Pair Approximation Research Articles

Evolutionary dynamics of cooperation driven by a mixed update rule in structured prisoner's dilemma games.

How to understand the evolution of cooperation remains a scientific challenge. Individual strategy update rule plays an important role in the evolution of cooperation in a population. Previous works mainly assume that individuals adopt one single update rule during the evolutionary process. Indeed, individuals may adopt a mixed update rule influenced by different preferences such as payoff-driven and conformity-driven factors. It is still unclear how such mixed update rules influence the evolutionary dynamics of cooperation from a theoretical analysis perspective. In this work, in combination with the pairwise comparison rule and the conformity rule, we consider a mixed updating procedure into the evolutionary prisoner's dilemma game. We assume that individuals adopt the conformity rule for strategy updating with a certain probability in a structured population. By means of the pair approximation and mean-field approaches, we obtain the dynamical equations for the fraction of cooperators in the population. We prove that under weak selection, there exists one unique interior equilibrium point, which is stable, in the system. Accordingly, cooperators can survive with defectors under the mixed update rule in the structured population. In addition, we find that the stationary fraction of cooperators increases as the conformity strength increases, but is independent of the benefit parameter. Furthermore, we perform numerical calculations and computer simulations to confirm our theoretical predictions.

Simplified Multireference Coupled-Cluster Methods: Hybrid Approaches With Averaged Coupled Pair Theories.

We define an approximation to the internally contracted multireference coupled-cluster method with single and double excitations by a hybrid approach. The rationale is to treat the external pair energy contributions by the coupled-cluster method, which provides accurate results for a large part of the correlation energy while being tractable as the involved pair cluster operators commute. For the internal and semi-internal contributions, for which the coupled-cluster part becomes involved due to non-commuting operators, a linearized approach based on the coupled-electron pair approximation (CEPA) is used. For the latter, the CEPA(0) method, the averaged coupled pair functional (ACPF), the averaged quadratic coupled-cluster (AQCC) method, and the averaged CEPA method are tested. We test the methods concerning size consistency, potential energy curves for C2, N2, CN, and O3 and for the singlet-triplet splitting of ortho-, meta-, and para-benzynes. Our results show that AQCC provides the most accurate results and stable performance. The main drawback of the method is that it shows small violations of size consistency.

Q-deformed evolutionary dynamics in simple matrix games.

We consider evolutionary games in which the agent selected for update compares strategies with q neighbors, rather than a single neighbor as in standard evolutionary game theory. If all q sampled neighbors agree, then agents update their strategy using a smoothed best-response rule. Following ideas from social impact theory, we also study mathematical generalizations of the dynamics to noninteger q>0. Looking at fixed point stability and fixation times for 2×2 games with all-to-all interactions, we find that the flow changes significantly as a function of q. Further, we develop the pair approximation for the q-deformed dynamics on uncorrelated graphs. We also study games with more than two strategies, such as the rock-paper-scissors game where we show that changing q leads to the emergence of new types of flow within the simplex.

Enhancing the electron pair approximation with measurements on trapped-ion quantum computers

The electron pair approximation offers an efficient variational quantum eigensolver (VQE) approach for chemistry simulations on quantum computers. With the number of entangling gates scaling quadratically with system size and a constant measurement overhead, the orbital optimized unitary pair coupled cluster double (oo-upCCD) ansatz strikes a balance between accuracy and efficiency. However, the electron pair approximation prevents the method from achieving quantitative accuracy. To improve it, we explore the theory of second order perturbation (PT2) correction to oo-upCCD. PT2 accounts for the missing broken-pair contributions in oo-upCCD, while retaining its efficiencies. For molecular bond stretching and chemical reactions, the method significantly improves the predicted energy accuracy, reducing oo-upCCD’s error by up to 90%. On IonQ’s quantum computers, we find that the PT2 energy correction is highly noise-resilient. The predicted VQE-PT2 reaction energies are in excellent agreement with noise-free simulators after applying simple error mitigations solely on the VQE energies.

Predicting Heteropolymer Interactions: Demixing and Hypermixing of Disordered Protein Sequences

