Solvable Model Research Articles

Robust optimization for a dynamic emergency materials supply chain network under major infectious disease epidemics

ABSTRACT The COVID-19 outbreak has posed serious challenges to logistics and supply chain management. This study examines how living materials can be provided to a population under quarantine control due to the spread of infectious diseases. First, we present a modified SEIR model for predicting the number of people and materials demanded. Considering the dynamic evolution and other characteristics of infectious diseases, and the resulting demand uncertainty, we develop a robust optimization model for a multi-period dynamic emergency materials supply chain network with the objective of minimising total cost. We then transform it into a solvable robust counterpart model, propose a dynamic adjustment strategy, and cope with it using the Rolling Horizon approach to obtain dynamic location and materials distribution decisions. Finally, we conduct a case study of the Shanghai epidemic in 2022 to verify the feasibility and effectiveness of our model and method.

Towards non-invertible anomalies from generalized Ising models

We present a general approach to the bulk-boundary correspondence of noninvertible topological phases, including both topological and fracton orders. This is achieved by a novel bulk construction protocol where solvable (d+1)(d+1)-dimensional bulk models with noninvertible topology are constructed from the so-called generalized Ising (GI) models in dd dimensions. The GI models can then terminate on the boundaries of the bulk models. The construction generates abundant examples, including not only prototype ones such as \mathbb{Z}_2ℤ2 toric code models in any dimensions no less than two, and the X-cube fracton model, but also more diverse ones such as the \mathbb{Z}_2x \mathbb{Z}_2ℤ2xℤ2 topological order, the 4d \mathbb{Z}_2ℤ2 topological order with pure-loop excitations, etc. The boundary of the solvable model is potentially anomalous and corresponds to precisely only sectors of the GI model that host certain total symmetry charges and/or satisfy certain boundary conditions. We derive a concrete condition for such bulk-boundary correspondence. The condition is violated only when the bulk model is either trivial or fracton ordered. A generalized notion of Kramers-Wannier duality plays an important role in the construction. Also, utilizing the duality, we find an example where a single anomalous theory can be realized on the boundaries of two distinct bulk fracton models, a phenomenon not expected in the case of topological orders. More generally, topological orders may also be generated starting with lattice models beyond the GI models, such as those with symmetry protected topological orders, through a variant bulk construction, which we provide in an appendix.

Spherical densities and potentials in exactly solvable model molecules.

A recently initiated variant of density functional theory utilizes a set of spherically symmetric densities instead of the density. The exact functionals are unknown in the new theory akin to the standard density functional theory. In order to test approximate functionals exactly solvable models are introduced. A harmonic molecular ion, the analogue to the hydrogen molecule ion and a harmonic two-electron molecule showing analogy to the hydrogen molecule are proposed. It has been found that the wave function and the density can be given analytically. The exact spherical densities and the effective potentials of the Euler equations also have analytical form. It has been shown that the models can be easily extended to several "nuclei."

Improving Estimation of the Koopman Operator with Kolmogorov-Smirnov Indicator Functions.

It has become common to perform kinetic analysis using approximate Koopman operators that transform high-dimensional timeseries of observables into ranked dynamical modes. The key to the practical success of the approach is the identification of a set of observables that form a good basis on which to expand the slow relaxation modes. Good observables are, however, difficult to identify a priori and suboptimal choices can lead to significant underestimations of characteristic time scales. Leveraging the representation of slow dynamics in terms of Hidden Markov Models (HMM), we propose a simple and computationally efficient clustering procedure to infer surrogate observables that form a good basis for slow modes. We apply the approach to an analytically solvable model system as well as on three protein systems of different complexities. We consistently demonstrate that the inferred indicator functions can significantly improve the estimation of the leading eigenvalues of Koopman operators and correctly identify key states and transition time scales of stochastic systems, even when good observables are not known a priori.

