高分子系統非平衡群集弛釋過程的統計力學與其拓展之研究

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高分子系統非平衡群集弛釋過程的統計力學與其拓展之研

究

計畫類別：個別型

計畫編號： NSC 100-2112-M-004-001-

執行期間： 100 年 05 月 01 日至 101 年 09 月 30 日

執行單位：國立政治大學應用物理研究所

計畫主持人：馬文忠

計畫參與人員：碩士班研究生-兼任助理人員：張景婷

碩士班研究生-兼任助理人員：王柏淵

碩士班研究生-兼任助理人員：曾嘉瑤

公開資訊：本計畫可公開查詢

中華民國 101 年 12 月 31 日

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已脫離了平衡態統計力學的基本假設所涵蓋的範疇、仍需要

更多的數據與理論探討；我們除了分子系統的模擬分析，也

將金融、社會現象，這些本質上呈現強烈的非高斯統計的系

統，當作熱統計的標的系統予以初步的探討。我們在這一期

計畫中，進行了一項金融數據的統計與動力性質的分析，以

及一項社會合作現象的動力模型之初步建構與模擬。我們將

同一股票市場的一籃子股票當作是一個多體系統，我們發現

股價的變動與多體物理系統的粒子之位移都同樣具有時間尺

度的多樣性，隨時間變動呈現一次冪的表現之隨機行走行為

只在一定的時間尺度有效，不同的時間尺度的時間變動行式

均隱含有動力性質的線索；我們可藉由兩系統間的對照類比

建立起一籃子股票的巨觀參數。對於一個社會其成員與其他

成員間的合作態度(策略)的模擬中，我們將策略的採納看作

是個體的狀態，隨時間演進每個成員會根據整個社會和他採

用相同或不同合作策略的其他成員的綜合經驗判定各種策略

當下的優劣而決定是否改變其態度。這如同是一個動態多體

系統、其個別成員的狀態取決於整個系統的(能量)狀態,只不

過每兩次狀態改變間是一系列個別成員間的對局的動力過

程。

中文關鍵詞：非平衡, 高分子凝聚現象, 時間尺度, 蛋白質, 金融, 合作

策略

英文摘要： In this project, we carry out extensive computer

simulations to study the aggregation dynamics in

model polymer systems, to find out the mechanical and

thermo-statistical properties of those

backbone-connectivity-induced ＇quasi-steady＇ situations. In

a detailed analysis on the contents of chain

connectivity, we summarize their effects on the

dynamic processes of aggregation, as a preparation

for the comparison with refined models and fine-tuned

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aggregation processes have been realized not all

in ＇quasi-steady＇ situations, the latter

constraints will be greatly helpful in tracking the

processes among different proteins or peptides or

those under various conditions, by macroscopic

parameters. It is similar to the advantage in

employing the idea of ＇quasi-static＇ process in

conventional thermodynamics. Under the circumstance

that it still requires more data and theoretical

analysis to clarify whether, or how far, has the

thermo-statistical nature of the quasi-steady

situations been deviated from the conventional

equilibrium statistical mechanics, we have carried

out some studies on the collective financial or

social phenomena, where the statistics is known to be

highly non-Gaussian. We have studied the time scale

dependence of the price changes in collections of

stocks in real market and initiated simulation

studies on the dynamics of cooperation strategies of

simplified model societies.

英文關鍵詞： aggregation, backbone-connectivity, non-equilibrium,

time scale, quasi-steady, non-Gaussian, financial

market, cooperation strategy

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計性質。我們以分子鍊的背瘠連結性對這些模型系統的凝聚過程產生的影響作有一歸

納整理,作為將模型進一步細化、比較與對照的參考;並釐清了凝聚過程並不必然是在

準穩的狀態這一事實。如果能以特別設計動力模型將電腦模擬下的凝聚過程控制在準

穩的狀態、類似傳統熱力學的準靜(quasi-static) 過程的觀念，將有助於總合對各種

不同的蛋白質或牲肽之凝聚過程、將其描述予以參數化。由於其背後的熱統計性是不

是已脫離了平衡態統計力學的基本假設所涵蓋的範疇、仍需要更多的數據與理論探討;

