Infinite System Size Research Articles

Predicting properties of the stationary probability currents for two-species reaction systems without solving the Fokker-Planck equation.

We derive methods for estimating the topology of the stationary probability current j[over ⃗]_{s} of the two-species Fokker-Planck equation (FPE) without the need to solve the FPE. These methods are chosen such that they become exact in certain limits, such as infinite system size or vanishing coupling between species in the diffusion matrix. The methods make predictions about the fixed points of j[over ⃗]_{s} and their relation to extrema of the stationary probability distribution and to fixed points of the convective field, which is related to the deterministic drift of the system. Furthermore, they predict the rotation sense of j[over ⃗]_{s} around extrema of the stationary probability distribution. Even though these methods cannot be proven to be valid away from extrema, the boundary lines between regions with different rotation senses are obtained with surprising accuracy. We illustrate and test these methods, using simple reaction systems with only one coupling term between the two species as well as a few generic reaction networks taken from the literature. We use it also to investigate the shape of nonphysical probability currents occurring in reaction systems with detailed balance due to the approximations involved in deriving the Fokker-Planck equation.

Clifford Boundary Conditions: A Simple Direct-Sum Evaluation of Madelung Constants.

We propose a simple direct-sum method for the efficient evaluation of lattice sums in periodic solids. It consists of two main principles: (i) the creation of a supercell that has the topology of a Clifford torus, which is a flat, finite, and borderless manifold; (ii) the renormalization of the distance between two points on the Clifford torus by defining it as the Euclidean distance in the embedding space of the Clifford torus. Our approach does not require any integral transformations nor any renormalization of the charges. We illustrate our approach by applying it to the calculation of the Madelung constants of ionic crystals. We show that the convergence toward the system of infinite size is monotonic, which allows for a straightforward extrapolation of the Madelung constant. We are able to recover the Madelung constants with a remarkable accuracy, and at an almost negligible computational cost, i.e., a few seconds on a laptop computer.

Numerical Verification of the Fluctuation-Dissipation Theorem for Isolated Quantum Systems.

The fluctuation-dissipation theorem (FDT) is a hallmark of thermal equilibrium systems in the Gibbs state. We address the question whether the FDT is obeyed by isolated quantum systems in an energy eigenstate. In the framework of the eigenstate thermalization hypothesis, we derive the formal expression for two-time correlation functions in the energy eigenstates or in the diagonal ensemble. They satisfy the Kubo-Martin-Schwinger condition, which is the sufficient and necessary condition for the FDT, in the infinite system size limit. We also obtain the finite size correction to the FDT for finite-sized systems. With extensive numerical works for the XXZ spin chain model, we confirm our theory for the FDT and the finite size correction. Our results can serve as a guide line for an experimental study of the FDT on a finite-sized system.

Infinite ergodic theory meets Boltzmann statistics

We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite ergodic theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. A generalized virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances. When the process is non-recurrent, e.g. diffusion in three dimensions with a Coulomb-like potential, we use weighted time averages to restore basic canonical relations between time and ensemble averages.

Level statistics of extended states in random non-Hermitian Hamiltonians

Absence of level repulsion between extended states in random non-Hermitian systems is demonstrated. As a result, the general Wigner-Dyson distributions of level spacing of diffusive metals in the usual Hermitian systems is replaced by the Poisson distribution for quasiparticle level spacing of non-Hermitian disordered metals in the thermodynamic limit of infinite system size. This is a very surprising result because Poisson statistics is universally true for the Anderson insulators where energy eigenstates do not overlap with each other so that energy levels are independent from each other.For disordered metals where different eigenstates overlap with each other, one should expect different levels trying to stay away from each other so that the Poisson distribution should not apply there. Our results show that the larger non-Hermitian energy (dissipation) can invalidate level repulsion principle that holds dearly in quantum mechanics. Thus, our theory provides a unified picture for recent discovery of so called "level attraction" in various systems. It provides also a theoretical basis for manipulating energy levels.

