The Steady State Fluctuation Relation for the Dissipation Function

We give a proof of transient fluctuation relations for the entropy production (dissipation function) in nonequilibrium systems, which is valid for most time reversible dynamics. We then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax, producing a unique nonequilibrium steady state. Although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown, if such a steady state exists, the generation of the steady state fluctuation relation from the transient relation is shown to be very general. It is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation, which is needed also for, e.g., the existence of transport coefficients. Because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities. The steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor, whose convergence is problematic, for systems close to equilibrium. We examine some model dynamics that have been considered previously, and show how they are described in the context of this work.     

Nonequilibrium work fluctuations in a driven Ising model

Monte Carlo simulations of a sheared Ising model are used to study nonequilibrium fluctuations of mechanical work. The validity of the transient (starting from equilibrium) and the steady state fluctuation relations is verified. A fluctuation relation has been also shown to hold for the mechanical work done on the system, during the transition between two nonequilibrium steady states corresponding to different drivings.     

Response of firm agent network to exogenous shock

This paper describes an agent-based model of interacting firms, in which interacting firm agents rationally invest capital and labor in order to maximize payoff. Both transactions and production are taken into account in this model. First, the performance of individual firms on a real transaction network was simulated. The simulation quantitatively reproduced the cumulative probability distribution of revenue, material cost, capital, and labor. Then, the response of the firms to a given exogenous shock, defined as a sudden change of gross domestic product, is discussed. The longer tail in cumulative probability and skewed distribution of growth rate are observed for a high growth scenario.     

On the Fluctuation Relation for Nosé-Hoover Boundary Thermostated Systems

We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nosé-Hoover thermostats applied at the boundaries of the system. The transient fluctuation relation, which holds only for a precise choice of the initial ensemble, is verified at all times, as expected. Times longer than the mesoscopic scale, needed for local equilibrium to be settled, are required if a different initial ensemble is considered. This shows how the transient fluctuation relation asymptotically leads to the steady state relation when, as explicitly checked in our systems, the condition found in (D.J. Searles, et al., J. Stat. Phys. 128:1337, 2007), for the validity of the steady state fluctuation relation, is verified. For the steady state fluctuations of the phase space contraction rate Λ and of the dissipation function Ω, a similar relaxation regime at shorter averaging times is found. The quantity Ω satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale; the quantity Λ appears to begin a monotonic convergence after such times. This is consistent with the fact that Ω and Λ differ by a total time derivative, and that the tails of the probability distribution function of Λ are Gaussian.     

Large-scale structure of a nation-wide production network

Production in an economy is a set of firms’ activities as suppliers and customers; a firm buys goods from other firms, puts value added and sells products to others in a giant network of production. Empirical study is lacking despite the fact that the structure of the production network is important to understand and make models for many aspects of dynamics in economy. We study a nation-wide production network comprising a million firms and millions of supplier-customer links by using recent statistical methods developed in physics. We show in the empirical analysis scale-free degree distribution, disassortativity, correlation of degree to firm-size, and community structure having sectoral and regional modules. Since suppliers usually provide credit to their customers, who supply it to theirs in turn, each link is actually a creditor-debtor relationship. We also study chains of failures or bankruptcies that take place along those links in the network, and corresponding avalanche-size distribution.     

Effect of coagulation of nodes in an evolving complex network

We propose a new type of stochastic network evolution model based on annihilation, creation, and coagulation of nodes, together with the preferential attachment rule. The system reaches a unique quasistatistically steady state in which the distribution of links follows a power law, lifetime of nodes follows an exponential distribution, and the mean number of links grows exponentially with time. The master equation of the model is solved analytically by applying Smoluchowski's coagulation equation for aerosols. The results indicate that coagulation of nodes in complex networks and mean field analysis of aerosols are similar in both the growth dynamics with irreversible processes and in the steady state statistics. We confirm that the basic properties of the model are consistent with the empirical results of a business transaction network having about 1×10(6) firms.     

