Equilibrium Behavior Research Articles

Understanding Far from Equilibrium Behavior in Economic Markets

Management theories rely on the Walrasian assumption of equilibrium as a persistently accessible market state. Any environmental disruption that induces deviation from equilibrium is deemed temporary; most organizations promptly return to the equilibrium. But what happens when the market is pushed too far away from equilibrium? In this study, I advance the discussion of post-Walrasian economics to discuss how markets operate far from equilibrium. I discuss how disruptions caused by extreme threats such as climate change and pandemic introduce instabilities that undermine market’s capacity for self-regulation, limit information dissemination, make it difficult to account for externalities, and misdirect tâtonnement. The resulting far-from-equilibrium dynamics expose organizations to multiple equilibria, localized logics of exchange, and non-price coordination mechanisms. They require a choice between replication, exploitation, and integration of which only the latter is mutually beneficial for organizations and the society. The study teases apart the new normal by examining the shifts in markets and how organizations can use these shifts for a positive impact on the society.

Sistematic Review: Mathematics Model Epidemiology of Dengue Fever

Dengue Hemorrhagic Fever (DHF) is a disease caused by an arbovirus that enters the human body through the Aedes aegypti mosquito. Dengue Hemorrhagic Fever is characterized by symptoms, headache; reddish skin that looks like measles; and muscle and joint pain. The spread of dengue hemorrhagic fever (DHF) globally has tended to be higher in the last 50 years, thus giving rise to ideas for systematic prevention. Many factors cause this dengue fever, climate factors, weather, residential density, and other factors. The Epidemiological Model of DHF can provide a pattern of prevention against dengue outbreaks so that this problem can be modeled mathematically and through the stability of the equilibrium point, the dynamics or behavior of the model can be determined. The spread of DHF can be suppressed by providing control in the form of vaccines, eradication with insecticides and treatment. This review article aims to provide a systematic description of the causative factors, a mathematical model of DHF with an epidemiological approach. To get the right model related to the SIR epidemiological mathematical model and its modifications which can be an alternative solution to describe and analyze the model mathematically. The methodology used is to systematically search for journals and articles from August 2021 to April 2022. This is done by accessing electronic journal portals such as: Elsevier, Springer, ResearchGate, google scholar, Scient direct and so on. Other references also use national journals and information on the Ministry of Health's website. The results of a review of factors that cause dengue fever are physical environmental factors that have a significant influence on rainfall and air temperature. Mathematical models with Optimal Control can be used as an alternative to mathematical analysis which is expected to help solutions to reduce the spread of dengue fever. The recommendation given is to investigate the disease-causing factors from the chemical aspect related to the chemical effect in water that influences the growth of mosquito larvae in water which can indirectly affect DHF disease and involves these parameters in mathematical models and optimal control.

Adsorption separation of guanosine 5′-Monophosphate and cytidine 5′-Monophosphate by mixed-mode Resin HD-1: Experimental study and mathematical modeling

The preparative separation of guanosine 5′-monophosphate (GMP) and cytidine 5′-monophosphate (CMP) on mixed-mode resin HD-1 was experimentally and theoretically investigated. The adsorption mechanisms of the two nucleotides were elucidated by analyzing adsorption equilibria and kinetics at different pH values. The adsorption dynamics of GMP and CMP in a fixed bed packed with resin HD-1 were studied. All nucleotide monovalent cations, zwitterions, and monovalent anions were adsorbed by the resin. Further, a general adsorption isotherm model was developed by considering the adsorption of different nucleotide species and the dissociation equilibrium behaviors of resin ligands. The model fit the adsorption isotherm data of GMP and CMP well. A modified liquid-film linear driving force model with the combined physical adsorption of nucleotides in different dissociation states and ion exchange of Na+ was established. The dissociation equilibria of resin ligands and nucleotides were considered. The model satisfactorily predicted the adsorption kinetic data at different pH values. The values of the efficient diffusion coefficients for GMP and CMP were not significantly influenced by the solution pH. A modified transport-diffusion model with pH gradient elution was proposed. The model accurately predicted the elution chromatographic peaks of GMP and CMP, as well as the pH at the outlet of the fixed bed packed with resin HD-1.

