Optimal Transition Path Research Articles

A Note on the Optimal Speed of Transition: Aghion and Blanchard Revisited

Abstract This note illustrates, by reconsidering the seminal optimal speed-of-transition model of Aghion, P., and O. J. Blanchard. (1994. “On the Speed of Transition in Central Europe.” NBER Macroeconomics Annual 9: 283–319), that optimal transition paths, in general, exhibit nonlinearities and discontinuities. Aghion and Blanchard consider only an approximate solution with a constant unemployment rate over the transition process. The exact solution features an increasing unemployment rate with a discontinuity when the state sector is closed down at the optimally chosen endpoint of transition. Economic transition problems bear many similarities to scrap value problems with free terminal time, often encountered in resource economics. In relation to the transition to a green economy, the discussion in this note therefore casts doubt on the optimality of a green transition discussed in, e.g., the European Union in terms of politically specified rather than optimally designed milestones for emissions reductions, i.e., by −55 % compared to 1990 levels until 2030 and net zero until 2050.

Optimal Transition Paths for AMOC Collapse and Recovery in a Stochastic Box Model

Optimal Transition Paths for AMOC Collapse and Recovery in a Stochastic Box Model

Longitudinal dynamic envelope solution and transition path optimization based on reachability analysis for fixed-wing VTOL UAVs

Flight velocity and tilt angle are commonly employed to define the safe region for fixed-wing vertical take-off and landing (FW-VTOL) UAVs as they undergo tilting, referred to as the transition corridor. Nevertheless, the transition corridor only considers the aircraft's limitations in a quasi-stationary condition and ignores the additional constraints introduced by the aircraft's kinematic behavior. To obtain a more accurate safe state space, the concept of the dynamic envelope was proposed by previous researchers. The dynamic envelope of UAV represents the safe state space taking into account aerodynamic constraints and kinematic constraints. Solving the dynamic envelope has consistently remained a focal point of research in the field. This study introduces an innovative approach that employs reachability analysis to solve the dynamic envelope of the FW-VTOL UAV during the transition stage. Building upon this, it proposes a safety index for the transition path, optimized to determine the safest transition path. Subsequently, specific case studies verified the effectiveness of the solution process and optimization method. The designed optimal path enhances safety by 9.71% compared to the basic path, signifying its practical significance in mitigating risk during the transition stage of FW-VTOL UAVs. Furthermore, the attributes of the transition path are analyzed based on two aspects: (A) a comparison between the optimal transition path and the two-dimensional transition path with a constant angle of attack; (B) an examination of how key UAV design parameters influence the transition path's safety. Overall, the methodology and general rules proposed in this study provide a theoretical basis and technical support for improving the security of FW-VTOL UAVs in the transition stage.

Irreversibility of stochastic state transitions in Langevin systems with odd elasticity.

Active microscopic objects, such as an enzyme molecule, are modeled by the Langevin system with the odd elasticity, in which energy injection from the substrate to the enzyme is described by the antisymmetric part of the elastic matrix. By applying the Onsager-Machlup integral and large deviation theory to the Langevin system with odd elasticity, we can calculate the cumulant generating function of the irreversibility of the state transition. For an N-component system, we obtain a formal expression of the cumulant generating function and demonstrate that the oddness λ, which quantifies the antisymmetric part of the elastic matrix, leads to higher-order cumulants that do not appear in a passive elastic system. To demonstrate the effect of the oddness under the concrete parameter, we analyze the simplest two-component system and obtain the optimal transition path and cumulant generating function. The cumulants obtained from expansion of the cumulant generating function increase monotonically with the oddness. This implies that the oddness causes the uncertainty of stochastic state transitions.

