurations. Particles in regions of low local order generate tail values according to different dynamical mechanisms: they move towards empty regions or bounce back and forth into close neighbours. The tail structures are reproduced by a simple single-particle model including an intermittent harmonic potential.

We study the non-equilibrium response function Rij(t,t '), namely the variation of the local magnetization < Si(t)> on site i at time t as an effect of a perturbation applied at the earlier time t ' on site j, in a class of solvable spin models characterized by the vanishing of the so-called asymmetry. This class encompasses both systems brought out of equilibrium by the variation of a thermodynamic control parameter, as after a temperature quench, or intrinsically out of equilibrium models with violation of detailed balance. The one-dimensional Ising model and the voter model (on an arbitrary graph) are prototypical examples of these two situations which are used here as guiding examples. Defining the fluctuation-dissipation ratio Xij(t,t ')=beta Rij/(partial derivative Gij/partial derivative t '), where Gij(t,t ')=< Si(t)Sj(t ')> is the spin-spin correlation function and beta is a parameter regulating the strength of the perturbation (corresponding to the inverse temperature when detailed balance holds), we show that, in the quite general case of a kinetics obeying dynamical scaling, on equal sites this quantity has a universal form Xii(t,t ')=(t+t ')/(2t), whereas limt ->infinity Xij(t,t ')=1/2 for any ij couple. The specific case of voter models with long-range interactions is thoroughly discussed.

We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, S-i = +/- 1, positioned at a lattice vertex i, copies the state of another one located at a distance r, selected randomly with a probability P(r) proportional to r(-alpha). Employing both analytical and numerical methods, we compute the two-time correlation function G (r; t, s) (t >= s) between the state of a variable S-i at time s and that of another one, at distance r, at time t. At time t, the memory of an agent of its former state at time s, expressed by the autocorrelation function A(t, s) = G(r = 0; t, s), decays algebraically for alpha > 1 as [L(t)/L(s)](-lambda), where Lis a time-increasing coherence length and lambda is the Fisher-Huse exponent. We find lambda = 1 for alpha > 2, and lambda = 1/(alpha - 1) for 1 < alpha <= 2 . For alpha <= 1, instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that lambda increases upon decreasing alpha. The space-dependent correlation G (r; t, s) obeys a scaling symmetry G (r; t, s) = g [r/L(s); L(t)/L(s)] for alpha > 2. Similarly, for 1 < alpha <= 2, one has G (r; t, s) = g [r/L(t); L(t)/L(s)], where the length L regulating two-time correlations now differs from the coherence length as L proportional to L-delta, with delta = 1 + 2 (2 - alpha).

We study analytically the ordering kinetics and the final metastable states in the three-dimensional long-range voter model where N agents described by a Boolean spin variable S_{i} can be found in two states (or opinion) ±1. The kinetics is such that each agent copies the opinion of another at distance r chosen with probability P(r)∝r^{-α} (α>0). In the thermodynamic limit N→∞ the system approaches a correlated metastable state without consensus, namely without full spin alignment. In such states the equal-time correlation function C(r)=〈S_{i}S_{j}〉 (where r is the i-j distance) decreases algebraically in a slow, nonintegrable way. Specifically, we find C(r)∼r^{-1}, or C(r)∼r^{-(6-α)}, or C(r)∼r^{-α} for α>5, 3<α≤5, and 0≤α≤3, respectively. In a finite system metastability is escaped after a time of order N and full ordering is eventually achieved. The dynamics leading to metastability is of the coarsening type, with an ever-increasing correlation length L(t) (for N→∞). We find L(t)∼t^{1/2} for α>5, L(t)∼t^{5/2α} for 4<α≤5, and L(t)∼t^{5/8} for 3≤α≤4. For 0≤α<3 there is not macroscopic coarsening because stationarity is reached in a microscopic time. Such results allow us to conjecture the behavior of the model for generic spatial dimension.

