Discontinuous transitions in globally coupled potential systems with additive noise.

There are conflicting reports in the literature regarding the finite-size scaling of the Liouvillian gap and dynamical fluctuations at discontinuous phase transitions, with various studies reporting either exponential or power-law behavior. We clarify this issue by employing large deviation theory. We distinguish two distinct classes of discontinuous phase transitions that have different dynamical properties. The first class is associated with phase coexistence, i.e., the presence of multiple stable attractors of the system dynamics (e.g., local minima of the free-energy functional) in a finite phase diagram region around the phase transition point. In that case, one observes asymptotic exponential scaling related to stochastic switching between attractors (though the onset of exponential scaling may sometimes occur for very large system sizes). In the second class, there is no phase coexistence away from the phase transition point, while at the phase transition point itself there are infinitely many attractors. In that case, one observes power-law scaling related to the diffusive nature of the system relaxation to the stationary state.

Recently, the learning by confusion (LbC) approach has been proposed as a machine learning tool to determine the critical temperature T_{c} of phase transitions without any prior knowledge of its even approximate value. The method has been proven effective, but it has been used only for continuous phase transitions, where the confusion results only from deliberate incorrect labeling of the data. However, in the case of a discontinuous phase transition, additional confusion can result from the coexistence of different phases. To verify whether the confusion scheme can also be used for discontinuous phase transitions, we apply the LbC method to three microscopic models, the Blume-Capel, the q-state Potts, and the Falicov-Kimball models, which undergo continuous or discontinuous phase transitions depending on model parameters. With the help of a simple model, we predict that the phase coexistence present in discontinuous phase transitions can indeed make the neural network more confused and thus decrease its performance. However, numerical calculations performed for the models mentioned above indicate that other aspects of this kind of phase transition are more important and can render the LbC method even less effective. Nevertheless, we demonstrate that in some cases the same aspects allow us to use the LbC method to identify the order of a phase transition.

Lateral predictive coding is a recurrent neural network that creates energy-efficient internal representations by exploiting statistical regularity in sensory inputs. Here, we analytically investigate the trade-off between information robustness and energy in a linear model of lateral predictive coding and numerically minimize a free energy quantity. We observed several phase transitions in the synaptic weight matrix, particularly a continuous transition that breaks reciprocity and permutation symmetry and builds cyclic dominance and a discontinuous transition with the associated sudden emergence of tight balance between excitatory and inhibitory interactions. The optimal network follows an ideal gas law over an extended temperature range and saturates the efficiency upper bound of energy use. These results provide theoretical insights into the emergence and evolution of complex internal models in predictive processing systems.

We define the characteristic temperature T C of a material by using the behaviour of the specific heat in the vicinity of this temperature: C p,N (T ) = f(1 − (T/T C ),p,N ) for T < T C and C p,N (T ) = g((T/T C ) − 1,p,N ) for T > T C where p is the external pressure and N is the number of particles. The characteristic temperature T C = T C ( p, N ) can e.g. be identified with the transition temperature in the case of continuous phase transitions or with one of the two characteristic temperatures (on heating) and (on cooling) in the case of discontinuous phase transitions. Using the idea of the characteristic temperature, we present a new method on how to calculate the Gibbs potential G = G(T, p, N) in the vicinity of the characteristic temperature, without it being a necessity to know the explicite expressions for f and g. This general and uniform approach allows to derive many thermodynamical relations connecting measurable quantities at the characterisic point as e.g. Clausius–Clapeyron equation and its analogs, valid for discontinuous phase transitions and 15 relations valid for both continuous and discontinuous phase transitions, 6 of them are new. As a special case of these latter relations, we find six relations valid for continuous phase transitions (Ehrenfest equation and its analogs). The derived relations result exclusively from the definition of the characteristic point and are independent from the underlying microscopical mechanism leading to phase transitions (as e.g. solid–liquid, liquid–vapour, magnetic, ferroelectric, structural and superconducting transitions) and therefore, they are generally valid. These relations can be very important for material science. Using some of them it is possible to predict (without performing additional experiments) whether the transition or characteristic temperature T C of a given material sample will increase or decrease under doping (injection) or applied pressure. Similar 24 thermodynamical relations can also be found with ease for systems described by the Helmholtz potential F = F(T,V, N ).