A Bayesian nonatomic game and its applicability to finite-player situations

We investigate a nonatomic game (NG) involving a random state of the world. Every player receives only a signal of the state’s realization. Not knowing the true state, the player would strive for a higher average payoff conditioned on her received signal. We demonstrate that equilibria exist for the game under reasonable conditions. Not only are they in existence, but these equilibrium points are also useful. When the state space is finite, such an equilibrium would help induce asymptotically equilibrium behaviors as n tends to +∞ in n-player Bayesian games whose player profiles are randomly generated from the original NG’s player distribution. More important, pure ϵ-equilibria would likely emerge even though the NG equilibrium might well be mixed to start with. Pure versions of the latter would exist when the NG is anonymous.

Equilibrium Behavior of a Tethered Autogyro: Application in Extended Flight and Power Generation

Abstract In this article, we study the characteristics of steady autorotation of a tethered autogyro. The phenomenon of autorotation refers to the natural spinning of a rotor in a wind field. We explore the viability of tethered autogyros as unmanned aerial vehicles (UAVs) for long-duration and energy efficient hovering applications, such as in monitoring or surveillance. The tether provides mooring and can be used to power the rotor and to transmit wind power to the ground when suitable. This is a novel application of autorotation. It requires a generalized formulation and modeling of autorotation, beyond what is reported in the literature. We adopt a model-based approach where the blade element momentum (BEM) method and catenary mechanics are used to model the aerodynamics and the tether, respectively. The resulting model is highly nonlinear and numerical methods are proposed to solve for the equilibria. The model is validated against existing simulation and experimental results in the literature. It is extended to incorporate new features that are pertinent to our application, such as low rotor speeds, regenerative torque for power generation, combining catenary mechanics with aerodynamics, and varying atmospheric conditions with altitude. We characterize the autorotational equilibria over a range of operating conditions involving multiple independent variables. The analysis reveals an optimal operating range of the tip speed ratio of the autogyro under equilibrium. It also indicates the possibility of power generation in large autogyros stationed at high altitudes.

The impacts of investor network and herd behavior on market stability: Social learning, network structure, and heterogeneity

The networked structure of investor relationship has been documented by many researches. This paper proposes a dynamic model based on investor network, and investigate the impacts of investor interactions on market stability. Two learning patterns are taken into account. The herd behavior is characterized with the imitation and diffusion of trading strategies in investor network. The behavioral equilibria and their stability conditions are studied with mathematical analysis. As the findings show, herd behavior can affect trader structure as well as market stability. Although it is a potential source of bubbles, there exist some cases where herd behavior is beneficial to market stability. The impacts of network structure are closely related with the imitation mechanism. If investors mainly care about the absolute number of neighbors who adopt different strategies, the network effect could weaken market stability. To guarantee the stability conditions, the variance of degree distribution should fit with other parameters. A high variance of degree distribution might lead to market instability.

A numerical approach to determine the optimal condition of the gas anti-solvent supercritical process for nanoparticles production

Supercritical gas antisolvent (GAS) process is an efficient method for nanoparticles production, in which accurate selection of operational condition is essential. Thermodynamic models can be applied for evaluation the phase equilibrium behavior and determination the required precipitation pressure of GAS process. In this research, thermodynamic behavior of (CO2—dimethyl sulfoxide (DMSO)) binary system and both of (CO2–DMSO-anthraquinone Violet 3RN (AV3RN)) and (CO2–DMSO-solvent Yellow 33 (SY33)) ternary systems in the GAS process were studied at different temperatures (308, 318, 328 and 338) K and pressures (1.0–14.0) MPa, using Peng–Robinson equation of state (PR-EoS). The minimum precipitation pressure of AV3RN and SY33 at 308, 318, 328 and 338 K were 7.80, 8.57, 9.78 and 11 MPa and 8, 8.63, 9.5 and 10.77 MPa, respectively. Also, the mole fraction of substances in liquid phase of ternary systems were determined by PR-EoS, at 328 K versus pressure. The accuracy of the obtained results were investigated using the experimental data reported in the literatures.

