Validity of annealed approximation in a high-dimensional system.

This study investigates the suitability of the annealed approximation in high-dimensional systems characterized by dense networks with quenched link disorder, employing models of coupled oscillators. We demonstrate that dynamic equations governing dense-network systems converge to those of the complete-graph version in the thermodynamic limit, where link disorder fluctuations vanish entirely. Consequently, the annealed-network systems, where fluctuations are attenuated, also exhibit the same dynamic behavior in the thermodynamic limit. However, a significant discrepancy arises in the incoherent (disordered) phase wherein the finite-size behavior becomes critical in determining the steady-state pattern. To explicitly elucidate this discrepancy, we focus on identical oscillators subject to competitive attractive and repulsive couplings. In the incoherent phase of dense networks, we observe the manifestation of random irregular states. In contrast, the annealed approximation yields a symmetric (regular) incoherent state where two oppositely coherent clusters of oscillators coexist, accompanied by the vanishing order parameter. Our findings imply that the annealed approximation should be employed with caution even in dense-network systems, particularly in the disordered phase.

Extended mean-field approach for chimera states in random complex networks.

Identical oscillators in the chimera state exhibit a mixture of coherent and incoherent patterns simultaneously. Nonlocal interactions and phase lag are critical factors in forming a chimera state within the Kuramoto model in Euclidean space. Here, we investigate the contributions of nonlocal interactions and phase lag to the formation of the chimera state in random networks. By developing an extended mean-field approximation and using a numerical approach, we find that the emergence of a chimera state in the Erdös-Rényi network is due mainly to degree heterogeneity with nonzero phase lag. For a regularly random network, although all nodes have the same degree, we find that disordered connections may yield the chimera state in the presence of long-range interactions. Furthermore, we show a nontrivial dynamic state in which all the oscillators drift more slowly than a defined frequency due to connectivity disorder at large phase lags beyond the mean-field solutions.

Effective-potential approach to hybrid synchronization transitions.

The Kuramoto model exhibits different types of synchronization transitions depending on the type of natural frequency distribution. To obtain these results, the Kuramoto self-consistency equation (SCE) approach has been used successfully. However, this approach affords only limited understanding of more detailed properties such as the stability. Here we extend the SCE approach by introducing an effective potential, that is, an integral version of the SCE. We examine the landscape of this effective potential for second-order, first-order, and hybrid synchronization transitions in the thermodynamic limit. In particular, for the hybrid transition, we find that the minimum of effective potential displays a plateau across the region in which the order parameter jumps. This result suggests that the effective potential can be used to determine a type of synchronization transition.

Exactly solvable two-terminal heat engine with asymmetric Onsager coefficients: Origin of the power-efficiency bound.

An engine producing a finite power at the ideal (Carnot) efficiency is a dream engine which is not prohibited by the thermodynamic second law. Some years ago, a two-terminal heat engine with asymmetric Onsager coefficients in the linear response regime was suggested by Benenti et al. [Phys. Rev. Lett. 106, 230602 (2011)10.1103/PhysRevLett.106.230602], as a prototypical system to make such a dream come true with nondivergent system parameter values. However, such a system has never been realized, in spite of many trials. Here, we introduce an exactly solvable two-terminal Brownian heat engine with the asymmetric Onsager coefficients in the presence of a Lorenz (magnetic) force. Nevertheless, we show that the dream engine regime cannot be accessible, even with the asymmetric Onsager coefficients, due to an instability keeping the engine from reaching its steady state. This is consistent with recent tradeoff relations between the engine power and efficiency, where the (cyclic) steady-state condition is implicitly presumed. We conclude that the inaccessibility to the dream engine originates from the steady-state constraint on the engine.

Entropy production and fluctuation theorems on complex networks.

Entropy production (EP) is a fundamental quantity useful for understanding irreversible process. In stochastic thermodynamics, EP is more evident in probability density functions of trajectories of a particle in the state space. Here, inspired by a previous result that complex networks can serve as state spaces, we consider a data packet transport problem on complex networks. EP is generated owing to the complexity of pathways as the packet travels back and forth between two nodes along the same pathway. The total EPs are exactly enumerated along all possible shortest paths between every pair of nodes, and the functional form of the EP distribution is proposed based on our numerical results. We confirm that the EP distribution satisfies the detailed and integral fluctuation theorems. Our results should be pedagogically helpful for understanding trajectory-dependent EP in stochastic processes and exploring nonequilibrium fluctuations associated with the entanglement of dividing and merging among the shortest pathways in complex networks.

Three heats in a strongly coupled system and bath.