High-order network models have emerged as powerful frameworks for modeling complex dynamical systems that involve multi-body interactions in recent years. Despite their widespread adoption in prior studies, the subtle difference between high-order structures, even those of the same order, has frequently been overlooked. To fill this gap in between, this study takes four typical triadic (second-order) configurations as examples and examines their influence on complex contagion process described by the Susceptible-Infected-Susceptible (SIS) model. A network construction model is first proposed to generate high-order networks with one of the four triadic configurations while maintaining the average second-order degree as well as the first-order structure. Numerical simulations of contagions on these networks reveal that, while different triadic configurations do not alter the qualitative characteristics of the phase transition (i.e., from continuous to discontinuous phase transitions) in contagion processes, they have a noticeable impact on the second-order epidemic threshold for discontinuous phase transitions. Specifically, networks with 2-simplices exhibit the smallest threshold for discontinuous phase transitions, whereas those with 3-cycle structures display the largest threshold. By employing the heterogeneous mean-field (HMF) approach, the critical second-order epidemic threshold for the emergence of discontinuous phase transitions is analytically derived, providing a theoretical explanation for the numerical findings.&amp;#xD;

Depending on the parity of the rotational quantum number J, solid hydrogens exhibit either pressure-driven broken symmetry phase (BSP) transitions (even-J species – para-hydrogen (p-H2), and ortho-deuterium (o-D2)) or usual order-disorder phase transitions (odd-J species – o-H2 and p-D2) with the phase transition temperature increasing monotonically with pressure from the phase transition point at zero pressure. At the same time, for solid HD the BSP phase transition line displays a minimum, indicating that the disordered phase is reentrant. In this work a model of quantum linear rotors is used to study characteristic features of the P–T phase diagrams for ortho–para mixtures in solid H2 and D2. We developed a mean-field theory of even-J – odd-J mixtures of quantum linear rotors on a 3D lattice. Two limiting cases are considered: mixtures at thermodynamic equilibrium, where the conversion time is small or comparable with the thermalization time, and the opposite case, frozen mixtures, when the conversion time is large compared with other relevant times. We found that for all equilibrium linear rotor systems — the even-J – odd-J mixtures — the reentrant behavior of the phase transition lines is an entropy-driven phenomenon, as was previously found for the all-J system. Experimentally, conditions to find the reentrant BSP transition line are most favorable for H2 mixtures, whereas the frozen monotonic phase lines should be the case for D2 even-J – odd-J mixtures. These results may therefore be particularly useful for understanding the anomalous transition region of the BSP transition documented for D2 mixtures.

Beta oscillations (13-30 Hz) of the local field potential are a hallmark of Parkinson's disease induced by dopamine deficiency, yet their underlying mechanism remains unclear. This study investigates the complex dynamics underlying beta oscillations of striatal medium spiny neurons in a theoretical model, where the conductance of the M-current (gM) is reduced to reflect dopamine deficiency. For an isolated neuron, codimension-1 and -2 bifurcations underlie excitability transition and the shift of phase response curves (PRCs) from Type 2 to Type 1. In a network with inhibitory coupling, interactions between synchronization regimes and firing frequency of individual neurons generate beta oscillations. Under weak coupling, anti-phase synchronization (AS) changes to global synchronization (GS) via a discontinuous phase transition with a bistable regime. Typical AS and GS are explained by the match between PRC types and inhibitory post-synaptic currents. In the AS, the network frequency is about twice the firing frequency of individual neurons at intermediate gM, resulting in beta network oscillations. As gM decreases further, neurons show higher firing frequencies, and GS replaces AS via a discontinuous transition, yielding beta oscillations with paradoxical frequency reduction. Under strong coupling, partial synchronization emerges instead of AS or GS, accompanied by a continuous transition. The partial synchronization induces beta oscillations across a wide parameter region, where the network frequency exceeds the low firing frequency of individual neurons. These nonlinear dynamics of single neurons and synchronization transitions in the network provide new insights into the generation and modulation of beta oscillations in Parkinson's disease.