Cells contain multiple condensates which spontaneously form due to the heterotypic interactions between their components. Although the proteins and disordered region sequences that are responsible for condensate formation have been extensively studied, the rule of interactions between the components that allow demixing, i.e., the coexistence of multiple condensates, is yet to be elucidated. Here, we construct an effective theory of the interaction between heteropolymers by fitting it to the molecular dynamics simulation results obtained for more than 200 sequences sampled from the disordered regions of human proteins. We find that the sum of amino acid pair interactions across two heteropolymers predicts the Boyle temperature qualitatively well, which can be quantitatively improved by the dimer pair approximation, where we incorporate the effect of neighboring amino acids in the sequences. The improved theory, combined with the finding of a metric that captures the effective interaction strength between distinct sequences, allowed the selection of up to three disordered region sequences that demix with each other in multicomponent simulations, as well as the generation of artificial sequences that demix with a given sequence. The theory points to a generic sequence design strategy to demix or hypermix thanks to the low-dimensional nature of the space of the interactions that we identify. As a consequence of the geometric arguments in the space of interactions, we find that the number of distinct sequences that can demix with each other is strongly constrained, irrespective of the choice of the coarse-grained model. Altogether, we construct a theoretical basis for methods to estimate the effective interaction between heteropolymers, which can be utilized in predicting phase separation properties as well as rules of assignment in the localization and functions of disordered proteins. Published by the American Physical Society 2024

Approximations to Study the Impact of the Service Discipline in Systems with Redundancy

In this paper we develop the first methods to approximate the queue length distribution in a queueing system with redundancy under various service disciplines. We focus on a system with exponential job sizes, i.i.d. copies, and a large number of servers. In the case of Processor Sharing, we provide a pair and a triplet approximation. We also develop a pair approximation for First-Come-First-Served, Limited Processor Sharing and Last-Come-First-Served servers. We present numerical evidence that shows that all the approximations presented in the paper are highly accurate, but that none of them are asymptotically exact (as the number of servers goes to infinity).

Ordering dynamics of nonlinear voter models.

We study the ordering dynamics of nonlinear voter models with multiple states, also providing a discussion of the two-state model. The rate with which an individual adopts an opinion scales as the qth power of the number of the individual's neighbors in that state. For q>1 the dynamics favor the opinion held by the most agents. The ordering to consensus is driven by deterministic drift, and noise plays only a minor role. For q<1 the dynamics favors minority opinions, and for multistate models the ordering proceeds through a noise-driven succession of metastable states. Unlike linear multistate systems, the nonlinear model cannot be reduced to an effective two-state model. We find that the average density of active interfaces in the model with multiple opinion states does not show a single exponential decay in time for q<1, again at variance with the linear model. This highlights the special character of the conventional (linear) voter model, in which deterministic drift is absent. As part of our analysis, we develop a pair approximation for the multistate model on graphs, valid for any positive real value of q, improving on previous approximations for nonlinear two-state voter models.

Approximations to Study the Impact of the Service Discipline in Systems with Redundancy

As job redundancy has been recognized as an effective means to improve performance of large-scale computer systems, queueing systems with redundancy have been studied by various authors. Existing results include methods to compute the queue length distribution and response time but only when the service discipline is First-Come-First-Served (FCFS). For other service disciplines, such as Processor Sharing (PS), or Last-Come-First-Served (LCFS), only the stability conditions are known. In this paper we develop the first methods to approximate the queue length distribution in a queueing system with redundancy under various service disciplines. We focus on a system with exponential job sizes, i.i.d. copies, and a large number of servers. We first derive a mean field approximation that is independent of the scheduling policy. In order to study the impact of service discipline, we then derive refinements of this approximation to specific scheduling policies. In the case of Processor Sharing, we provide a pair and a triplet approximation. The pair approximation can be regarded as a refinement of the classic mean field approximation and takes the service discipline into account, while the triplet approximation further refines the pair approximation. We also develop a pair approximation for three other service disciplines: First-Come-First-Served, Limited Processor Sharing and Last-Come-First-Served. We present numerical evidence that shows that all the approximations presented in the paper are highly accurate, but that none of them are asymptotically exact (as the number of servers goes to infinity). This makes these approximations suitable to study the impact of the service discipline on the queue length distribution. Our results show that FCFS yields the shortest queue length, and that the differences are more substantial at higher loads.

A Simple Electron-Density Based Force Field Model for High-Energy Interactions between Atoms and Molecules.

In high-energy molecular dynamics or Monte Carlo simulations, standard force fields optimized for simulations at ambient temperatures are inadequate. This is largely because their repulsive parts have been regarded as not very significant, even well below zero interaction energies. It is, therefore, not obvious which force fields to resort to for simulating hot gases or plasmas. A force field model that uses the electronic densities of noninteracting atoms or molecules within the pair approximation is introduced. We start by deriving a naïve model that neglects any exchange and correlation effects between the electronic clouds and then correct this model by adding a term calibrated from ab initio calculations using the CCSD(T)/cc-pVTZ level of theory. The resulting expression for this term can be regarded as a simple exchange-correlation function. We compare the results for the repulsive part of the potential energy hypersurfaces with the force fields commonly used on some dimers of small molecules.

Second quantization-based symmetry-adapted perturbation theory: Generalizing exchange beyond single electron pair approximation.