We investigate three kinds of heat produced in a system and a bath strongly coupled via an interaction Hamiltonian. By studying the energy flows between the system, the bath, and their interaction, we provide rigorous definitions of two types of heat, Q_{S} and Q_{B}, from the energy loss of the system and the energy gain of the bath, respectively. This is in contrast to the equivalence of Q_{S} and Q_{B}, which is commonly assumed to hold in the weak-coupling regime. The bath we consider is equipped with a thermostat which enables it to reach an equilibrium. We identify another kind of heat Q_{SB} from the energy dissipation of the bath into the superbath that provides the thermostat. We derive the fluctuation theorems (FTs) for the system variables and various heats, which are discussed in comparison with the FT for the total entropy production. We take an example of a sliding harmonic potential of a single Brownian particle in a fluid and calculate the three heats in a simplified model. These heats are found to equal, on average, in the steady state of energy, but show different fluctuations at all times.

Nonequilibrium phase transition in an open quantum spin system with long-range interaction.

We investigate a nonequilibrium phase transition in a dissipative and coherent quantum spin system using the quantum Langevin equation and mean-field theory. Recently, the quantum contact process (QCP) was theoretically investigated using the Rydberg antiblockade effect, in particular, when the Rydberg atoms were excited in s states so that their interactions were regarded as being between the nearest neighbors. However, when the atoms are excited to d states, the dipole-dipole interactions become effective, and long-range interactions must be considered. Here we consider a quantum spin model with a long-range QCP, where the branching and coagulation processes are allowed not only for the nearest-neighbor pairs but also for long-distance pairs, coherently and incoherently. Using the semiclassical approach, we show that the mean-field phase diagram of our long-range model is similar to that of the nearest-neighbor QCP, where the continuous (discontinuous) transition is found in the weak (strong) quantum regime. However, at the tricritical point, we find a new universality class, which was neither that of the QCP at the tricritical point nor that of the classical directed percolation model with long-range interactions. Implementation of the long-range QCP using interacting cold gases is discussed.

An experimentally-achieved information-driven Brownian motor shows maximum power at the relaxation time.

We present an experimental realization of an information-driven Brownian motor by periodically cooling a Brownian particle trapped in a harmonic potential connected to a single heat bath, where cooling is carried out by the information process consisting of measurement and feedback control. We show that the random motion of the particle is rectified by symmetry-broken feedback cooling where the particle is cooled only when it resides on the specific side of the potential center at the instant of measurement. Studying how the motor thermodynamics depends on cycle period τ relative to the relaxation time τB of the Brownian particle, we find that the ratcheting of thermal noise produces the maximum work extraction when τ ≥ 5τB, while the extracted power is maximum near τ = τB, implying the optimal operating time for the ratcheting process. In addition, we find that the average transport velocity is monotonically decreased as τ increases and present the upper bound for the velocity.

Nature of synchronization transitions in random networks of coupled oscillators.

We consider a system of phase oscillators with random intrinsic frequencies coupled through sparse random networks and investigate how the connectivity disorder affects the nature of collective synchronization transitions. Various distribution types of intrinsic frequencies are considered: uniform, unimodal, and bimodal distribution. We employ a heterogeneous mean-field approximation based on the annealed networks and also perform numerical simulations on the quenched Erdös-Rényi networks. We find that the connectivity disorder drastically changes the nature of the synchronization transitions. In particular, the quenched randomness completely wipes away the diversity of the transition nature, and only a continuous transition appears with the same mean-field exponent for all types of frequency distributions. The physical origin of this unexpected result is discussed.

Link-disorder fluctuation effects on synchronization in random networks.

We consider one typical system of oscillators coupled through disordered link configurations in networks, i.e., a finite population of coupled phase oscillators with distributed intrinsic frequencies on a random network. We investigate the collective synchronization behavior, paying particular attention to link-disorder fluctuation effects on the synchronization transition and its finite-size scaling (FSS). Extensive numerical simulations as well as the mean-field analysis have been performed. We find that link-disorder fluctuations effectively induce uncorrelated random fluctuations in frequency, resulting in the FSS exponent ν[over ¯]=5/2, which is identical to that in the globally coupled case (no link disorder) with frequency-disorder fluctuations.

Double resonance in the infinite-range quantum Ising model.

We study quantum resonance behavior of the infinite-range kinetic Ising model at zero temperature. Numerical integration of the time-dependent Schrödinger equation in the presence of an external magnetic field in the z direction is performed at various transverse field strengths g. It is revealed that two resonance peaks occur when the energy gap matches the external driving frequency at two distinct values of g, one below and the other above the quantum phase transition. From the similar observations already made in classical systems with phase transitions, we propose that the double resonance peaks should be a generic feature of continuous transitions, for both quantum and classical many-body systems.

Rectification of spatial disorder.