Effects of consumer loyalty and product web compatibility on cooperative advertising and pricing policies in a dual-channel supply chain

A common practice for brand manufacturers is to operate dual distribution channels in which they offer an online channel for direct sales to end consumers and an independently-managed retail channel for sales in physical stores. This structure enables the manufacturers to reach multiple segments of consumers with different online and offline shopping preferences, but it may create channel conflicts due to the manufacturers’ competitive position in the end market. Cooperative advertising programs can be implemented in response to the emerging competitive dynamics between the manufacturers and the retailers. We investigate the impact of the consumers’ sales channel preference (i.e., “consumer loyalty”) and the product compatibility with online shopping (i.e., “product web compatibility or web fit”) on the cooperative advertising and pricing decisions of a manufacturer and a retailer in a dual-channel supply chain. We use game-theoretical models and characterize the firms’ equilibrium behaviors under different power structures in the channel. Our results indicate that the level of the retailer’s advertising investment and the manufacturer’s reimbursement in the cooperative advertising program depend critically on consumer loyalty, product web compatibility, and the power distribution among the channel members. For example, when the channel power is symmetrically distributed or held asymmetrically by the retailer, the retailer’s local advertising level increases as the product web compatibility decreases or the proportion of store-loyal consumers increases; whereas this trend is reversed when the manufacturer is the channel leader. We examine how the introduction of the direct channel affects the profits, and we generate additional managerial insights from numerical experiments.

Experimental and modeling analyses of COD removal from industrial wastewater using the TiO2–chitosan nanocomposites

In the present study, titanium oxide (TiO2) nanoparticles, chitosan, and several nanocomposites containing different mass dosages of TiO2 and chitosan have been applied as the adsorbent for COD removal from the industrial wastewater (Bouali Sina Petrochemical Company, Iran). The FESEM, XRD, and FTIR tests have been employed to characterize TiO2 nanoparticles, chitosan, and fabricated nanocomposites. Then, the effect of adsorption parameters, including TiO2–chitosan mass ratio (1:1, 1:2, and 2:1), adsorbent content (0.25–2.5 g), temperature (20–50 °C), pH (3–11), solution volume (100–500 mL), and contact time (30–180 min) on the COD reduction has also been monitored both experimentally and numerically. The Box–Behnken design of the experiment approves that TiO2–chitosan (1:1), adsorbent content of 2.5 g, temperature = 20 °C, pH 7.4, solution volume of 100 mL, and contact time = 180 min are the condition that maximizes the COD removal (i.e., 94.5%). Moreover, the Redlich–Peterson and Pseudo-second order models are the best isotherm and kinetic scenarios to describe COD removal’s transient and equilibrium behaviors. The maximum monolayer COD adsorption capacity of the TiO2–chitosan nanocomposite is 89.5 mg g−1. The results revealed that the industrial wastewater COD is better to remove using the TiO2–chitosan (1:1) at temperature = 20 °C.

Kinetics and Adsorption Equilibrium Study of Free Fatty Acid (FFA) from Crude Palm Oil (CPO) on Anionic Resin

Crude Palm Oil (CPO) is obtained from palm fruit extraction. Free Fatty Acid (FFA) level is one of the most important parameters to determine the CPO quality. The standard FFA level in the industry is less than 5%. In the palm oil processing industry, FFA levels in CPO have been minimized through vacuum distillation processes. However, due to the emulsified water in oil and long departure waiting time in port, FFA levels in CPO raise to above 5% resulting in lower quality and selling price. This study has developed a novel approach to mitigate this problem by using anionic resin as an adsorbent to decrease the FFA level. We investigated the kinetics and equilibrium behavior of FFA adsorption in CPO on the anionic resin. The adsorption was conducted in several resin dose (17.5%; 25%; 33.33%; 43%). The kinetic study shows that the FFA adsorption kinetics on anionic resin follows the Pseudo-second-order rate model with the value of k2 vary between 2.1034×10-4 – 1.7375×10-3 and the qe value is in the range of 103.627 - 163.880 (mg/g). The equilibrium study shows that the Freundlich isotherm model is the fittest. The Kf values are obtained between 4.16-15.02, and the n values are in the range of 1.62-2.05. The thermodynamic observation was also involved in this study. The value of and are 28.59 kJ/mol and 91.25 J/mol/K respectively. The results show that the reaction is endothermic, spontaneous, and feasible.

Simulating domain architecture evolution.