Thermal Conductivity of Gold–Phenylethanethiol (Au144PET60) Nanoarrays: A Molecular Dynamics Study

Solvated nanoarrays of Au144 nanoparticles capped with 60 phenylethanethiol (PET) moieties were studied using reverse nonequilibrium molecular dynamics (RNEMD) simulations. The thermal conductivities of the nanoarrays were computed for two chemically dissimilar solvents. These were compared to an effective medium theory (EMT) model based solely on bulk and interfacial properties and the volume fraction taken up by the particles. The interfacial thermal conductance for isolated particles was also estimated via RNEMD using individual solvated particles in nonperiodic geometries. A strong system-size dependence was found in bulk conductivity calculations, particularly in the polar solvent, dichloromethane, but was found to be unimportant in a second nonpolar solvent (toluene). However, the bulk conductivity of solvated nanoarrays also requires projection to infinite system size. The Au144PET60 particles were found to display a molten core at the temperatures used in this study. Our primary finding is that the volume fraction of the metal particles is the primary variable controlling the thermal conductivity of the nanoarrays and that simulated conductivities are predicted well by the EMT model.

Nucleation of spatiotemporal structures from defect turbulence in the two-dimensional complex Ginzburg-Landau equation.

We numerically investigate nucleation processes in the transient dynamics of the two-dimensional complex Ginzburg-Landau equation toward its "frozen" state with quasistationary spiral structures. We study the transition kinetics from either the defect turbulence regime or random initial configurations to the frozen state with a well-defined low density of quasistationary topological defects. Nucleation events of spiral structures are monitored using the characteristic length between the emerging shock fronts. We study two distinct situations, namely when the system is quenched either far from the transition limit or near it. In the former deeply quenched case, the average nucleation time for different system sizes is measured over many independent realizations. We employ an extrapolation method as well as a phenomenological formula to account for and eliminate finite-size effects. The nonzero (dimensionless) barrier for the nucleation of single spiral droplets in the extrapolated infinite system size limit suggests that the transition to the frozen state is discontinuous. We also investigate the nucleation of spirals for systems that are quenched close to but beyond the crossover limit and of target waves which emerge if a specific spatial inhomogeneity is introduced. In either of these cases, we observe long, "fat" tails in the distribution of nucleation times, which also supports a discontinuous transition scenario.

On the Hydrodynamic Stability of a Lennard-Jones Molecular Fluid

As a central concept in fluid dynamics stability is fundamental in understanding transitions from laminar to turbulent flow. In continuum flows, it is well-established that a transition to turbulence can occur at subcritical Reynolds numbers, in contrast to theoretical predictions. In non-equilibrium molecular dynamics (NEMD), it has been widely observed that at a critical Reynolds number the fluid undergoes an ordering transition from an amorphous phase to a ‘string’ phase. Using the fluctuation theorem (FT) and the dissipation function, we generalize the classical continuum Reynolds-Orr equation to sheared molecular fluids by ascribing a natural description to the nature of stochastic perturbations, i.e. fluctuations in shear stress. Via the Poincare inequality, we arrive at a new stability criterion by providing a lower bound on the exponential decay of perturbations, which reduces to the classical continuum result in the limit of infinite system size. We investigate the nature of these velocity perturbations and conditions necessary for growth in the kinetic energy of perturbations. We obtain a fluid dependent estimate for the critical Reynolds number by which one may estimate the critical Reynolds number at which the fluid transitions to the string phase, thus providing a framework for generalizing classical continuum theories to the microscale.

Determining a local Hamiltonian from a single eigenstate

We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is ``yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the ``k-correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.

Three-Spin System in the Presence of the Quantum Cluster Interaction

We consider a three-spin system under Heisenberg and quantum cluster interactions. Firstly, by focusing on the entanglement between each pair of them at zero temperature, we show that where the quantum phase transition (QPT) of the infinite-size system might happen can be observed even for a such finite spin system. In fact, where the QPT happened, the ground state of the mentioned three-spin system will become the W-state, which is the main reason of possibility of detecting the place of an infinite-size system. Secondly, we investigate the thermal entanglement and show that corresponding to the temperature increase, the mentioned point goes toward the larger exchange coupling constant values, whereas the amount of entanglement decreases, that the latter meets our expectation. Then by using the one-tangle method, we show that at zero temperature, QPT is taken place at the same point. Finally, by using the logarithmic negativity measure, we investigate the role of thermal fluctuations on the previously mentioned finite spin system.

Efficient quantum simulation for thermodynamics of infinite-size many-body systems in arbitrary dimensions

In this work we propose to simulate many-body thermodynamics of infinite-size quantum lattice models in one, two, and three dimensions, in terms of few-body models of only O(10) sites, which we coin as quantum entanglement simulators (QES's). The QES is described by a temperature-independent Hamiltonian, with the boundary interactions optimized by the tensor network methods to mimic the entanglement between the bulk and environment in a finite-size canonical ensemble. The reduced density matrix of the physical bulk then gives that of the infinite-size canonical ensemble under interest. We show that the QES can, for instance, accurately simulate varieties of many-body phenomena, including finite-temperature crossover and algebraic excitations of the one-dimensional spin liquid, the phase transitions and low-temperature physics of the two- and three-dimensional antiferromagnets, and the crossovers of the two-dimensional topological system. Our work provides an efficient way to explore the thermodynamics of intractable quantum many-body systems with easily accessible systems.