The role of BRICS countries in the optimal low-carbon transition path for sustainable development

With the development of industrialization, greenhouse gas emissions caused by human activities, especially carbon dioxide emissions, continue to increase. The BRICS countries (Brazil, Russia, India, China, and South Africa) account for a significant share of global carbon emissions. This study focuses on the BRICS countries as the subject of inquiry, employing carbon emission-related data spanning the temporal range from 2000 to 2020. This study systematically assesses the interrelationships among variables using the Pearson correlation analysis and Granger causality test. Variables exhibiting pronounced correlation and causality, as discerned from the analytical outcomes, are subsequently identified and selected for further examination. Use the GCAM (Global Change Assessment Model) to simulate the Rcp-ref scenario (baseline scenario) and the Rcp2.6 scenario (assuming that the temperature will not rise by 2 °C), and compare the optimal paths of key variables in the two scenarios, and draw the following conclusions. (1) By the end of this century, the total carbon emissions of the BRICS countries under Rcp2.6 scenario will be 6139MTC less than those under Rcp-ref scenario. (2) Compared with the Rcp-ref scenario, in the Rcp2.6 scenario, the proportion of fossil fuels such as oil, natural gas, and coal in the main energy consumption significantly reduces, while the proportion of renewable energy sources such as wind, solar, and biomass increases. It is worth noting that the proportion of nuclear is significantly rising. (3) The demand for electricity increases significantly in the Rcp2.6 scenario. This study reveals the specific directions for improving the energy structure of BRICS countries to achieve the global average temperature rise of no more than 2 °C. It provides suggestions for the future policies of these countries' governments. In addition, BRICS countries should focus on observing changes in some factors, such as the increase in nuclear energy consumption and the increase in electricity demand. The reasons behind the changes in these factors are the direction of future research.

Constant advance replicas method for locating minimum energy paths and transition states.

The chain-of-states (CoS) constant advance replicas (CAR) method and its climbing image variant (CI-CAR) for locating minimum energy paths (MEPs) and transition states are reported. The CAR algorithm applies the Lagrange multiplier method for imposing holonomic constraints on a chain-of-replicas, aiming to maintain equal mass-weighted/scaled root-mean-square (RMS) distances between the adjacent replicas by removing the sliding-down displacements contributed by the potential gradients during path optimization. Two contextual regularization schemes with clear geometrical interpretations are implemented to jointly promote high convergence and numerical robustness of the CAR algorithm. We show that the constrained reaction path can be solved normally within 5 steps of Lagrange multiplier updates with remarkably high numerical precision via the CAR approach. The efficacy of the CAR methods is demonstrated by testing on multiple analytical, classical, and quantum mechanical transition paths: the Müller potential, the alanine dipeptide isomerization, the helix unwinding of the VIVITLVMLKKK 12-mer peptide, and the Baker set of reactions. We also explore the potential of applying adaptive momentum (AdaM) optimizers for locating optimal transition paths under complex conformational changes. Most importantly, we discuss extensively the differences and connections between our newly proposed CAR methods and several related methods, with focuses on the reaction path with holonomic constraints (RPCons) approach of Brokaw et al. [J. Chem. Theory Comput. 2009, 5 (8), 2050-2061] and the state-of-the-art string method (SM) of E et al. [J. Chem. Phys. 2007, 126 (16), 164103]. The CAR approach represents a latest update to the general theoretical framework of reaction path finding algorithms in the two-ended CoS regime.

Welfare Consequences of Sustainable Finance

Abstract We model the welfare consequences of mandates that restrict investors to hold firms with net-zero carbon emissions. To qualify for these mandates, value-maximizing firms have to accumulate decarbonization capital. Qualification lowers a firm’s required return by its decarbonization investments divided by Tobin’s q, that is, the greenium or the dividend yield shareholders forgo to address the global-warming externality. The welfare-maximizing mandate approximates the first-best solution, yielding welfare gains compared to laissez-faire by mitigating the weather disaster risks resulting from carbon emissions. Our model generates optimal transition paths for decarbonization that we use to evaluate proposed net-zero targets. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online

Optimal ecological transition path of a credit portfolio distribution, based on multidate Monge–Kantorovich formulation

Optimal ecological transition path of a credit portfolio distribution, based on multidate Monge–Kantorovich formulation

The business-economic energy system modelling tool energyPRO

The transition towards carbon-neutral energy systems requires the identification of solutions that are optimal at a societal level, however, market actors operate under a different logic where each individual investment option is required to show business-economic feasibility. Thus, while there are energy system analysis modelling tools that can help identify optimal transition paths for a given society in general through the minimisation of national or regional energy system costs, there is also a need for modelling tools that take a closer focus on how energy system actors and their investment considerations perceive and are affected by the economic and technical environment. The modelling tool energyPRO is of the latter type developed and evolved over the last decades to assist in the assessment of the feasibility of different energy units in the energy systems, but which can also model larger complex systems. This article presents energyPRO with a focus on its system understanding and general model characteristics and a thorough view into its two optimisation approaches based on analytical programming or mixed integer linear programming. Finally, a comparison with other models as well as a review of its characterisation and application in the academic literature is supplied.