MotivationSimulation is an essential technique for generating biomolecular data with a ‘known’ history for use in validating phylogenetic inference and other evolutionary methods. On longer time scales, simulation supports investigations of equilibrium behavior and provides a formal framework for testing competing evolutionary hypotheses. Twenty years of molecular evolution research have produced a rich repertoire of simulation methods. However, current models do not capture the stringent constraints acting on the domain insertions, duplications, and deletions by which multidomain architectures evolve. Although these processes have the potential to generate any combination of domains, only a tiny fraction of possible domain combinations are observed in nature. Modeling these stringent constraints on domain order and co-occurrence is a fundamental challenge in domain architecture simulation that does not arise with sequence and gene family simulation.ResultsHere, we introduce a stochastic model of domain architecture evolution to simulate evolutionary trajectories that reflect the constraints on domain order and co-occurrence observed in nature. This framework is implemented in a novel domain architecture simulator, DomArchov, using the Metropolis–Hastings algorithm with data-driven transition probabilities. The use of a data-driven event module enables quick and easy redeployment of the simulator for use in different taxonomic and protein function contexts. Using empirical evaluation with metazoan datasets, we demonstrate that domain architectures simulated by DomArchov recapitulate properties of genuine domain architectures that reflect the constraints on domain order and adjacency seen in nature. This work expands the realm of evolutionary processes that are amenable to simulation.Availability and implementationDomArchov is written in Python 3 and is available at http://www.cs.cmu.edu/~durand/DomArchov. The data underlying this article are available via the same link.Supplementary information Supplementary data are available at Bioinformatics online.

Energy Dissipation and Stress Equilibrium Behavior of Granite under Dynamic Impact

Stress equilibrium time is an important index to judge the homogeneity of rocks. In order to study the relationship between stress equilibrium time and crushing energy consumption before rock destruction, Hopkinson tests were conducted on granite specimens with different length-to-diameter ratios. In this paper, by studying the size and strain rate effects of rocks, five different sizes of granite specimens with different aspect ratios were prepared and Hopkinson impact tests were conducted under four strain rate conditions. Data analysis and processing using the three-wave method to investigate the stress uniformity of granite specimens under impact loading. The energy balance factor was introduced to compare and analyze the stress equilibrium time of five kinds of long-diameter granite specimens, and it was found that the stress equilibrium time of rocks with the same length–diameter ratio decreased with the increase of loading strain rate, while the granite specimens with length–diameter ratio of 0.8 showed a better stress equilibrium time. In order to better find the aspect ratio and loading strain rate that can crush better and maintain a long equilibrium time, the energy consumption of rock crushing is further analyzed. The energy dissipation of granite specimens with loading strain rate of 156.8 s−1 and 253.2 s−1 was found to be more concentrated, and the energy dissipation rate was stable at about 48%. Subsequently, the relationship between stress equilibrium time and energy dissipation was established, and it was proved that the fastest growing time period of the energy dissipation curve was approximately equal to the rock stress equilibrium time, while the length-to-diameter ratio of the granite specimen that could better maintain the stress uniformity before rock crushing was 0.8 and the loading strain rate was 156.8 s−1.

Effect of solvent and temperature on the phase equilibrium behavior of hydroxylamine sulfate in water-ethanol mixed solvents: Solubility and ternary liquid-liquid equilibrium diagram with liquid-liquid equilibrium

The thermodynamic properties of hydroxylamine sulfate (HAS) in binary mixed solvents of water-ethanol were investigated in this work. Experimental results indicated that a temperature above 323.15 K will bring about the occurrence of liquid-liquid phase separation in the water-ethanol-HAS system. Solubility of HAS in water and ethanol-water binary mixed solvents was experimentally determined from 283.15 to 323.15 K under atmospheric pressure (101.3 kPa) by employing a gravimetric method. As the mass fraction of water in the mixed solvent increase, there is an increase in the solubility of HAS. In addition, the solubility of HAS is positively dependent on the temperature. Furthermore, the modified van’t Hoff model, Apelblat model, van’t Hoff − Jouyban − Acree model, and Apelblat − Jouyban − Acree model were used to correlate the experimental solubility of HAS, and the Apelblat model showed the best agreement. Besides, the van't Hoff equation was utilized to evaluate the thermodynamic parameters of HAS in binary solvents, calculation results confirm that these dissolving processes are endothermic and entropy-driving. Ternary phase diagrams with liquid-liquid equilibrium at 324.15, 328.15 and 333.15 K were constructed, and the boundaries of each phase area of the ternary phase diagram were verified at 333.15 K. As for the water-ethanol-HAS system, the temperature has an influence on the phase zone of the ternary phase diagram by affecting solid-liquid phase equilibrium. However, the temperature has no effect on the liquid-liquid phase equilibrium. The nucleation process of oiling-out crystallization of HAS was determined to follow a two-step mechanism: the liquid-liquid phase separation is generated at first, and then nucleation occurred and crystals subsequently grow in the aqueous phase.