The temperature-induced high-spin(HS)-low-spin(LS) transition of the 3d4-3d7 transition-metal ions in crystals (oxides and sulphides) are investigated on a semiquantitative basis by taking into account the elastic interaction between HS (or LS) clusters produced in a LS (or HS) crystal. The critical temperature at which the discontinuous phase transition takes place is evaluated as a function of the ligand-field strength Delta in units of spin-pairing energy Pi . It is shown that the 'crossover range' where this critical temperature exists is restricted to a finite range of Delta / Pi around the crossover point. The crossover ranges are evaluated for the crystals MO, MS2, M2O3, and LaMO3 where M represents transition-metal ions. The temperature variation of the equilibrium concentrations of HS and LS clusters in LS and HS crystals, respectively, is calculated for several values of Delta / Pi in perovskite LaMO3 crystals. It is pointed out that two discontinuous transitions from HS to LS and LS to HS with decreasing temperature may be expected in the d6 and the d7 systems.

The ferromagnetic (J>0) version of the Blume-Emery-Griffiths model in the region of repulsive biquadratic couplings (K<0) is considered on a Cayley tree of coordination z, reducing the statistical problem to the analysis of a two-dimensional nonlinear discrete map. In order to investigate the effect of the coordination z on the system multicritical behavior, we study the particular case K/J=-3.5 with the inclusion of crystal fields (D≠0), but vanishing external magnetic fields (H=0), for two distinct lattice coordinations (z=4 and z=6). The thermodynamic solutions on the Bethe lattice (the central region of a large Cayley tree) are associated with the attractors of the two-dimensional map. The phase diagrams display several thermodynamic phases (paramagnetic, ferromagnetic, ferrimagnetic, and staggered quadrupolar). In some cases, there are regions of numerical costability of two different attractors of the map, associated with discontinuous phase transitions between the corresponding phases. To verify the thermodynamic stability of the phases and to locate the first-order boundaries, the analytical expression of the Gibbs free energy was obtained by the method proposed by Gujrati [Phys. Rev. Lett. 74, 809 (1995)PRLTAO0031-900710.1103/PhysRevLett.74.809]. For lower coordinations (z=4) the transition between the ferrimagnetic and the staggered quadrupolar phases is always continuous, while the transition between the ferromagnetic and the ferrimagnetic phases is discontinuous at low temperatures, turning into continuous for temperatures above a tricritical point. On the other hand, for higher coordinations (z=6), the transition between the ferromagnetic and the ferrimagnetic phases is always continuous. However, the transition between the ferrimagnetic and the staggered quadrupolar phases is continuous for higher temperatures and discontinuous for temperatures below a tricritical point, in agreement with previous results obtained in the mean-field approximation (infinity-coordination limit). In both cases, the occurrence and the thermodynamic stability of the ferrimagnetic phase is confirmed.

Neutron Diffraction Study of the Low-Temperature Phases of Sodium Oxide Nitrite, Na3ONO2

Based on quasistationary distribution ideas, a general finite size scaling theory is proposed for discontinuous nonequilibrium phase transitions into absorbing states. Analogously to the equilibrium case, we show that quantities such as response functions, cumulants, and equal area probability distributions all scale with the volume, thus allowing proper estimates for the thermodynamic limit. To illustrate these results, five very distinct lattice models displaying nonequilibrium transitions-to single and infinitely many absorbing states-are investigated. The innate difficulties in analyzing absorbing phase transitions are circumvented through quasistationary simulation methods. Our findings (allied to numerical studies in the literature) strongly point to a unifying discontinuous phase transition scaling behavior for equilibrium and this important class of nonequilibrium systems.

