Stochastic Reservoirs Research Articles

Non-equilibrium fluctuations for SEP([formula omitted]) with open boundary

We analyze the non-equilibrium fluctuations of the partial symmetric simple exclusion process, SEP(α), which allows at most α∈N particles per site, and we put it in contact with stochastic reservoirs whose strength is regulated by a parameter θ∈R. Setting α=1, we find the results of Landim et al. (2008), Franco et al. (2019) and Gonçalves et al. (2020) and extend the known results to cover all range of θ.

Numerical simulation of geothermal energy production from hot dry rocks under the interplay between the heterogeneous fracture and stimulated reservoir volume

Efficient development of hot dry rock (HDR) relies on a thorough assessment of the extraction potential. Stimulated reservoir volumes (SRV) and hydraulic fractures are the primary sites for water circulation and heat exchange in the heat mining process, and thus are crucial to HDR development. The objective of this study is to investigate the heat extraction performance of HDR reservoir under the interplay between the heterogeneous fracture and SRV, which can enhance the thorough understanding of the heat extraction process of stochastic HDR reservoirs and contribute to optimizing the design of the HDR development. A series of 3D thermal-hydraulic-mechanical coupling models with different permeability distributions were established innovatively. Results indicated that the homogeneous reservoir with a 35-year operation time yields an outlet temperature ranging from 161 to 131.8 °C, a thermal breakthrough time of 18 years, a production thermal power between 14.5 and 10.5 MW, and a maximum heat extraction ratio of 32%. Spatial heterogeneity in the fracture permeability significantly impacted the reservoir productivity by creating preferential flow channels. Introducing heterogeneity with a variance of 1 and correlation length of 60 m resulted in a reduction of 4.4 °C in the outlet temperature, 1 year in thermal breakthrough time, 0.6 MW in production thermal power, and 1% in heat extraction ratio compared to homogeneous reservoirs. Moreover, the heterogeneous permeability in SRV further decreased heat mining capability, with a reduction of approximately 14.8 °C, 10 years, 1.9 MW, and 5.17% observed when incorporating SRV heterogeneity (variance of 2 and correlation length of 120 m) into the reservoir already subjected to heterogeneous fracture. This study also provided valuable insights into how stress conditions affect the performance of a reservoir with the heterogeneous fracture and SRV. The results pointed out that applying in situ stress lowered reservoir attenuation rate and allowed for greater heat extraction compared to the reservoir without in situ stress. This can be attributed to the less dispersed fracture permeability occurring under in-situ stress, which caused by the reduced sensitivity of permeability to stress under high-stress conditions. Results also demonstrated that the thermal breakthrough time is the most sensitive performance metric to the variations of the fracture and SRV permeability, followed by the output thermal power, heat extraction ratio, and finally the produced temperature.

Hydrodynamical Behavior for the Symmetric Simple Partial Exclusion with Open Boundary

We analyze the generalized symmetric exclusion process, which allows at most $\alpha$ particles per site, and we put it in contact with stochastic reservoirs whose strength is regulated by a parameter $\theta\in\mathbb R$. We prove that the hydrodynamic behavior is given by the heat equation and depending on the value of $\theta$, the equation is supplemented with different boundary conditions. Setting $\alpha = 1$ we find the results known in [1] for the symmetric simple exclusion process.

An efficient and accurate Mixed‐integer linear programming model for long‐term operations of large‐scale hydropower systems

Mixed-integer linear programming is widely used in hydropower reservoirs operation optimisation because of its modelling flexibility, solution stability, and global search capability. A major obstacle to applying mixed-integer linear programming to hydropower reservoirs operation optimisation, especially for long-term operations of large-scale hydropower systems, is the non-linear phenomenon while converting water energy into electricity, which is normally expressed as a non-linear bivariate function, that is, hydropower production function. To cross this technological barrier, a novel linearisation method for hydropower production function is proposed. The method uses rectangular meshing techniques to approximate hydropower production function. Then Special Ordered Sets of type 2 (SOS2)constraints are adopted for sub-rectangle selection, which are formulated based on the binary branching schemes generated by binary reflected grey code. The proposed method can linearise the hydropower production function with only logarithmic-sized binary variables of the traditional methods and maintain high accuracy. The method is then applied to optimising the operation of the Lancang River Basin hydropower system, which has 13 head-dependent reservoirs with a deterministic model and a two-stage stochastic model. The results show that the authors' method significantly outperforms conventional linearisation methods and can obtain accurate solutions within an acceptable time for deterministic and stochastic large-scale hydropower reservoirs operation optimisation problems.

Insights from quantitative and mathematical modelling on the proposed 2030 goal for gambiense human African trypanosomiasis (gHAT).