Adsorption of crystal violet and methylene blue dyes using a cellulose-based adsorbent from sugercane bagasse: characterization, kinetic and isotherm studies

Biodegradable cellulose based on sugarcane bagasse is produced using alkaline treatment followed by bleaching method for dye removal. FTIR, Raman, SEM, EDS, XRD, and N2 adsorption–desorption isotherm analysis were utilized to characterize the surface morphology and functional groups of the sugarcane bagasse (SCB) and treated sugarcane bagasse (TSCB). The TSCB was investigated to remove of Crystal violet (CV) and Methylene blue (MB). Different factors influence the adsorption processes such as contact time, adsorbent dose, dye concentration, temperature, pH, and NaCl dose were studied. The Langmuir model was investigated to study the adsorption equilibrium behavior of CV and MB dyes by TSCB. A maximum adsorption capacities (qmax) of 107.5 and 112.9 mgg−1 was calculated for CV and MB-dyes, respectively. Also, the adsorption processes are physiosorption with average energy values of E = 1.151 and 1.101 KJmol-1 for CV and MB-dyes, respectively. The kinetic data of the investigated processes were controlled by the pseudo-second order (PSO) model with qe,cal (mgg−1) values of 59.3 and 58.9 mgg−1 for CV and MB-dyes, respectively. Suggesting that chemical mechanisms may influence the removal of CV and MB dyes. Overall, TSCB was found to be an effective, environmentally friendly, and low-cost material to remove of cationic dyes in this research.

Study on the Isobaric Vapor–Liquid Equilibrium Behavior of the Methanol + Methyl Ethyl Ketone System Using Ionic Liquids as Extractants and Model Correlation

Study on the Isobaric Vapor–Liquid Equilibrium Behavior of the Methanol + Methyl Ethyl Ketone System Using Ionic Liquids as Extractants and Model Correlation

Measurement and thermodynamic modelling of Liquid-liquid equilibrium for extraction of acrylonitrile from water with n-Alkyl acetates

In order to recover the acrylonitrile from wastewater solution, liquid–liquid equilibrium behaviors in the systems {water + acrylonitrile + n-alkyl acetates} were explored at 303.2 K and 101.3 kPa., respectively. The partition coefficients and separation factors in each system were used to compare and discuss the effects of extraction. The experimental data were correlated with the NRTL and UNIQUAC models, and the binary interaction parameters required for each model were obtained. The mixing surface analysis of Gibbs energy topology was used to test the parameters in models. The UNIFAC model was used to predict LLE data compared with experimental data. The deviation between the calculated and experimental values of the three thermodynamic models was evaluated by the root mean square deviation, and the values were lower than 0.0082, 0.0088, and 0.0143, respectively. The σ-profiles were used to discuss the interactions of substances in this work.

Modeling colloidal interactions that predict equilibrium and non-equilibrium states.

Modulating the interaction potential between colloids suspended in a fluid can trigger equilibrium phase transitions as well as the formation of non-equilibrium "arrested states," such as gels and glasses. Faithful representation of such interactions is essential for using simulation to interrogate the microscopic details of non-equilibrium behavior and for extrapolating observations to new regions of phase space that are difficult to explore in experiments. Although the extended law of corresponding states predicts equilibrium phases for systems with short-ranged interactions, it proves inadequate for equilibrium predictions of systems with longer-ranged interactions and for predicting non-equilibrium phenomena in systems with either short- or long-ranged interactions. These shortcomings highlight the need for new approaches to represent and disambiguate interaction potentials that replicate both equilibrium and non-equilibrium phase behavior. In this work, we use experiments and simulations to study a system with long-ranged thermoresponsive colloidal interactions and explore whether a resolution to this challenge can be found in regions of the phase diagram where temporal effects influence material state. We demonstrate that the conditions for non-equilibrium arrest by colloidal gelation are sensitive to both the shape of the interaction potential and the thermal quench rate. We exploit this sensitivity to propose a kinetics-based algorithm to extract distinct arrest conditions for candidate potentials that accurately selects between potentials that differ in shape but share the same predicted equilibrium structure. The algorithm selects the candidate that best matches the non-equilibrium behavior between simulation and experiments. Because non-equilibrium behavior in simulation is encoded entirely by the interparticle potential, the results are agnostic to the particular mechanism(s) by which arrest occurs, and so we expect our method to apply to a range of arrested states, including gels and glasses. Beyond its utility in constructing models, the method reveals that each potential has a quantitatively distinct arrest line, providing insight into how the shape of longer-ranged potentials influences the conditions for colloidal gelation.