Threshold versions of Schloegl's model on a lattice, which involve autocatalytic creation and spontaneous annihilation of particles, can provide a simple prototype for discontinuous non-equilibrium phase transitions. These models are equivalent to so-called threshold contact processes. A discontinuous transition between populated and vacuum states can occur selecting a threshold of N ≥ 2 for the minimum number, N, of neighboring particles enabling autocatalytic creation at an empty site. Fundamental open questions remain given the lack of a thermodynamic framework for analysis. For a square lattice with N = 2, we show that phase coexistence occurs not at a unique value but for a finite range of particle annihilation rate (the natural control parameter). This generic two-phase coexistence also persists when perturbing the model to allow spontaneous particle creation. Such behavior contrasts both the Gibbs phase rule for thermodynamic systems and also previous analysis for this model. We find metastability near the transition corresponding to a non-zero effective line tension, also contrasting previously suggested critical behavior. Mean-field type analysis, extended to treat spatially heterogeneous states, further elucidates model behavior.

We apply a spin-coherent states formalism to study the simplest version of a central-spin model, the so-called Mermin model, in which a given (central) spin 1/2 interacts symmetrically with a "bath" composed of N spins 1/2, characterized by a common frequency. The symmetric interaction allows one to restrict the problem to the fully symmetric sector of the Hilbert space associated with these N spins and therefore to treat these as one large N/2 spin. In particular, we derive analytic expressions for the integrated density of states in the thermodynamic limit when the number of bath spins is taken to infinity. From the thermodynamic limit spectrum we show the phase diagram for the system can be divided into four regions, partitioned, on the one hand, into a symmetric (nondegenerate) phase or a broken symmetry (degenerate) phase, and, on the other hand, by the case of overlapping or nonoverlapping energy surfaces. The nature and position of singularities appearing in the energy surfaces change as one moves from region to region. Our spin-coherent states formalism naturally leads us to the Majorana representation, which is useful to transform the Schrödinger equation into a Ricatti-like form that can be solved in the thermodynamic limit to obtain closed-form expressions for the integrated density of states. The energy surface singularities correspond with critical points in the density of states. We then use our results to compute expectation values for the system that help to characterize the nature of the quantum phase transition between the symmetric and broken phases.

Motivated by recent findings, we discuss the existence of a direct and robust mechanism providing discontinuous absorbing transitions in short-range systems with single species, with no extra symmetries or conservation laws. We consider variants of the contact process, in which at least two adjacent particles (instead of one, as commonly assumed) are required to create a new species. Many interaction rules are analyzed, including distinct cluster annihilations and a modified version of the original pair contact process. Through detailed time-dependent numerical simulations, we find that for our modified models, the phase transitions are of first order, hence contrasting with their corresponding usual formulations in the literature, which are of second order. By calculating the order-parameter distributions, the obtained bimodal shapes as well as the finite-scale analysis reinforce coexisting phases and thus a discontinuous transition. These findings strongly suggest that the above particle creation requirements constitute a minimum and fundamental mechanism determining the phase coexistence in short-range contact processes.

We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice ℤ d . Our essential assumption is that these models satisfy the condition of reflection positivity. We prove that whenever the associated mean-field theory predicts a discontinuous transition, the actual model also undergoes a discontinuous transition (which occurs near the mean-field transition temperature), provided the dimension is sufficiently large or the first-order transition in the mean- field model is sufficiently strong. As an application of our general theory, we show that for d sufficiently large, the 3-state Potts ferromagnet on ℤ d undergoes a first-order phase transition as the temperature varies. Similar results are established for all q-state Potts models with q≥3, the r-component cubic models with r≥4 and the O(N)-nematic liquid-crystal models with N≥3.