Gambiense human African trypanosomiasis (gHAT) is a parasitic, vector-borne neglected tropical disease that has historically affected populations across West and Central Africa and can result in death if untreated. Following from the success of recent intervention programmes against gHAT, the World Health Organization (WHO) has defined a 2030 goal of global elimination of transmission (EOT). The key proposed indicator to measure achievement of the goal is zero reported cases. Results of previous mathematical modelling and quantitative analyses are brought together to explore both the implications of the proposed indicator and the feasibility of achieving the WHO goal. Whilst the indicator of zero case reporting is clear and measurable, it is an imperfect proxy for EOT and could arise either before or after EOT is achieved. Lagging reporting of infection and imperfect diagnostic specificity could result in case reporting after EOT, whereas the converse could be true due to underreporting, lack of coverage, and cryptic human and animal reservoirs. At the village-scale, the WHO recommendation of continuing active screening until there are three years of zero cases yields a high probability of local EOT, but extrapolating this result to larger spatial scales is complex. Predictive modelling of gHAT has consistently found that EOT by 2030 is unlikely across key endemic regions if current medical-only strategies are not bolstered by improved coverage, reduced time to detection and/or complementary vector control. Unfortunately, projected costs for strategies expected to meet EOT are high in the short term and strategies that are cost-effective in reducing burden are unlikely to result in EOT by 2030. Future modelling work should aim to provide predictions while taking into account uncertainties in stochastic dynamics and infection reservoirs, as well as assessment of multiple spatial scales, reactive strategies, and measurable proxies of EOT.

Hydrodynamics of a particle model in contact with stochastic reservoirs

We consider an exclusion process with finite-range interactions in the microscopic interval [0, N]. The process is coupled with the simple symmetric exclusion processes in the intervals [−N, −1] and [N + 1, 2N], which simulate reservoirs. We show that an average of the empirical densities of the processes speeded up by the factor N2 converge to solutions of parabolic partial differential equations inside [−N, −1], [0, N], and [N + 1, 2N], which correspond to the macroscopic intervals (−1, 0), (0, 1), and (1, 2). Since the total number of particles is preserved by the evolution, we obtain the Neumann boundary conditions on the external boundaries u = −1, u = 2 of the reservoirs. Finally, a system of Neumann and Dirichlet boundary conditions is derived at the interior boundaries u = 0, u = 1 of the reservoirs.

Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

We consider the weakly asymmetric simple exclusion process on the discrete space {1,⋯,n-1}(n∈ℕ), in contact with stochastic reservoirs, both with density ρ∈(0,1) at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter ρ. Under time diffusive scaling tn 2 and for ρ=1 2, when the asymmetry parameter is taken of order 1/n, we prove that the density fluctuations at stationarity are macroscopically governed by the energy solution of the stochastic Burgers equation with Dirichlet boundary conditions, which is shown to be unique and to exhibit different boundary behavior than the Cole–Hopf solution.

Non-equilibrium and stationary fluctuations for the SSEP with slow boundary

We derive the non-equilibrium fluctuations of one-dimensional symmetric simple exclusion processes in contact with stochastic reservoirs which are regulated by a factor n−θ. Depending on the range of θ we obtain processes with various boundary conditions. Moreover, as a consequence of the previous result, we deduce the non-equilibrium stationary fluctuations.The main ingredient to prove these results is the derivation of precise bounds on the two-point space–time correlation function, which are a consequence of precise bounds on the transition probabilities of some underlying random walks.

Insights from quantitative and mathematical modelling on the proposed 2030 goal for gambiense human African trypanosomiasis (gHAT).

Gambiense human African trypanosomiasis (gHAT) is a parasitic, vector-borne neglected tropical disease that has historically affected populations across West and Central Africa and can result in death if untreated. Following from the success of recent intervention programmes against gHAT, the World Health Organization (WHO) has defined a 2030 goal of global elimination of transmission (EOT). The key proposed indicator to measure achievement of the goal is to have zero reported cases. Results of previous mathematical modelling and quantitative analyses are brought together to explore both the implications of the proposed indicator and the feasibility of achieving the WHO goal. Whilst the indicator of zero case reporting is clear and measurable, it is an imperfect proxy for EOT and could arise either before or after EOT is achieved. Lagging reporting of infection and imperfect diagnostic specificity could result in case reporting after EOT, whereas the converse could be true due to underreporting, lack of coverage, and cryptic human and animal reservoirs. At the village-scale, the WHO recommendation of continuing active screening until there are three years of zero cases yields a high probability of local EOT, but extrapolating this result to larger spatial scales is complex. Predictive modelling of gHAT has consistently found that EOT by 2030 is unlikely across key endemic regions if current medical-only strategies are not bolstered by improved coverage, reduced time to detection and/or complementary vector control. Unfortunately, projected costs for strategies expected to meet EOT are high in the short term and strategies that are cost-effective in reducing burden are unlikely to result in EOT by 2030. Future modelling work should aim to provide predictions while taking into account uncertainties in stochastic dynamics and infection reservoirs, as well as assessment of multiple spatial scales, reactive strategies, and measurable proxies of EOT.