This paper presents a general second-quantized form of a permutation operator interchanging n pairs of electrons between interacting subsystems in the framework of the symmetry-adapted perturbation theory (SAPT). We detail the procedure for constructing this operator through the consecutive multiplication of single-pair permutation operators. This generalized form of the permutation operator has enabled the derivation of universal formulas for S2n approximations of the exchange energies in the first and second order of the interaction operator. We present expressions for corrections of S4 approximations and assess its efficacy on a selection of systems anticipated to exhibit a slowly converging overlap expansion. Additionally, we outline a method to sum the overlap expansion series to infinity in second-quantization, up to the second order in V. This new approach offers an alternative to the existing formalism based on density-matrix formulations. When combined with a symbolic algebra program for automated derivations, it paves the way for advancements in SAPT theory, particularly for intricate wavefunction theories.

Cooperativity, Information Gain, and Energy Cost During Early LTP in Dendritic Spines.

We investigate a mutual relationship between information and energy during the early phase of LTP induction and maintenance in a large-scale system of mutually coupled dendritic spines, with discrete internal states and probabilistic dynamics, within the framework of nonequilibrium stochastic thermodynamics. In order to analyze this computationally intractable stochastic multidimensional system, we introduce a pair approximation, which allows us to reduce the spine dynamics into a lower-dimensional manageable system of closed equations. We found that the rates of information gain and energy attain their maximal values during an initial period of LTP (i.e., during stimulation), and after that, they recover to their baseline low values, as opposed to a memory trace that lasts much longer. This suggests that the learning phase is much more energy demanding than the memory phase. We show that positive correlations between neighboring spines increase both a duration of memory trace and energy cost during LTP, but the memory time per invested energy increases dramatically for very strong, positive synaptic cooperativity, suggesting a beneficial role of synaptic clustering on memory duration. In contrast, information gain after LTP is the largest for negative correlations, and energy efficiency of that information generally declines with increasing synaptic cooperativity. We also find that dendritic spines can use sparse representations for encoding long-term information, as both energetic and structural efficiencies of retained information and its lifetime exhibit maxima for low fractions of stimulated synapses during LTP. Moreover, we find that such efficiencies drop significantly with increasing the number of spines. In general, our stochastic thermodynamics approach provides a unifying framework for studying, from first principles, information encoding, and its energy cost during learning and memory in stochastic systems of interacting synapses.

Q-voter model with independence on signed random graphs: Homogeneous approximations.

The q-voter model with independence is generalized to signed random graphs and studied by means of Monte Carlo simulations and theoretically using the mean-field approximation and different forms of the pair approximation. In the signed network with quenched disorder, positive and negative signs associated randomly with the links correspond to reinforcing and antagonistic interactions, promoting, respectively, the same or opposite orientations of two-state spins representing agents' opinions; otherwise, the opinions are called mismatched. With probability 1-p, the agents change their opinions if the opinions of all members of a randomly selected q neighborhood are mismatched, and with probability p, they choose an opinion randomly. The model on networks with finite mean degree 〈k〉 and fixed fraction of the antagonistic interactions r exhibits ferromagnetic transition with varying the independence parameter p, which can be first or second order, depending on q and r, and disappears for large r. Besides, numerical evidence is provided for the occurrence of the spin-glass-like transition for large r. The order and critical lines for the ferromagnetic transition on the p vs r phase diagram obtained in Monte Carlo simulations are reproduced qualitatively by the mean-field approximation. Within the range of applicability of the pair approximation, for the model with 〈k〉 finite but 〈k〉≫q, predictions of the homogeneous pair approximation concerning the ferromagnetic transition show much better quantitative agreement with numerical results for small r but fail for larger r. A more advanced signed homogeneous pair approximation is formulated which distinguishes between classes of active links with a given sign connecting nodes occupied by agents with mismatched opinions; for the model with 〈k〉≫q its predictions agree quantitatively with numerical results in a whole range of r where the ferromagnetic transition occurs.

Interaction state Q-learning promotes cooperation in the spatial prisoner's dilemma game

Interaction state Q-learning promotes cooperation in the spatial prisoner's dilemma game

Molecular Oxygen Trimer: Multiplet Structures and Stability.

We present a detailed theoretical study of the molecular oxygen trimer where the potential energy surfaces of the seven multiplet states have been calculated by means of a pair approximation with very accurate dimer ab initio potentials. In order to obtain all the states a matrix representation of the potential using the uncoupled spin representation has been applied. The and states are nearly degenerate and low-lying isomers appear for most multiplicities. A crucial point in deciding the relative stabilities is the zero-point energy which represents a sizable fraction of the electronic well-depth. Therefore, we have performed accurate diffusion Monte Carlo studies of the lowest state in each multiplicity. Analysis of the wavefunction allows a deeper interpretation of the cluster structures, finding that they are significantly floppy in most cases.