Note on the Rényi entropy of 2D perturbed fermions

In this paper we study the R\'{e}nyi entropy of 2D massless free fermions perturbed by the $T\bar{T}$ term and the $J\bar{T}$ term at the first order perturbation. Three cases, the vacuum state of infinite size system with $T\bar{T}$ perturbation, the excited states of finite size system with $T\bar{T}$ perturbation, and the vacuum state of infinite size system with $J\bar{T}$ perturbation, are analyzed. We use the bosonization approach to calculate the perturbative expansions of R\'{e}nyi entropy. In the bosonization language the twist operator is known explicitly, with which the computation of correlators in the perturbative expansion can be simply performed. Moreover, we show that the $T\bar{T}$ and $J\bar{T}$ terms have a simple form and are similar with each other. For the first and third cases, we reproduce the known results for R\'{e}nyi entropy using the bosonization method. While for the second case, we obtain new results for the excited states.

A random walk perspective on hide-and-seek games

We investigate hide-and-seek games on complex networks using a random walk framework. Specifically, we investigate the efficiency of various degree-biased random walk search strategies to locate items that are randomly hidden on a subset of vertices of a random graph. Vertices at which items are hidden in the network are chosen at random as well, though with probabilities that may depend on degree. We pitch various hide and seek strategies against each other, and determine the efficiency of search strategies by computing the average number of hidden items that a searcher will uncover in a random walk of n steps. Our analysis is based on the cavity method for finite single instances of the problem, and generalises previous work of De Bacco et al (2015 J. Phys. A: Math. Theor. 48 205004) so as to cover degree-biased random walks. We also extend the analysis to deal with the thermodynamic limit of infinite system size. We study a broad spectrum of functional forms for the degree bias of both the hiding and the search strategy and investigate the efficiency of families of search strategies for cases where their functional form is either matched or unmatched to that of the hiding strategy. Our results are in excellent agreement with those of numerical simulations. We propose two simple approximations for predicting efficient search strategies. One is based on an equilibrium analysis of the random walk search strategy. While not exact, it produces correct orders of magnitude for parameters characterising optimal search strategies. The second exploits the existence of an effective drift in random walks on networks, and is expected to be efficient in systems with low concentration of small degree nodes.

Finite-size effects for spiking neural networks with spatially dependent coupling.

We study finite-size fluctuations in a network of spiking deterministic neurons coupled with nonuniform synaptic coupling. We generalize a previously developed theory of finite-size effects for globally coupled neurons with a uniform coupling function. In the uniform coupling case, mean-field theory is well defined by averaging over the network as the number of neurons in the network goes to infinity. However, for nonuniform coupling it is no longer possible to average over the entire network if we are interested in fluctuations at a particular location within the network. We show that if the coupling function approaches a continuous function in the infinite system size limit, then an average over a local neighborhood can be defined such that mean-field theory is well defined for a spatially dependent field. We then use a path-integral formalism to derive a perturbation expansion in the inverse system size around the mean-field limit for the covariance of the input to a neuron (synaptic drive) and firing rate fluctuations due to dynamical deterministic finite-size effects.

Heating in Integrable Time-Periodic Systems.

We investigate a heating phenomenon in periodically driven integrable systems that can be mapped to free-fermion models. We find that heating to the high-temperature state, which is a typical scenario in nonintegrable systems, can also appear in integrable time-periodic systems; the amount of energy absorption rises drastically near a frequency threshold where the Floquet-Magnus expansion diverges. As the driving period increases, we also observe that the effective temperatures of the generalized Gibbs ensemble for conserved quantities go to infinity. By the use of the scaling analysis, we reveal that, in the limit of infinite system size and driving period, the steady state after a long time is equivalent to the infinite-temperature state. We obtain the asymptotic behavior L^{-1} and T^{-2} as to how the steady state approaches the infinite-temperature state as the system size L and the driving period T increase.

Superslow relaxation in identical phase oscillators with random and frustrated interactions.