Generalized self-consistent reservoir model for normal and anomalous heat transport in quantum harmonic chains.

Fictitious stochastic reservoirs incorporate scattering and dephasing mechanisms into the system in contact with these reservoirs. The reservoir-system coupling is described by the related self-energy in terms of the nonequilibrium Green's function formalism or equivalently the quantum Langevin equation formalism. In this study, we investigate thermal transport in a finite segment of an infinitely extended quantum harmonic chain with an equal self-energy at each site by using the self-consistent reservoir approach. In this setup, the entire system is lattice translation invariant so that mismatched boundaries are excluded from the model. Solving the Landauer-Büttiker equations under the self-consistent adiabatic condition, we quantitatively elucidate a thermally induced crossover of ballistic-to-diffusive transport and its scaling relation prescribed by a temperature-dependent mean free path. It is also shown that normal transport emerges in the diffusive limit for a linear self-energy, while nonlinear higher-order ones generically lead to anomalous transport. Physical implications of these observations are discussed in terms of the persistence of a massless Goldstone mode as well as the conservation of total linear momentum.

Nambu mechanics for stochastic magnetization dynamics

The Landau–Lifshitz–Gilbert (LLG) equation describes the dynamics of a damped magnetization vector that can be understood as a generalization of Larmor spin precession. The LLG equation cannot be deduced from the Hamiltonian framework, by introducing a coupling to a usual bath, but requires the introduction of additional constraints. It is shown that these constraints can be formulated elegantly and consistently in the framework of dissipative Nambu mechanics. This has many consequences for both the variational principle and for topological aspects of hidden symmetries that control conserved quantities. We particularly study how the damping terms of dissipative Nambu mechanics affect the consistent interaction of magnetic systems with stochastic reservoirs and derive a master equation for the magnetization. The proposals are supported by numerical studies using symplectic integrators that preserve the topological structure of Nambu equations. These results are compared to computations performed by direct sampling of the stochastic equations and by using closure assumptions for the moment equations, deduced from the master equation.

Heat flow in anharmonic crystals with internal and external stochastic baths: a convergent polymer expansion for a model with discrete time and long range interparticle interaction

We investigate a chain of oscillators with anharmonic on-site potentials, with long range interparticle interactions, and coupled both to external and internal stochastic thermal reservoirs of Ornstein–Uhlenbeck type. We develop an integral representation, a` la Feynman–Kac, for the correlations and the heat current. We assume the approximation of discrete times in the integral formalism (together with a simplification in a subdominant part of the harmonic interaction) and develop a suitable polymer expansion for the model. In the regime of strong anharmonicity, strong harmonic pinning, and for the interparticle interaction with integrable polynomial decay, we prove the convergence of the polymer expansion uniformly in volume (number of sites and time). We also show that the two-point correlation decays in space such as the interparticle interaction. The existence of a convergent polymer expansion is of practical interest: it establishes a rigorous support for a perturbative analysis of the heat flow problem and for the computation of the thermal conductivity in related anharmonic crystals, including those with inhomogeneous potentials and long range interparticle interactions. To show the usefulness and trustworthiness of our approach, we compute the thermal conductivity of a specific anharmonic chain, and make a comparison with related numerical results presented in the literature.

Well-test response in stochastic permeable media

Abstract In this paper, different approaches are employed to study the ramifications of various statistical and geostatistical parameters (i.e. mean and variance, vertical and horizontal correlation lengths and covariance function types) on the well-test results. These approaches include analytical, numerical, and semi-analytical methods, each of which explains the ensemble and/or volume average permeability of heterogeneous reservoirs from transient tests. In particular, in the numerical method, a few thousand pixel-based geostatistical permeability realizations, with various parameters, are generated; and the single-phase flow simulations are performed for each realization. The extracted permeability from the ensemble well-test response is then related to the underlying statistical and geostatistical parameters of permeability distribution. In this paper special attention is paid to a well-test response (i.e. a ramp effect) in a particular heterogeneous reservoir, that is manifested as a steep increase of the well-test derivative curve. A real test example is also presented to give an intuition as to how the permeable heterogeneity is reflected in a real problem. This paper provides the necessary practical insight for constructing a link between the static geological model and the dynamic test data found in highly heterogeneous stochastic reservoirs.

Thermal conductivity of anharmonic crystals with self-consistent baths: Analytical computation with discrete time

We analytically compute the thermal conductivity of anharmonic crystals with self-consistent stochastic reservoirs. We develop an integral representation for the heat current, assume the approximation of discrete times, and in a perturbative analysis that is rigorously supported by a convergent cluster expansion compute the thermal conductivity for a chain with quartic anharmonic on-site potential. In the high anharmonicity regime, the result for the dependence of the conductivity on temperature is the same as for the system without inner reservoirs. The presented formalism is quite general and is also valid for inhomogeneous systems in any space dimension.