Phase equilibria and thermodynamic modeling of the Sn–S, Ag–S, and Sb–S systems

Phase equilibria and thermodynamic modeling of the Sn–S, Ag–S, and Sb–S systems

Linear Scaling Calculations of Excitation Energies with Active-Space Particle-Particle Random-Phase Approximation.

We developed an efficient active-space particle-particle random-phase approximation (ppRPA) approach to calculate accurate charge-neutral excitation energies of molecular systems. The active-space ppRPA approach constrains both indexes in particle and hole pairs in the ppRPA matrix, which only selects frontier orbitals with dominant contributions to low-lying excitation energies. It employs the truncation in both orbital indexes in the particle-particle and the hole-hole spaces. The resulting matrix, whose eigenvalues are excitation energies, has a dimension that is independent of the size of the systems. The computational effort for the excitation energy calculation, therefore, scales linearly with system size and is negligible compared with the ground-state calculation of the (N - 2)-electron system, where N is the electron number of the molecule. With the active space consisting of 30 occupied and 30 virtual orbitals, the active-space ppRPA approach predicts the excitation energies of valence, charge-transfer, Rydberg, double, and diradical excitations with the mean absolute errors (MAEs) smaller than 0.03 eV compared with the full-space ppRPA results. As a side product, we also applied the active-space ppRPA approach in the renormalized singles (RS) T-matrix approach. Combining the non-interacting pair approximation that approximates the contribution to the self-energy outside the active space, the active-space GRSTRS@PBE approach predicts accurate absolute and relative core-level binding energies with the MAEs around 1.58 and 0.3 eV, respectively. The developed linear scaling calculation of excitation energies is promising for applications to large and complex systems.

Thermodynamic evaluation and optimization of (LiF + ReF3) (Re = La, Nd, Gd, Yb, Y, Lu) binary systems

Thermodynamic evaluation and optimization of (LiF + ReF3) (Re = La, Nd, Gd, Yb, Y, Lu) binary systems

A proton pairing isomer in 174Hf: Reinterpretation of data and support for the split prolate–oblate pairing approximation for deformed nuclei

The available data on the even–even isotope 174Hf are reinterpreted to conclude that the low-lying 02+ level, usually regarded as a “β vibration”, is actually a proton (2p–2 h) excitation known as a Pairing Isomer. This is the first identification of a Pairing Isomer based on Proton Nilsson Orbitals. Its identification demonstrates that Pairing Isomers can exist anywhere in the Nuclear Chart if the positioning of the Fermi Surface is favourable with respect to relevant Nilsson orbitals.

Q-neighbor Ising model on multiplex networks with partial overlap of nodes.

The q-neighbor Ising model for the opinion formation on multiplex networks with two layers in the form of random graphs (duplex networks), the partial overlap of nodes, and LOCAL&AND spin update rule was investigated by means of the pair approximation and approximate master equationsas well as Monte Carlo simulations. Both analytic and numerical results show that for different fixed sizes of the q-neighborhood and finite mean degrees of nodes within the layers the model exhibits qualitatively similar critical behavior as the analogous model on multiplex networks with layers in the form of complete graphs. However, as the mean degree of nodes is decreased the discontinuous ferromagnetic transition, the tricritical point separating it from the continuous transition, and the possible coexistence of the paramagnetic and ferromagnetic phases at zero temperature occur for smaller relative sizes of the overlap. Predictions of the simple homogeneous pair approximation concerning the critical behavior of the model under study show good qualitative agreement with numerical results; predictions based on the approximate master equationsare usually quantitatively more accurate but yet not exact. Two versions of the heterogeneous pair approximation are also derived for the model under study, which, surprisingly, yield predictions only marginally different or even identical to those of the simple homogeneous pair approximation. In general, predictions of all approximations show better agreement with the results of Monte Carlo simulations in the case of continuous than discontinuous ferromagnetic transition.

From complete to selected model spaces in determinant-based multi-reference second-order perturbation treatments.

The present paper reformulates and improves a previously proposed determinant-based second-order multi-reference perturbative formalism. Through a rather simple modification of the energy denominators, this formalism takes into account the interactions between the model space determinants, which are repeated in outer space. The method has been shown to be size-consistent when the model space is a complete active space, which is a severe limit. It is shown here that the completeness of the model space is not necessary to keep this property, provided that the zero-order function satisfies some conditions. For instance, size consistency may be obtained from truncated complete active spaces. It may even be satisfied from Singles and Doubles Configuration Interactions, provided that a coupled electron pair approximation is used in the definition of the model space wave function. The physical content of the method is illustrated by a series of model problems, showing its robustness. A major benefit of the fact that the perturbers are single determinants is the possibility to revise with full flexibility the model-space component of the wave function, i.e., to treat the feedback effect of the dynamic correlation on the valence component of the wave function.