我們除了分子系統的模擬分析，也將金融、社會現象，這些本質上呈現強烈的非高斯

統計的系統，當作熱統計的標的系統予以初步的探討。我們在這一期計畫中，進行了

一項金融數據的統計與動力性質的分析，以及一項社會合作現象的動力模型之初步建

構與模擬。我們將同一股票市場的一籃子股票當作是一個多體系統，我們發現股價的

變動與多體物理系統的粒子之位移都同樣具有時間尺度的多樣性，隨時間變動呈現一

次冪的表現之隨機行走行為只在一定的時間尺度有效，不同的時間尺度的時間變動行

式均隱含有動力性質的線索;我們可藉由兩系統間的對照類比建立起一籃子股票的巨

觀參數。對於一個社會其成員與其他成員間的合作態度(策略)的模擬中，我們將策略

的採納看作是個體的狀態，隨時間演進每個成員會根據整個社會和他採用相同或不同

合作策略的其他成員的綜合經驗判定各種策略當下的優劣而決定是否改變其態度。這

如同是一個動態多體系統、其個別成員的狀態取決於整個系統的(能量)狀態,只不過每

兩次狀態改變間是一系列個別成員間的對局的動力過程。

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Collective Relaxation Processes in Polymer Systems and Its Extension”

Project No. NSC100-2112-M-004-001, Part I (2011.5.1-2012.9.30)

Wen-Jong Ma

Graduate Institute of Applied Physics

National Chengchi University

ABSTRACT

In this project, we carry out extensive computer simulations to study the aggregation

dynamics in model polymer systems, to find out the mechanical and thermo-statistical

properties of those backbone-connectivity-induced “quasi-steady” situations. In a detailed

analysis on the contents of chain connectivity, we summarize their effects on the dynamic

processes of aggregation, as a preparation for the comparison with refined models and

fine-tuned dynamic process for further studies. While the aggregation processes have been

realized not all in “quasi-steady” situations, the latter constraints will be greatly helpful in

tracking the processes among different proteins or peptides or those under various conditions,

by macroscopic parameters. It is similar to the advantage in employing the idea of

“quasi-static” process in conventional thermodynamics. Under the circumstance that it still

requires more data and theoretical analysis to clarify whether, or how far, has the

thermo-statistical nature of the quasi-steady situations been deviated from the conventional

equilibrium statistical mechanics, we have carried out some studies on the collective financial

or social phenomena, where the statistics is known to be highly non-Gaussian. We have

studied the time scale dependence of the price changes in collections of stocks in real market

and initiated simulation studies on the dynamics of cooperation strategies of simplified model

societies.

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1. Aggregation Phenomena and non Maxwell-Boltzmann Behavior Induced by Dynamic

Anisotropy of Backbone Connectivity in non Equilibrium Collective Relaxing Polymer

Model Systems

We summarize our molecular dynamics simulation studies that reveal the relationship

between the backbone connectivity of polymer chains and some benchmark features

displayed in the aggregation processes of model polymer chains [1]. It provides the

information to tackle with the protein aggregation problem, from the viewpoint of polymer

physics, enquiring how the underlying factors, such as the geometric packing, the topology

and their mutual interplay with the presence of solvent, play the roles in the processes. Such a

viewpoint has been adapted effectively in our analysis of protein data bases [2]. To facilitate

such an approach further in computer simulation, we intend to track on the dynamic processes

by rendering the aggregation processes under “quasi-steady” situations which are

characterized by steady velocity distribution of monomers and the distribution deviates from

the standard Maxwell-Boltzmann behavior to follow Tsallis q-statistics, with the value of q

deviating and larger than unity (the Maxwell-Boltzmann type) in increasing the strength of the

n.n. bonding. Most importantly, the state of the system can be characterized by one single q,

allowing the instantaneous temperature to vary in time. We have obtained the evidences [3]

that the situations are related to the presence of dynamical balance for the underlying curved

frames, which leads to certain symmetries in statistical functions over long range correlations.