Treating disorder in introductory solid state physics

Introductory textbooks in solid state physics present solvable models for illustrating the occurrence of allowed bands and forbidden gaps in the energy spectrum of Bloch electrons. However, the quantum mechanical description of electrons in non-periodic solids, such as amorphous materials, is beyond the scope of introductory courses because of its intrinsic complexity. The tight-binding approximation can account for such a scenario by letting the atomic levels vary at random from lattice site to site. We theoretically tackle the study of the average properties of the energy spectrum by introducing a transfer matrix method that allows us to obtain closed expressions for the so-called coherent potential. The coherent potential is energy-dependent and constant in space. It replaces the actual atomic random potential, thus generating a periodic effective medium with the same average properties as the non-periodic solid. We demonstrate that the average density of states can be calculated within this framework without relying on heavy mathematical machinery. Thus, our approach is suitable for introductory courses in solid state physics and materials science.

Asymmetric space-dependent systems: partial stabilization through the addition of noise and exact solutions for the corresponding nonlinear Langevin equations

In many instances, the dynamical richness and complexity observed in natural phenomena can be related to stochastic drives influencing their temporal evolution. For example, random noise allied to spatial asymmetries may induce stabilization of otherwise diverging trajectories in dynamical systems. However, to identify how exactly this takes place in actual processes usually is not a simple task. Here we unveil a few trends leading to dynamical stabilization and diversity of behavior by introducing Gaussian white noise to a class of exactly solvable non-linear deterministic models displaying space-dependent drifts. For the resulting nonlinear Langevin equations, the associated Fokker-Planck equations can be solved through the similarity method or the Fourier transform technique. By comparing the cases with and without noise, we discuss the changes in the systems dynamical characteristics. Simple examples of drift and diffusion coefficients are explicitly analyzed and comparisons with some other models in the literature are made. Our study illustrates the rich phenomenology originated from spatially heterogeneous dynamical systems under the influence of white noise.

Attenuating dynamics of strongly interacting fermionic superfluids in SYK solvable models

Quench dynamics of fermionic superfluids are an active topic both experimentally and theoretically. Using the BCS theory, such non-equilibrium problems can be reduced to nearly independent spin dynamics, only with a time-dependent mean-field pairing term. This results in persisting oscillations of the pairing strength in certain parameter regimes. However, experiments have observed that the oscillations decay rapidly when the interaction becomes strong, such as in the unitary Fermi gas [Phys. Rev. Res. 3, 023205 (2021)]. Theoretical analysis on this matter is still absent. In this work, we construct an SYK-like model to analyze the effect of strong interactions in a one-dimensional BCS system. We employ the large-NN approximation and a Green’s function-based technique to solve the equilibrium problem and quench dynamics. Our findings reveal that a strong SYK interaction suppresses the pairing order. Additionally, we verify that the system quickly thermalizes with SYK interactions, whether it involves intrinsic pairing order or proximity effect, resulting in a rapid decay of the oscillation strength. The decay rates exhibit different scaling laws against SYK interaction, which can be understood in terms of the Boltzmann equation. This work represents a first step towards understanding the attenuating dynamics of strongly interacting fermionic superfluids.

Simply solvable model capturing the approach to statistical self-similarity for the diffusive coarsening of bubbles, droplets, and grains.

Aqueous foams and a wide range of related systems are believed to coarsen by diffusion between neighboring domains into a statistically self-similar scaling state, after the decay of initial transients, such that dimensionless domain size and shape distributions become time independent and the average grows as a power law. Partial integrodifferential equationsfor the time evolution of the size distribution for such phase separating systems can be formulated for arbitrary initial conditions, but these are cumbersome for analyzing data on nonscaling state preparations. Here we show that essential features of the approach to the scaling state are captured by an exactly-solvable ordinary differential equationfor the evolution of the average bubble size. The key ingredient is to characterize the bubble size distribution approximately, using the average size of all bubbles and the average size of the critical bubbles, which instantaneously neither grow nor shrink. The difference between these two averages serves as a proxy for the width of the size distribution. Solution of our model shows that behavior is controlled by a signed length δ that is proportional to the width of the initial distribution relative to that in the scaling state. In particular, δ is negative if the initial preparation is too monodisperse, and is positive if it is too polydisperse. To test our approach, we compare with data for quasi-two dimensional dry foams created with three different initial amounts of polydispersity. This allows us to readily identify the critical radius from the average area of six-sided bubbles, whose growth rate is zero by the von Neumann law. The growth of the average and critical radii agree quite well with exact solution, though the most monodisperse sample crosses over to the scaling state faster than expected. A simpler approximate solution of our model performs equally well. Our approach is applicable to 3d foams, which we demonstrate by re-analyzing prior data, as well as to froths of dilute droplets and to phase separation kinetics for more general systems such as emulsions, binary mixtures, and alloys.