Modelo para o despacho de usinas individualizadas no planejamento hidrotérmico de médio prazo baseado em PNL

Este trabalho apresenta um modelo de despacho hidrotérmico à usinas individualizadas, utilizando métodos de otimização baseados em programação não linear. Para tanto, considera-se funções de custo futuro geradas por um modelo de decisão estratégica baseado em programação dinâmica e sistemas equivalentes de energia. O modelo proposto considera diversos cenários históricos de afluências hidrológicas às usinas hidrelétricas, os quais são simulados para um horizonte de planejamento da operação de médio/longo prazo. Os resultados obtidos através do modelo proposto, denominado Modelo Individualizado de Usinas Hidráulicas (MIUH), são comparados com os resultados obtidos a partir da utilização do modelo SUISHI-O adotado pelo Operador Nacional do Sistema Elétrico Brasileiro (ONS).

Superdiffusive Heat Transport in a Class of Deterministic One-dimensional Many-Particle Lorentz Gases

We study heat transport in a one-dimensional chain of a finite number N of identical cells, coupled at its boundaries to stochastic particle reservoirs. At the center of each cell, tracer particles collide with fixed scatterers, exchanging momentum. In a recent paper (Collet and Eckmann in Commun. Math. Phys. 287:1015, 2009), a spatially continuous version of this model was derived in a scaling regime where the scattering probability of the tracers is γ∼1/N, corresponding to the Grad limit. A Boltzmann-like equation describing the transport of heat was obtained. In this paper, we show numerically that the Boltzmann description obtained in Collet and Eckmann (Commun. Math. Phys. 287:1015, 2009) is indeed a bona fide limit of the particle model. Furthermore, we study the heat transport of the model when the scattering probability is 1, corresponding to deterministic dynamics. Thought as a lattice model in which particles jump between different scatterers the motion is persistent, with a persistence probability determined by the mass ratio among particles and scatterers, and a waiting time probability distribution with algebraic tails. We find that the heat and particle currents scale slower than 1/N, implying that this model exhibits anomalous heat and particle transport.

Scaling and the thermal conductivity of the Frenkel-Kontorova model

We have studied the dependence of the thermal conductivity kappa on the strength of the interparticle potential lambda and the strength of the external potential beta in the Frenkel-Kontorova model. We found that the functional relation can be expressed in a scaling form, kappa proportional, variantlambda;{32}/beta;{2} . This result is first obtained by nonequilibrium molecular dynamics. It is then confirmed by two analytical methods, the self-consistent phonon theory and the self-consistent stochastic reservoirs method. The thermal conductivity kappa is therefore a decreasing functon of beta and an increasing function of lambda.

Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs

We investigate a class of anharmonic crystals in d dimensions, d≥1, coupled to both external and internal heat baths of the Ornstein-Uhlenbeck type. The external heat baths, applied at the boundaries in the 1-direction, are at specified, unequal, temperatures T l and T r . The temperatures of the internal baths are determined in a self-consistent way by the requirement that there be no net energy exchange with the system in the non-equilibrium stationary state (NESS). We prove the existence of such a stationary self-consistent profile of temperatures for a finite system and show that it minimizes the entropy production to leading order in (T l −T r ). In the NESS the heat conductivity κ is defined as the heat flux per unit area divided by the length of the system and (T l −T r ). In the limit when the temperatures of the external reservoirs go to the same temperature T, κ(T) is given by the Green-Kubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature T. This κ(T) remains bounded as the size of the system goes to infinity. We also show that the corresponding infinite system Green-Kubo formula yields a finite result. Stronger results are obtained under the assumption that the self-consistent profile remains bounded.

Analytic approach to the (an)harmonic crystal chains with self-consistent stochastic reservoirs

We consider the harmonic and anharmonic chains of oscillators with self-consistent stochastic reservoirs and derive an integral representation (à la Feynman-Kac) for the correlations, in particular, for the heat flow. For the harmonic chain, we give a new proof that its thermal conductivity is finite in the steady state. Based on this integral representation for the correlations and a perturbative analysis, the approach is quite general and can be extended to more intricate systems.

Effect of Phonon-Phonon Interactions on Localization

We study the heat current J in a classical one-dimensional disordered chain with on-site pinning and with ends connected to stochastic thermal reservoirs at different temperatures. In the absence of anharmonicity all modes are localized and there is a gap in the spectrum. Consequently J decays exponentially with system size N. Using simulations we find that even a small amount of anharmonicity leads to a J approximately 1/N dependence, implying diffusive transport of energy.