A study of carbon peaking and carbon neutral pathways in China's power sector under a 1.5°C temperature control target.

The clean and low-carbon transition of China's power sector is of great importance to the achievement of dual carbon targets and the control of global warming. This paper first estimates the remaining carbon budget of the power sector under a 1.5°C temperature control target and on this basis constructs 1.5°C and 2°C power transition scenarios, examining key boundary conditions such as economic development and changes in the cost of power generation technologies. Second, the Genetic Algorithm-Extreme Learning Machine (GA-ELM) model is used to forecast the electricity demand for the next forty years. Finally, with the objective of minimising the total planning cost, a pathway optimisation model of the power system is constructed to explore the optimal transition path for the power system using the dual carbon target, carbon budget and electricity demand as the main constraints. The results of the study show that the carbon budget of the Chinese power sector is approximately 7.1 × 1010 t CO2 for a 1.5°C temperature control target. The electricity demand tends to saturate after 2050 and reaches 1.58 × 1013 kWh in 2060. The time of the carbon peak and carbon neutralisation in the power sector is 5years ahead of the double carbon target. By 2060, the power system will be dominated by new energy sources, with the proportion of installed non-fossil energy capacity at over 90% and the proportion of non-fossil energy generation at over 85%. Compared to that under the 2°C temperature control target, the power sector under the 1.5°C temperature control target needs to accelerate the pace of the low-carbon transition of electricity and deal with key issues such as the orderly withdrawal of coal power, the construction of a diversified clean energy system and the application of carbon capture devices. This study recommends that the process of building a zero-emissions power sector requires a good pace of the construction of new power systems at a suitable pace, increased efforts to tackle key technologies and improved relevant market mechanisms. China's carbon-neutral pathway in the power sector also has implications for other countries' clean, low-carbon transitions of their power systems.

How to advance China’s carbon emission peak?— A comparative analysis of energy transition in China and the USA

This paper investigates carbon emission peak in China based on a comparative analysis of energy transition in China and the United States (US). The LMDI model is adopted to decompose carbon emissions into several driving factors in 2000-2018 for China and the US. Gray forecasting and NAR neural network are combined to predict peak time and identify optimal transition paths. The factor decomposition indicates that energy intensity is the main inhibitory factor for increased carbon emissions, while economic growth and population size are contributors for increased carbon emissions. There are significant differences in the impact of structure effect on carbon emissions in the two countries. The industry decomposition indicates that industry development is a critical inhibitor for increased carbon emissions after 2014 in China. The growth of transport and agriculture are basically contributing to increase carbon emissions in China and the US. The forecast results illustrate that China could complete carbon emission peak by 2030 under the baseline scenario, with a peak volume of 11354.72Mt CO2. Under the industrial structure adjustment scenario, the carbon peak year may be advanced to 2028. While adjusting industrial structure and energy consumption structure at the same time, China could achieve carbon emission peak at 9918.21Mt CO2 in 2025.

Optimal transition paths of phenotypic switching in a non-Markovian self-regulation gene expression.

Gene expression is a complex biochemical process involving multiple reaction steps, creating molecular memory because the probability of waiting time between consecutive reaction steps no longer follows exponential distributions. What effect the molecular memory has on metastable states in gene expression remains not fully understood. Here, we study transition paths of switching between bistable states for a non-Markovian model of gene expression equipped with a self-regulation. Employing the large deviation theory for this model, we analyze the optimal transition paths of switching between bistable states in gene expression, interestingly finding that dynamic behaviors in gene expression along the optimal transition paths significantly depend on the molecular memory. Moreover, we discover that the molecular memory can prolong the time of switching between bistable states in gene expression along the optimal transition paths. Our results imply that the molecular memory may be an unneglectable factor to affect switching between metastable states in gene expression.