We study the ordering kinetics of a generalization of the voter model with long-range interactions, the $p$-voter model, in one dimension. It is defined in terms of boolean variables $S_{i}$, agents or spins, located on sites $i$ of a lattice, each of which takes in an elementary move the state of the majority of $p$ other agents at distances $r$ chosen with probability $P(r)\propto r^{-\alpha}$. For $p=2$ the model can be exactly mapped onto the case with $p=1$, which amounts to the voter model with long-range interactions decaying algebraically. For $3\le p<\infty$, instead, the dynamics falls into the universality class of the one-dimensional Ising model with long-ranged coupling constant $J(r)=P(r)$ quenched to small finite temperatures. In the limit $p\to \infty$, a crossover to the (different) behavior of the long-range Ising model quenched to zero temperature is observed.

The one-dimensional long-range voter model, where an agent takes the opinion of another at distance $r$ with probability $\propto r^{-\alpha}$, is studied analytically. The model displays rich and diverse features as $\alpha$ is changed. For $\alpha >3$ the behavior is similar to the one of the nearest-neighbor version, with the formation of ordered domains whose typical size grows as $R(t)\propto t^{1/2}$ until consensus (a fully ordered configuration) is reached. The correlation function $C(r,t)$ between two agents at distance $r$ obeys dynamical scaling with sizeable corrections at large distances $r>r^*(t)$, slowly fading away in time. For $2< \alpha \le 3$ violations of scaling appear, due to the simultaneous presence of two lengh-scales, the size of domains growing as $t^{(\alpha-2)/(\alpha-1)}$, and the distance $L(t)\propto t^{1/(\alpha-1)}$ over which correlations extend. For $\alpha \le 2$ the system reaches a partially ordered stationary state, characterised by an algebraic correlator, % $C(r)\propto r^{-(2-\alpha)}$, whose lifetime diverges in the thermodynamic limit of infinitely many agents, so that consensus is not reached. For a finite system escape towards the fully ordered configuration is finally promoted by development of large distance correlations. In a system of $N$ sites, global consensus is achieved after a time $T \propto N^2$ for $\alpha>3$, $T \propto N^{\alpha-1}$ for $2<\alpha \le 3$, and $T \propto N$ for $\alpha \le 2$.

We study the ordering kinetics of a generalization of the voter model with long-range interactions, the p-voter model, in one dimension. It is defined in terms of Boolean variables S-i, agents or spins, located on sites i of a lattice, each of which takes in an elementary move the state of the majority of p other agents at distances r chosen with probability P(r)proportional to r(-alpha). For p = 2 the model can be exactly mapped onto the case with p = 1, which amounts to the voter model with long-range interactions decaying algebraically. For 3 <= p < infinity, instead, the dynamics falls into the universality class of the one-dimensional Ising model with long-ranged coupling constant J(r) = P(r) quenched to small finite temperatures. In the limit p -> infinity, a crossover to the (different) behavior of the long-range Ising model quenched to zero temperature is observed. Since for p > 3 a closed set of differential equations cannot be found, we employed numerical simulations to address this case.

We investigate the influence of long-range (LR) interactions on the phase ordering dynamics of the one-dimensional random-field Ising model (RFIM). Unlike the usual RFIM, a spin interacts with all other spins through a ferromagnetic coupling that decays as r^{-(1+σ)}, where r is the distance between two spins. In the absence of LR interactions, the size of coarsening domains R(t) exhibits a crossover from pure system behavior R(t)∼t^{1/2} to an asymptotic regime characterized by logarithmic growth: R(t)∼(lnt)^{2}. The LR interactions affect the preasymptotic regime, which now exhibits ballistic growth R(t)∼t, followed by σ-dependent growth R(t)∼t^{1/(1+σ)}. Additionally, the LR interactions also affect the asymptotic logarithmic growth, which becomes R(t)∼(lnt)^{α(σ)} with α(σ)<2. Thus, LR interactions lead to faster growth than for the nearest-neighbor system at short times. Unexpectedly, this driving force causes a slowing down of the dynamics (α<2) in the asymptotic logarithmic regime. This is explained in terms of a nontrivial competition between the pinning force caused by the random field and the driving force introduced by LR interactions. We also study the spatial correlation function and the autocorrelation function of the magnetization field. The former exhibits superuniversality for all σ, i.e., a scaling function that is independent of the disorder strength. The same holds for the autocorrelation function when σ<1, whereas a signature of the violation of superuniversality is seen for σ>1.