Production, depreciation and the size distribution of firms

Many empirical researches indicate that firm size distributions in different industries or countries exhibit some similar characters. Among them the fact that many firm size distributions obey power-law especially for the upper end has been mostly discussed. Here we present an agent-based model to describe the evolution of manufacturing firms. Some basic economic behaviors are taken into account, which are production with decreasing marginal returns, preferential allocation of investments, and stochastic depreciation. The model gives a steady size distribution of firms which obey power-law. The effect of parameters on the power exponent is analyzed. The theoretical results are given based on both the Fokker-Planck equation and the Kesten process. They are well consistent with the numerical results.     

Effects of the turnover rate on the size distribution of firms: An application of the kinetic exchange models

We address the issue of the distribution of firm size. To this end we propose a model of firms in a closed, conserved economy populated with zero-intelligence agents who continuously move from one firm to another. We then analyze the size distribution and related statistics obtained from the model. There are three well known statistical features obtained from the panel study of the firms i.e., the power law in size (in terms of income and/or employment), the Laplace distribution in the growth rates and the slowly declining standard deviation of the growth rates conditional on the firm size. First, we show that the model generalizes the usual kinetic exchange models with binary interaction to interactions between an arbitrary number of agents. When the number of interacting agents is in the order of the system itself, it is possible to decouple the model. We provide exact results on the distributions which are not known yet for binary interactions. Our model easily reproduces the power law for the size distribution of firms (Zipf’s law). The fluctuations in the growth rate falls with increasing size following a power law (though the exponent does not match with the data). However, the distribution of the difference of the firm size in this model has Laplace distribution whereas the real data suggests that the difference of the log of sizes has the same distribution.     

Firm profitability and the network of organizational capabilities

A Laplace distribution for firm profit rates (or returns on assets) can be obtained through the sum of many independent shocks if the number of shocks is Poisson distributed. Interpreting this as a linear chain of events, we generalize the process to a hierarchical network structure. The hierarchical model reproduces the observed distributional patterns of firm profitability, which crucially depend on the life span of firms. While the profit rates of long-lived firms obey a symmetric Laplacian, short-lived firms display a different behavior depending on whether they are capable of generating positive profits or not. Successful short-lived firms exhibit a symmetric yet more leptokurtic pdf than long-lived firms. Our model suggests that these firms are more dynamic in their organizational capabilities, but on average also face more risk than long-lived firms. Finally, short-lived firms that fail to generate positive profits have the most leptokurtic distribution among the three classes, and on average lose slightly more than their total assets within a year.     

Transient magnetostriction in ferro- and ferri-magnetic materials

An acoustic impulse excited by a laser pulse generates some transient magnetostriction effects in various ferromagnetic and ferrimagnetic materials. A crystal of small magnetic anisotropy energy gives a steady magnetoelastic oscillation which is resonant with the sample dimension. The other materials show the surface effect only. The temperature dependence is studied up to the Curie temperature. Around the phase transition point the long-wavelength fluctuation of magnetization is excited by the elastic impulse, and some critical phenomena are observed. These effects are discussed by the usual theory of magnetoelastic interaction.     

Genotype selection model with two time-correlated white noises

We study the steady state properties of a genotype selection model in the presence of two time correlated Gaussian white noises. Based on the corresponding Fokker-Planck equation the steady state solution of the probability distribution has been investigated. The fluctuation of both the mutation factor and the genetic factor can break the balance of the gene selection. The fluctuation of the genetic factor can facilitate the gene separation which allows us to select one gene-type haploid from a gene group while the fluctuation of the mutation factor impedes the gene separation. Remarkably, the correlation time between the two noises can induce the system to change from two states to one.     

The firm as a bundle of barcodes

We empirically investigate the firm growth model proposed by Buldyrev et al. by using a unique dataset that contains the daily sales of more than 200 thousand products, which are collected from about 200 supermarkets in Japan over the last 20 years. We find that the empirical firm growth distribution is characterized by a Laplace distribution at the center and power-law at the tails, as predicted by the model. However, some of these characteristics disappear once we randomly reshuffle products across firms, implying that the shape of the empirical distribution is not produced as described by the model. Our simulation results suggest that the shape of the empirical distribution stems mainly from the presence of relationship between the size of a product and its growth rate.     