Waypoint Graph Generation with Growing Neural Gases for Pathfinding Applications in Molecular Dynamics Simulations

Path planning is a fundamental problem in the fields of autonomous robotics and molecular simulation. A large body of algorithms developed for vision-based robotic motion planning have been successfully applied to molecular pathfinding problems. We present the application of one such technique, the growing neural gas (GNG), an unsupervised machine learning procedure that estimates the optimal topological discretization of a collective variable space of arbitrary dimensionality. The GNG constructs an undirected graph representation of a probability density by utilizing a Hebb-like competitive learning rule, where individual neurons compete to represent different regions of the input space by moving and replicating in response to random samples drawn from the target density. The nodes of the resulting graph correspond to waypoint molecular states and its edges represent the potential transitions between states. Recent developments in autoencoder- and flow-based generative learning procedures have facilitated the unsupervised sampling and discovery of key collective variables in molecular systems, along which free energy differences can be calculated. Applying the GNG to such multidimensional free energy profiles allows for rapid generation of waypoint graphs within the collective variable space, which can be subsequently refined to determine optimal transition paths between selected states of interest. This proposed pipeline enables the unsupervised estimation of minimum free-energy transition paths from an initial molecular structure. We demonstrate our procedure on toy Gaussian-well systems and model molecular systems such as alanine tripeptide and show that the GNG can be a powerful tool for feature space discretization and path planning applications in molecular dynamics simulations.

First passage times and minimum actions for a stochastic minimal bistable system

Bistability is an ubiquitous phenomenon in biological systems, and always plays important roles in cell division, differentiation, cancer onset, apoptosis and so on. However, stochastic fluctuations in bistable systems are still hard to understand. To address this issue, we propose a chemical master equation model for a minimal bistable system, which underlies generally bistable systems. For this master equation model, we mainly focus on the mean first passage times (MFPTs) by respectively using Gillespie algorithm and an approximation method of the large deviation theory, and does on minimum actions along optimal transition paths from OFF to ON states by the large deviation theory. Further, we find that for this stochastic system the MFPTs have different change tendencies compared to the corresponding minimum actions. Our results of this minimal stochastic model can also well understand more general bistable systems.

Quasi-Potential Calculation and Minimum Action Method for Limit Cycle

We study the noise-induced escape from a stable limit cycle of a non-gradient dynamical system driven by a small additive noise. The fact that the optimal transition path in this case is infinitely long imposes a severe numerical challenge to resolve it in the minimum action method. We first consider the landscape of the quasi-potential near the limit cycle, which characterizes the minimal cost of the noise to drive the system far away form the limit cycle. We derive and compute the quadratic approximation of this quasi-potential near the limit cycle in the form of a positive definite solution to a matrix-valued periodic Riccati differential equation on the limit cycle. We then combine this local approximation in the neighborhood of the limit cycle with the minimum action method applied outside of the neighborhood. The neighborhood size is selected to be compatible with the path discretization error. By several numerical examples, we show that this strategy effectively improves the minimum action method to compute the spiral optimal escape path from limit cycles in various systems.

Protein Conformational Transitions from All-Atom Adaptively Biased Path Optimization.

Simulation methods are valuable for elucidating protein conformational transitions between functionally diverse states given that transition pathways are difficult to capture experimentally. Nonetheless, specific computational algorithms are required because of the high free energy barriers between these different protein conformational states. Adaptively biased path optimization (ABPO) is an unrestrained, transition-path optimization method that works in a reduced-variable space to construct an adaptive biasing potential to aid convergence. ABPO was previously applied using a coarse-grained Go̅-model to study conformational activation of Lyn, a Src family tyrosine kinase. How effectively ABPO can be applied at the higher resolution of an all-atom model to explore protein conformational transitions is not yet known. Here, we report the all-atom conformational transition paths of three protein systems constructed using the ABPO methodology. Two systems, triose phosphate isomerase and dihydrofolate reductase, undergo local flipping of a short loop that promotes ligand binding. The third system, estrogen receptor α ligand binding domain, has a helix that adopts different conformations when the protein is bound to an agonist or an antagonist. For each protein, distance-based or torsion-angle reduced variables were identified from unbiased trajectories. ABPO was computed in this reduced variable space to obtain the transition path between the two states. The all-atom ABPO is shown to successfully converge an optimal transition path for each of the three systems.