Under these circumstances, the deviation of q from unity depends on the degree of dynamic

anisotropy along the backbones of the chains. Our analysis reveals a scenario of dynamically

steady non-equilibrium situation, compatible with the inherent anisotropy along the

backbones that are useful in tracing the collective relaxation processes in various complex

systems of polymers.

2. Macroscopic Parameters and Time-Scale Dependence in Collective Dynamic and

Static Properties in Stock Price Changes

The financial time series and the trajectories of particle motion are, historically, the first

topics that had inspired the subject of "random walks". In this study, we show that the analogy

between the two systems can be extended to a wider range, to facilitate an effective analysis

on the collective properties behind the financial time series. We analyze the data of a

collection of 345 stocks listed in S&P 500 and that of the same number of stocks in Taiwan

market, for each month over the years 1996-1999. The high-frequency one-day moving

averages are shown to share similar features with the trajectories of particles. For the motion

of particles, the characteristic power-one time dependence in the mean square displacement

(MSD) for all particles are valid only over time scales restricted by spatial size of the system

from above and, from below, by the sufficiency of randomization via momentum exchanges

with the fellow particles or with the environment. Consequently, the MSD shows a crossover

at the lower bound of this regime, below which a power-two time dependent behavior prevails.

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correlations, we obtain a set of parameters [4] to describe the macroscopic states of the system.

Such information is further refined by a detailed analysis on the static probability density

functions used in the second moment (MSD) calculations and the dynamic correlations,

complemented by a comparison with the results obtained from the coupled random walk

model. The time scale dependence of the probability density functions of the log-returns and

the dependence of the autocorrelation functions for the 345 stocks of S&P 500 for each month

over the years 1996-1999 are analysed and compared with the outcomes from the simulations

on models of coupled random walks [5] where the couplings are built on scale-free

networks[6].

3. Zero-Temperature Simulation of Time Evolutions of Cooperation Strategies in a

Society

The effects of human cooperation on the societies and on the individuals have been an

important issue in social science. The dynamics of a model society of individuals with

adjustable cooperation strategies and with varying reputations gauged by social norms has

been recently proposed [7]. In order to refine the mean-field type analysis, we implement the

model in computer simulations, where the strategy adjustment of each individual is

determined by a social learning procedure, analogous to the Metropolis energy-driving state

transition in the Potts model. In between the consecutive strategy changes, one individual will

encounter a partner in a donor-recipient game, which results in the wealth changes in both

parties in form of cost, punishment or benefit and is followed by a reputation re-assignment to

the donor, taking into account the strategy of the donor and the reputation of the recipient.

The accumulated knowledge of wealth changes from sequences of games for all individuals in

the society weighs the strategy change transitions. In our zero-temperature simulations [8], we

obtain some primitive observations on the evolutions of strategies adapted by the individuals

of the model society.

[1]

W.-J. Ma and C. K. Hu, in Molecular Dynamics – Studies of Synthetic and Biological

Macromolecules, (2011, InTech on-line book) Chap. 3.

[2]

M.-C. Wu, M. S. Li, W.-J. Ma, M. Kouza4,6 and C.-K. Hu, Europhys. Lett. 96, 68005

(2011).

[3] W.-J. Ma and C.-K. Hu,

“non Maxwell-Boltzmann Behavior Induced by Dynamic

Anisotropy of Backbone Connectivity over non Equilibrium Collective Relaxation Processes

in Polymer Systems,“ preprint, see Appendix III.

[4]

W.-J. Ma, S.-C. Wang, C.-K. Hu and C.-N. Chen, “Macroscopic parameters in financial

fluctuations,“ preprint, see Appendix IV.

[5] W.-J. Ma and C.-K. Hu, R. E. Amritkar, Phys. Rev. E 70, 026101 (2004).