Device performance of solution-processed 2D semiconductors in printed electronics has been limited so far by structural defects and high interflake junction resistance. Covalently interconnected networks of transition metal dichalcogenides potentially represent an efficient strategy to overcome both limitations simultaneously. Yet, the charge-transport properties in such systems have not been systematically researched. Here, the charge-transport mechanisms of printed devices based on covalent MoS 2 networks are unveiled via multiscale analysis, comparing the effects of aromatic versus aliphatic dithiolated linkers. Temperature-dependent electrical measurements reveal hopping as the dominant transport mechanism: aliphatic systems lead to 3D variable range hopping, unlike the nearest neighbor hopping observed for aromatic linkers. The novel analysis based on percolation theory attributes the superior performance of devices functionalized with π-conjugated molecules to the improved interflake electronic connectivity and formation of additional percolation paths, as further corroborated by density functional calculations. Valuable guidelines for harnessing the charge-transport properties in MoS 2 devices based on covalent networks are provided.

We study analytically and numerically a couple of paradigmatic spin models, each described in terms of two sets of variables attached to two different thermal baths with characteristic timescales T and tau and inverse temperatures B and beta. In the limit in which one bath becomes extremely slow (tau -> infinity), such models amount to a paramagnet and to a one-dimensional ferromagnet in contact with a single bath. Our study is also motivated by analogies with disordered systems where widely separated timescales associated with different effective temperatures emerge. We show that these systems reach a stationary state in a finite time for any choice of B and beta. We determine the non-equilibrium fluctuation-dissipation relation between the autocorrelation and the response function in such a state and, from that, we discuss if and how thermalization with the two baths occurs and the emergence of a non-trivial fluctuation-dissipation ratio.

After a zero temperature quench, we study the kinetics of the one-dimensional Ising model with long-range interactions between spins at distance r decaying as r−α, with α≤1. As shown in our recent study (Corberi et al., 2021) that only a fraction of the non-equilibrium trajectories is characterised by the presence of coarsening domains while in the remaining ones the system is quickly driven towards a magnetised state. Restricting to realisations displaying coarsening we compute numerically the probability distribution of the size of the domains and find that it exhibits a scaling behaviour with an unusual α-dependent power-law decay. This peculiar behaviour is also related to the divergence of the average size of domains with system size at finite times. Such a scenario differs from the one observed when α>1, where the distribution decays exponentially. Finally, based on numerical results and on analytical calculations we argue that the average domain size grows asymptotically linearly in time.

It is known that, after a quench to zero temperature (T=0), two-dimensional (d=2) Ising ferromagnets with short-range interactions do not always relax to the ordered state. They can also fall in infinitely long-lived striped metastable states with a finite probability. In this paper, we study how the abundance of striped states is affected by long-range interactions. We investigate the relaxation of d=2 Ising ferromagnets with power-law interactions by means of Monte Carlo simulations at both T=0 and T≠0. For T=0 and the finite system size, the striped metastable states are suppressed by long-range interactions. In the thermodynamic limit, their occurrence probabilities are consistent with the short-range case. For T≠0, the final state is always ordered. Further, the equilibration occurs at earlier times with an increase in the strength of the interactions.

We study the kinetics of the two-dimensional q > 4-state Potts model after a shallow quench slightly below the critical temperature and above the pseudo spinodal. We use numerical methods and we focus on intermediate values of q, 4 < q < 100. We show that, initially, the system evolves as if it were quenched to the critical temperature. The further decay from the metastable state occurs by nucleation of k out of the q possible phases. For a given quench temperature, k is a logarithmically increasing function of the system size. This unusual finite size dependence is a consequence of a scaling symmetry underlying the nucleation phenomenon for these parameters.