Friedel oscillation in non-Fermi liquid: lesson from exactly solvable Hatsugai–Kohmoto model

When non-magnetic impurity immerses in Fermi sea, a regular modulation of charge density around impurity will appear and such phenomena is called Friedel oscillation (FO). Although both Luttinger liquid and Landau Fermi liquid show such characteristic oscillation, FO in generic non-Fermi liquid (NFL) phase is still largely unknown. Here, we show that FO indeed exists in NFL state of an exactly solvable model, i.e. the Hatsugai–Kohmoto model which has been intensively explored in recent years. Combining T-matrix approximation and linear-response-theory, an interesting picture emerges, if two interaction-induced quasi-particles bands in NFL are partially occupied, FO in this situation is determined by a novel structure in momentum space, i.e. the ‘average Fermi surface’ (average over two quasi-particle Fermi surface), which highlights the inter-band particle-hole excitation. We hope our study here provides a counterintuitive example in which FO with Fermi surface coexists with NFL quasi-particle, and it may be useful to detect hidden ‘average Fermi surface’ structure in other correlated electron systems.

Exactly solvable lattice models for interacting electronic insulators in two dimensions

Exactly solvable lattice models for interacting electronic insulators in two dimensions

Unified Algorithm of Factorization Method for Derivation of Exact Solutions from Schrödinger Equation with Potentials Constructed from a Set of Functions

We extend the scope of the unified factorization method to the solution of conditionally and unconditionally exactly solvable models of quantum mechanics, proposed in a previous paper [R.R. Nigmatullin, A.A. Khamzin, D. Baleanu, Results in Physics 41 (2022) 105945]. The possibilities of applying the unified approach in the factorization method are demonstrated by calculating the energy spectrum of a potential constructed in the form of a second-order polynomial in many of the linearly independent functions. We analyze the solutions in detail when the potential is constructed from two linearly independent functions. We show that in the general case, such kinds of potentials are conditionally exactly solvable. To verify the novel approach, we consider several known potentials. We show that the shape of the energy spectrum is invariant to the number of functions from which the potential is formed and is determined by the type of differential equations that the potential-generating functions obey.

A fully solvable model of fermionic interaction in 3 + 1d

Recently, Romatschke found that the poles in O(N) scalar theories do not affect observables such as temperature and pressure. Romatschke went on to show this result holds for marginal, relevant, and irrelevant operators in 3 + 1d O(N) scalar theories. We continue in this direction by studying large-N fermi-interactions in 3 + 1d. To do so, we produce a model of marginally coupled fermi-interactions which is fully renormalizable at large-N. This theory contains poles in the running coupling, however we argue these poles do not affect any physical observables. Further, our theory contains first order phase transition which separates a stable, meta-stable, and unstable phase.

A solvable model for symmetry-breaking phase transitions

Analytically solvable models are benchmarks in studies of phase transitions and pattern-forming bifurcations. Such models are known for phase transitions of the second kind in uniform media, but not for localized states (solitons), as integrable equations which produce solitons do not admit intrinsic transitions in them. We introduce a solvable model for symmetry-breaking phase transitions of both the first and second kinds (alias sub- and supercritical bifurcations) for solitons pinned to a combined linear-nonlinear double-well potential, represented by a symmetric pair of delta-functions. Both self-focusing and defocusing signs of the nonlinearity are considered. In the former case, exact solutions are produced for symmetric and asymmetric solitons. The solutions explicitly demonstrate a switch between the symmetry-breaking transitions of the first and second kinds (i.e., sub- and supercritical bifurcations, respectively). In the self-defocusing model, the solution demonstrates the transition of the second kind which breaks antisymmetry of the first excited state.

Solvable model for discrete time crystal enforced by nonsymmorphic dynamical symmetry

Solvable model for discrete time crystal enforced by nonsymmorphic dynamical symmetry

A comment on the solutions of the generalized Faddeev–Volkov model

We consider two recent solutions of the generalized Faddeev–Volkov model, which is an exactly solvable Ising-type lattice spin model. The first solution is obtained by using the noncompact quantum dilogarithm, and the second one is constructed in a recent study via the gauge/YBE correspondence. We show that the weight functions of these models obtained by different techniques are the same upto a constant.