Energy barriers in three-dimensional micromagnetic models and the physics of thermoviscous magnetization

A first principle micromagnetic and statistical calculation of viscous remanent magnetization (VRM) in an ensemble of cubic magnetite pseudo single-domain particles is presented. This is achieved by developing a fast relaxation algorithm for finding optimal transition paths between micromagnetic local energy minima. It combines a nudged elastic band technique with action minimization. Initial paths are obtained by repetitive minimizations of modified energy functions. For a cubic pseudo-single domain particle, 60 different local energy minima are identified and all optimal energy barriers between them are numerically calculated for zero external field. The results allow to estimate also the energy barriers in in weak external fields which are necessary to construct the time dependent transition matrices which describe the continuous homogeneous Markov processes of VRM acquisition and decay. By spherical averaging the remanence acquisition in an isotropic PSD ensemble was calculated over all time scales. The modelled particle ensemble shows a physically meaningful overshooting during VRM acquisition. The results also explain why VRM acquisition in PSD particles can occur much faster than VRM decay and therefore can explain for findings of extremely stable VRM in some paleomagnetic studies.

Robustness and relative stability of multiple attractors in\\ a stochastic Boolean network

Multiple phenotypic states and dynamic pathways always exist in biological networks, but their robustness against fluctuations and relative stability have not been fully recognized or carefully analyzed yet. Here we try to address these issues with a widely used stochastic Boolean network model. The stability of phenotypic states and dynamical pathways corresponds to the robustness and relative stability of attractors in such a stochastic Boolean network, which can be analyzed using the theory of exponentially perturbed Markov chains. It is already known that the logarithm of the expected exit time escaping from an attractor is proportional to the ``minimum activation energy barrier in the exponentially perturbed Markov chains. We first prove that within an exponentially perturbed Markov chain, all pathways between attractors with the minimum possible ``activation energy barrier have the same probability weights in the zero-noise limit, and therefore the relative stability just depends on the number of optimal paths. Another important implication of this theory is that, once the non-neglectable fluctuations are rather low, a phase transition phenomenon emerges: In one parameter region the probability weights of the coexisted phenotypic states and pathways are comparable with each other; whereas in some other parameter regions, the probability weight of certain phenotypic state or pathway can even dominate and become globally attractive. Finally, we apply the theory to an artificial model and the Siah-1/beta-catenin/p14/19 ARF loop of protein p53 dynamics. Our theory can also determine how the transition time and the number of optimal transition paths between the phenotypic states and parallel pathways depend on all the parameters, and help to identify those possibly more crucial nodes and the interactions between them in a biological network.

Transitions in genetic toggle switches driven by dynamic disorder in rate coefficients.

In biochemical systems, intrinsic noise may drive the system switch from one stable state to another. We investigate how kinetic switching between stable states in a bistable network is influenced by dynamic disorder, i.e., fluctuations in the rate coefficients. Using the geometric minimum action method, we first investigate the optimal transition paths and the corresponding minimum actions based on a genetic toggle switch model in which reaction coefficients draw from a discrete probability distribution. For the continuous probability distribution of the rate coefficient, we then consider two models of dynamic disorder in which reaction coefficients undergo different stochastic processes with the same stationary distribution. In one, the kinetic parameters follow a discrete Markov process and in the other they follow continuous Langevin dynamics. We find that regulation of the parameters modulating the dynamic disorder, as has been demonstrated to occur through allosteric control in bistable networks in the immune system, can be crucial in shaping the statistics of optimal transition paths, transition probabilities, and the stationary probability distribution of the network.

Optimal control strategy of working condition transition for copper flash smelting process

Because there are large state fluctuation of working conditions, excessive energy consumption caused by manual operation in the dynamic transient procedure for copper flash smelting process. Working conditions of copper flash smelting process must be adjusted to the complex and variable copper concentrate feeding to satisfy the smelting performance. Optimal control strategy based adjustment cost for copper flash smelting working condition transition is proposed, which can achieve the expected working condition by following the optimal working condition transition path. The Cauchy–Schwarz inequality-based two-level matching method is developed to set the expected working condition. Then with minimizing adjustment cost and restricting operation domain to optimize smelting performance indicators, the optimal control problem of working condition transition is converted into a multi-constraints optimization problem with two fixed ends. A Legendre pseudospectral-based optimization method is also presented to obtain the path of optimal working condition transition. The simulation results of actual production data collected are given to verify the effectiveness and feasibility of the proposed strategy.