The origin of asymmetric behavior of money flow in the business firm network

In the business firm network, the number of in-degrees and out-degrees show the same scale-free property, however, the distribution of authorities and hubs show asymmetric behavior. Here we show the result of an analysis of the two-link structure of the network to find the origin of this asymmetric behavior. We find the tendency for big construction firms intermediating small subcontracting firms to have higher hub degrees. By measuring the strength of preferential attachment rate of new companies, we also find a abnormally strong preferential attachment for which the exponent is 1.4 with respect to out-degree when a new company forms a business partnership with a construction company. We propose a new model that reproduces the asymmetric behavior of the degrees of authorities and hubs by changing the preferential attachment rate between the in-degree and the out-degree in the business firm network.     

Innovation flow through social networks: productivity distribution in France and Italy

From a detailed empirical analysis of the productivity of non financial firms across several countries and years we show that productivity follows a non-Gaussian distribution with `fat tails' in the large productivity region which are well mimicked by power law behaviors. We discuss how these empirical findings can be linked to a mechanism of exchanges in a social network where firms improve their productivity by direct innovation and/or by imitation of other firm's technological and organizational solutions. The type of network-connectivity determines how fast and how efficiently information can diffuse and how quickly innovation will permeate or behaviors will be imitated. From a model for innovation flow through a complex network we show that the expectation values of the productivity of each firm are proportional to its connectivity in the network of links between firms. The comparison with the empirical distributions in France and Italy reveals that in this model, such a network must be of a scale-free type with a power-law degree distribution in the large connectivity range.     

Fluctuation of Parameters in Tumor Cell Growth Model

We study the steady state properties of a logistic growth model in the presence of Gaussian white noise.Based on the corresponding Fokker-Planck equation the steady state solution of the probability distribution function and its extrema have been investigated. It is found that the fluctuation of the tumor birth rate reduces the population of the cells while the fluctuation of predation rate can prevent the population of tumor ceils from going into extinction.Noise in the system can induce the phase transition.     

Fluctuation of Parameters in Tumor Cell Growth Model

We study the steady state properties of a logistic growth model in the presence of Gaussian white noise.Based on the corresponding Fokker-Planck equation the steady state solution of the probability distribution functionand its extrema have been investigated. It is found that the fluctuation of the tumor birth rate reduces the populationof the cells while the fluctuation of predation rate can prevent the population of tumor cells from going into extinction.Noise in the system can induce the phase transition.     

Modeling sediment transport in river networks

A dynamical model is proposed to study sediment transport in river networks. A river can be divided into segments by the injection of branch streams of higher rank. The model is based on the fact that in a real river, the sediment-carrying capability of the stream in the ith segment may be modulated by the undergone state, which may be erosion or sedimentation, of the i−1th and ith segments, and also influenced by that of the ith injecting branch of higher rank. We select a database about the upper-middle reach of the Yellow River in the lower-water season to test the model. The result shows that the data, produced by averaging the erosion or sedimentation over the preceding transient process, are in good agreement with the observed average in a month. With this model, the steady state after transience can be predicted, and it indicates a scaling law that the quantity of erosion or sedimentation exponentially depends on the number of the segments along the reach of the channel. Our investigation suggests that fluctuation of the stream flow due to random rainfall will prevent this steady state from occurring. This is owing to the phenomenon that the varying trend of the quantity of erosion or sedimentation is opposite to that of sediment-carrying capability of the stream.     

The effect of adaptive behavior on risk propagation in industrial symbiosis networks

The complex symbiotic relationship in the industrial symbiosis network (ISN) may cause new risks for firms. In view of this problem, previous studies mainly regard the ISN as a static system, without considering the adaptive behavior of firms. This paper establishes a risk propagation model of the ISN based on the change of firm state, proposes four kinds of reconnection strategies to model the adaptive behavior, and uses numerical simulation to investigate the effect of adaptive behavior on risk propagation. The results demonstrate that all the reconnection strategies play an inhibitory role in the risk propagation. Therein, the effectiveness of PP strategy is the best, followed by RR strategy, and DP (SP) strategy. In any case, the effect of reconnection strategies on risk propagation will improve with the increase of the disconnection probability and network resilience. Additionally, the more decentralized weight distribution will weaken the inhibition of adaptive behavior on risk propagation.     

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.