We demonstrate that a large ensemble of noiseless globally coupled-pinned oscillators is capable of rectifying spatial disorder with spontaneous current activated through a dynamical phase transition mechanism, either of first or second order, depending on the profile of the pinning potential. In the presence of an external weak drive, the same collective mechanism can result in an absolute negative mobility, which, though not immediately related to symmetry breaking, is most prominent at the phase transition. Our results apply to a tug-of-war by competing molecular motors for bidirectional cargo transport.

Synchronization in interdependent networks.

We explore the synchronization behavior in interdependent systems, where the one-dimensional (1D) network (the intranetwork coupling strength J(I)) is ferromagnetically intercoupled (the strength J) to the Watts-Strogatz (WS) small-world network (the intranetwork coupling strength J(II)). In the absence of the internetwork coupling (J=0), the former network is well known not to exhibit the synchronized phase at any finite coupling strength, whereas the latter displays the mean-field transition. Through an analytic approach based on the mean-field approximation, it is found that for the weakly coupled 1D network (J(I)≪1) the increase of J suppresses synchrony, because the nonsynchronized 1D network becomes a heavier burden for the synchronization process of the WS network. As the coupling in the 1D network becomes stronger, it is revealed by the renormalization group (RG) argument that the synchronization is enhanced as J(I) is increased, implying that the more enhanced partial synchronization in the 1D network makes the burden lighter. Extensive numerical simulations confirm these expected behaviors, while exhibiting a reentrant behavior in the intermediate range of J(I). The nonmonotonic change of the critical value of J(II) is also compared with the result from the numerical RG calculation.

Voter model on a directed network: role of bidirectional opinion exchanges.

The voter model with the node update rule is numerically investigated on a directed network. We start from a directed hierarchical tree, and split and rewire each incoming arc at the probability p . In order to discriminate the better and worse opinions, we break the Z2 symmetry (σ=±1) by giving a little more preference to the opinion σ=1 . It is found that as p becomes larger, introducing more complicated pattern of information flow channels, and as the network size N becomes larger, the system eventually evolves to the state in which more voters agree on the better opinion, even though the voter at the top of the hierarchy keeps the worse opinion. We also find that the pure hierarchical tree makes opinion agreement very fast, while the final absorbing state can easily be influenced by voters at the higher ranks. On the other hand, although the ordering occurs much slower, the existence of complicated pattern of bidirectional information flow allows the system to agree on the better opinion.

Effective projection theory in a correlated electron system.

We propose an efficient method for nonperturbative calculation of Green's function in a correlated electron system. The key idea of the method is to project out irrelevant operators having zero norm in the ground state, which we refer to as effective projection theory. We apply the method to a mesoscopic Anderson model and show that for a given wavefunction ansatz, equations of motion can be closed only by relevant operators, allowing easy calculation of the zero-temperature Green's function. It turns out that the resulting Green's functions reproduce exact limits of both weak and strong interactions. The accuracy is also verified for small systems by comparison with exact diagonalization results, revealing that effective projection theory captures the essential correlated features in the entire regime of interactions.

Scaling laws between population and facility densities.

When a new facility like a grocery store, a school, or a fire station is planned, its location should ideally be determined by the necessities of people who live nearby. Empirically, it has been found that there exists a positive correlation between facility and population densities. In the present work, we investigate the ideal relation between the population and the facility densities within the framework of an economic mechanism governing microdynamics. In previous studies based on the global optimization of facility positions in minimizing the overall travel distance between people and facilities, it was shown that the density of facility D and that of population rho should follow a simple power law D approximately rho(2/3). In our empirical analysis, on the other hand, the power-law exponent alpha in D approximately rho(alpha) is not a fixed value but spreads in a broad range depending on facility types. To explain this discrepancy in alpha, we propose a model based on economic mechanisms that mimic the competitive balance between the profit of the facilities and the social opportunity cost for populations. Through our simple, microscopically driven model, we show that commercial facilities driven by the profit of the facilities have alpha = 1, whereas public facilities driven by the social opportunity cost have alpha = 2/3. We simulate this model to find the optimal positions of facilities on a real U.S. map and show that the results are consistent with the empirical data.

Instability of defensive alliances in the predator-prey model on complex networks.

A model of six-species food web is studied in the viewpoint of spatial interaction structures. Each species has two predators and two preys, and it was previously known that the defensive alliances of three cyclically predating species self-organize in two dimensions. The alliance-breaking transition occurs as either the mutation rate is increased or interaction topology is randomized in the scheme of the Watts-Strogatz model. In the former case of temporal disorder, via the finite-size scaling analysis, the transition is clearly shown to belong to the two-dimensional Ising universality class. In contrast, the geometric or spatial randomness for the latter case yields a discontinuous phase transition. The mean-field limit of the model is analytically solved and then compared with numerical results. The dynamic universality and the temporally periodic behaviors are also discussed.