Equilibrium and Dynamic Surface Tension Behavior in Colloidal Unimolecular Polymers (CUP).

Studies of the interfacial behavior of pure aqueous nanoparticles have been limited due tothe difficulty of making contaminant-free nanoparticles while also providing narrow size distribution. Colloidal unimolecular polymers (CUPs) are a new type of single-chain nanoparticle with a particle size ranging from 3 to 9 nm, which can be produced free of surfactants and volatile organic contents (VOCs). CUP particles of different sizes and surface charges were made. The surface tension behavior of these CUP particles in water was studied using a maximum bubble pressure tensiometer. The equilibrium surface tension decreased with increasing concentration and the number of charges present on the surface of the CUP particles influences the magnitude of the interfacial behavior. The effect of electrostatic repulsion between the particles on the surface tension was related. At higher concentrations, surface charge condensation started to dominate the surface tension behavior. The dynamic surface tension of CUP particles shows the influence of the diffusion of the particles to the interface on the relaxation time. The relaxation time of the CUP polymer was 0.401 s, which is closer to the diffusion-based relaxation time of 0.133s for SDS (sodium dodecyl sulfate).

Equilibrium Behavior of CO2 Adsorption from Biogas onto Zeolites

Biogas is one of the types of renewable energy that has a major gas content of CH4 and CO2. One of the methods for upgrading biogas is adsorption. The principle of adsorption’s work is to deliver biogas into a column with containing adsorbent, so that a CO2 gas is bound to zeolite, causing the CH4 to increase in the result product. The biogas used in research came from a cow feces that was dumped into a 10bar pressure vessel. Adsorbent used zeolite 13X that was activated. The biogas tube has a height 0.3 m and diameter 3 cm. It was made of stainless steel and there was a cartridge heater inside. Biogas was channeled into adsorbent column with pressure variations 0.8, 1.2, 1.6 and 2 bars and temperature variations 30, 50 and 100°C. The flow rate of biogas used 4 L/minute with a 4-minute data retrieval. Data retrieval was continuous, so that the composition of CO2 gas from beginning to end recorded on the Gas analyzer. The research aims to identify the balance of adsorption with variations in pressure and temperature by Langmuir equation which has been developed. The results show if the higher the pressure, then CO2 that was absorbed is inversely proportional to the temperature where the higher the temperature, the less CO2 was adsorbed. This research succeeded in getting the Langmuir constant, maximum amount adsorbed of CO2, and heat of adsorption.

Generalized Langevin equationwith a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme.

We introduce a hybrid projection scheme that combines linear Mori projection and conditional Zwanzig projection techniques and use it to derive a generalized Langevin equation(GLE) for a general interacting many-body system. The resulting GLE includes (i) explicitly the potential of mean force (PMF) that describes the equilibrium distribution of the system in the chosen space of reaction coordinates, (ii) a random force term that explicitly depends on the initial state of the system, and (iii) a memory friction contribution that splits into two parts: a part that is linear in the past reaction-coordinate velocity and a part that is in general nonlinear in the past reaction coordinates but does not depend on velocities. Our hybrid scheme thus combines all desirable properties of the Zwanzig and Mori projection schemes. The nonlinear memory friction contribution is shown to be related to correlations between the reaction-coordinate velocity and the random force. We present a numerical method to compute all parameters of our GLE, in particular the nonlinear memory friction function and the random force distribution, from a trajectory in reaction coordinate space. We apply our method on the dihedral-angle dynamics of a butane molecule in water obtained from atomistic molecular dynamics simulations. For this example, we demonstrate that nonlinear memory friction is present and that the random force exhibits significant non-Gaussian corrections. We also present the derivation of the GLE for multidimensional reaction coordinates that are general functions of all positions in the phase-space of the underlying many-body system; this corresponds to a systematic coarse-graining procedure that preserves not only the correct equilibrium behavior but also the correct dynamics of the coarse-grained system.