This paper is concerned with the relaxation dynamics of a large population of identical phase oscillators, each of which interacts with all the others through random couplings whose parameters obey the same Gaussian distribution with the average equal to zero and are mutually independent. The results obtained by numerical simulation suggest that for the infinite-size system, the absolute value of Kuramoto's order parameter exhibits superslow relaxation, i.e., 1/ln t as time t increases. Moreover, the statistics on both the transient time T for the system to reach a fixed point and the absolute value of Kuramoto's order parameter at t = T are also presented together with their distribution densities over many realizations of the coupling parameters.

Floquet Engineering in Quantum Chains

We consider a one-dimensional interacting spinless fermion model, which displays the well-known Luttinger liquid (LL) to charge density wave (CDW) transition as a function of the ratio between the strength of the interaction U and the hopping J. We subject this system to a spatially uniform drive which is ramped up over a finite time interval and becomes time periodic in the long-time limit. We show that by using a density matrix renormalization group approach formulated for infinite system sizes, we can access the large-time limit even when the drive induces finite heating. When both the initial and long-time states are in the gapless (LL) phase, the final state has power-law correlations for all ramp speeds. However, when the initial and final state are gapped (CDW phase), we find a pseudothermal state with an effective temperature that depends on the ramp rate, both for the Magnus regime in which the drive frequency is very large compared to other scales in the system and in the opposite limit where the drive frequency is less than the gap. Remarkably, quantum defects (instantons) appear when the drive tunes the system through the quantum critical point, in a realization of the Kibble-Zurek mechanism.

Cavitation transition in the energy landscape: Distinct tensile yielding behavior in strongly and weakly attractive systems.

Particles with cohesive interactions display a tensile instability in the energy landscape at the Sastry density ρS. The signature of this tensile limit is a minimum in the landscape equation of state, the pressure-density relationship of inherent structures sampled along a liquid isotherm. Our previous work [Y. E. Altabet, F. H. Stillinger, and P. G. Debenedetti, J. Chem. Phys. 145, 211905 (2016)] revisited the phenomenology of Sastry behavior and found that the evolution of the landscape equation of state with system size for particles with interactions typical of molecular liquids indicates the presence of an athermal first-order phase transition between homogeneous and fractured inherent structures, the latter containing several large voids. Here, we study how this tensile limit manifests itself for different interparticle cohesive strengths and identify two distinct regimes. Particles with sufficiently strong cohesion display an athermal first-order phase transition, consistent with our prior characterization. Weak cohesion also displays a tensile instability. However, the landscape equation of state for this regime is independent of system size, suggesting the absence of a first-order phase transition. An analysis of the voids suggests that yielding in the energy landscape of weakly cohesive systems is associated with the emergence of a highly interconnected network of small voids. While strongly cohesive systems transition from exclusively homogeneous to exclusively fractured configurations at ρS in the thermodynamic limit, this interconnected network develops gradually, starting at ρS, even at infinite system size.

Dynamic Measurement of Poverty: Modeling and Estimation

This study presents a model of income evolution from which dynamic versions of commonly used static poverty measures are derived. The dynamic indexes are calculated both for finite- and infinite-size economic systems. Probabilistic convergence results prove that the infinite-size system can be conveniently used to approximate the finite-size system in an effective way. Secondly, poverty indexes estimation based on micro-data are discussed under different sampling schemes and it is proved that they are strongly consistent. A hypothetical example is used to show the dynamic evolution of the poverty and the estimation methodologies.

System-Size Dependence of Electrolyte Activity Coefficients in Molecular Simulations.

A systematic study of the dependence of electrolyte activity coefficients on simulation system size has been undertaken. Using implicit-solvent simulations for which calculations with low statistical uncertainty are feasible, it was found that the chemical potential for a NaCl model depends strongly on simulation system size at concentrations up to about 0.3 mol/L; system-size effects at higher concentrations are much smaller. Similar trends were confirmed in systems with an explicit solvent. System-size effects on the chemical potential, when uncorrected, can lead to systematic errors in the activity coefficient greater than 10%. The rigorous method to correct for such system-size effects is to perform multiple simulations at each concentration and extrapolate to infinite system size. Unfortunately, this becomes impractical for explicit-solvent simulations at low concentrations, because of computational limitations that lead to large statistical uncertainties in the results. Somewhat counterintuitively, we find that lower systematic errors for the Henry's law reference chemical potential are obtained by using simulations at higher concentrations, for which system-size effects are much smaller, to obtain estimates for the reference chemical potential. This is the case even though at these higher concentrations deviations from the Debye-Hückel limiting law (or its empirical extensions) are greater than those at lower concentrations.