Accessing Position Space Wave Functions in Band Structure Calculations of Periodic Systems─A Generalized, Adapted Numerov Implementation for One-, Two-, and Three-Dimensional Quantum Problems.

In this work, a generalized, adapted Numerov implementation capable of determining band structures of periodic quantum systems is outlined. Based on the input potential, the presented approach numerically solves the Schrödinger equation in position space at each momentum space point. Thus, in addition to the band structure, the method inherently provides information about the state functions and probability densities in position space at each momentum space point considered. The generalized, adapted Numerov framework provided reliable estimates for a variety of increasingly complex test suites in one, two, and three dimensions. The accuracy of the proposed methodology was benchmarked against results obtained for the analytically solvable Kronig-Penney model. Furthermore, the presented numerical solver was applied to a model potential representing a 2D optical lattice being a challenging application relevant, for example, in the field of quantum computing.

Recent progress in understanding the microscopic origin of triboelectricity from mechanochemistry to thermoelectricity

Triboelectricity has long been discussed from ancient Greece to modern times in daily life experiences as well as in the earliest stages of planet formation and the cutting-edge triboelectric nanogenerator technology. Despite the numerous efforts from scientists and engineers, fundamental understanding of the friction-driven static electrification has remained elusive. Here, we review recent progress in understanding the microscopic origin of triboelectricity, directly associated with frictional energy dissipation at the interface, from mechanochemistry, strain-driven polarization, and tribo-tunneling to thermoelectricity. Noticeably, we note that the microscopic thermoelectric charging mechanism due to interfacial frictional heat offers a generally applicable, but exactly solvable triboelectric model in the weakly interacting regime, implying many opportunities in triboelectric based science and technology in the future.

Run-and-tumble motion in a linear ratchet potential: Analytic solution, power extraction, and first-passage properties.

We explore the properties of run-and-tumble particles moving in a piecewise-linear "ratchet" potential by deriving analytic results for the system's steady-state probability density, current, entropy production rate, extractable power, and thermodynamic efficiency. The ratchet's broken spatial symmetry rectifies the particles' self-propelled motion, resulting in a positive current that peaks at finite values of the diffusion strength, ratchet height, and particle self-propulsion speed. Similar nonmonotonic behavior is also observed for the extractable power and efficiency. We find the optimal apex position for generating maximum current varies with diffusion and that entropy production can have nonmonotonic dependence on diffusion. In particular, for vanishing diffusion, entropy production remains finite when particle self-propulsion is weaker than the ratchet force. Furthermore, power extraction with near-perfect efficiency is achievable in certain parameter regimes due to the simplifications afforded by modeling "dry" active particles. In the final part, we derive mean first-passage times and splitting probabilities for different boundary and initial conditions. This work connects the study of work extraction from active matter with exactly solvable active particle models and will therefore facilitate the design of active engines through these analytic results.

Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models

In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Θ-quasi-Hermitian) Hamiltonians H such that , the exactly solvable M-level bound-state models with arbitrary are rare. A new class of such models is proposed here, therefore. Its exact algebraic solvability (involving not only the closed formulae for wave functions but also the explicit description of all of the eligible metrics Θ) was achieved due to an extremely sparse (viz., just -parametric) but still nontrivial ‘zig-zag-matrix’ choice of the form of H.

QED Theory for Controlling the Molecule-Cavity Interaction: From Solvable Analytical Models to Realistic Ones.

The study of the interactions of chemical systems in a cavity and the ability to control the reactions inside the cavities become an evolving and hot field of research. Despite that, there is still a significant gap between experiment and theory. Herein, we aim to bridge this gap by starting with the analysis of solvable analytical models for reactions inside a cavity, then continuing to realistic models for many molecules inside a single mode and in a multimode cavity. In addition, we investigate different ways to control the strength of the molecule-cavity coupling term, which in turn allows controlling chemical reactions. Our analysis can benefit the development of ab initio computational methods to simulate molecular systems in polariton cavities; in addition, we show how to parameterize the model Hamiltonians in order to simulate a specific molecular system. Finally, we demonstrate the possibility of achieving isomerization, in case it is prohibited out of the cavity, by placing the reaction inside a cavity.