[6] B.-Y. Wang, Y.-F. Huang, W.-J. Ma, "Time Scale Dependence of Static and Dynamic

Quantities for High Frequency One-Day Moving Averages of a Collection of Stocks,"

(unpublished).

[7] T.k. Yu, S.-H. Chen, H. G. Li, "Social Norm, Costly Punishment and the Evolution to

Cooperation," 17th International Conference on Computing in Economics and Finance .

[8] C.-Y. Tseng, W.-J. Ma and S.-H. Chen, "Zero-Temperature Simulation of Time

Evolutions of Cooperation Strategies in a Society," (unpublished).

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Backbone Connectivity and Collective

Aggregation Phenomena in Polymer Systems

Wen-Jong Ma1,2and Chin-Kun Hu2 1_{Graduate Institute of Applied Physics, National Chengchi University,}

Taipei 11605

2_{Institute of Physics, Academia Sinica, Nankang, Taipei 11529}

Taiwan

1. Introduction

A convenient theoretical entry to tackle the protein aggregation problems is to track on the origin of the aggregation phenomena from the viewpoint of polymer physics, enquiring how the underlying factors, such as the geometric packing, the topology and their mutual interplays in presence of solvent, play their roles in the processes. Such a view has been inspired by recent experiments [1, 2] and molecular dynamics simulations [3, 4] which suggest the ubiquitous presence of ﬁbril formation [5] in various natural and laboratory prepared proteins or peptides. While the variety of amino acid sequences interferes with the occurrence of long-range structural ordering, a material-insensitive tendency of aggregation is observed [2, 3, 5]. By coarsening the sequence-sensitive details, the aggregation problem can be formulated in its minimal form as the clustering process of polymer chains [6, 7]. With such simpliﬁed models, the approach focuses more on the entropic effect caused by the constraint of chain connectivity [6–8], rather than on the material-dependent characteristics. In this chapter, we summarize our molecular dynamics simulation studies [6, 7] that reveal the relationship between the backbone connectivity of polymer chains and some benchmark features displayed in the aggregation processes of the model polymer chains.

In the model, the backbone connectivity of a polymer chain is realized by assigning a string of monomers with speciﬁc monomer-monomer two-body forces, perturbed with three-body and four-body angle dependent interactions [6], with their strengths measured by the parameters,

knn, Kband Kt, for the nearest neighbor (n.n.) interaction, the bending angle and the torsion

angle potentials, respectively. In a collection of polymer chains, all the non n.n. pairs along a chain and those pairs on different chains are subject to Lennard-Jones (L-J) pair interactions. The presence of angle potentials, with signiﬁcantly nonzero Kb or Kt values, breaks the

isotropy surround the backbone. In such a system, the local inter-chain hindrance prevents the chains from clustering into ordered domains. The situation can, however, be reverted by reducing the values of K_band Kt. We ﬁnd that the formation of bundle-like domains (Fig. 1)

is robust as soon as Kband Kt are small enough. The observations are assured even if we

introduce a small amount of dispelling background ﬂuid molecules or impose a small fraction of impurity monomer sites.

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EPL,96 (2011) 68005 www.epljournal.org doi: 10.1209/0295-5075/96/68005

Universal geometrical factor of protein conformations

as a consequence of energy minimization

Ming-Chya Wu1,2,3(a), Mai Suan Li4, Wen-Jong Ma2,5, Maksim Kouza4,6 _{and Chin-Kun Hu}2(b)

1_{Research Center for Adaptive Data Analysis, National Central University - Chungli 32001, Taiwan} 2_{Institute of Physics, Academia Sinica - Nankang, Taipei 11529, Taiwan}

3_{Department of Physics, National Central University - Chungli 32001, Taiwan}

4_{Institute of Physics, Polish Academy of Sciences - Al. Lotnikow 32/46, 02-668 Warsaw, Poland, EU} 5_{Graduate Institute of Applied Physics, National Chengchi University - Taipei 11605, Taiwan}

6_{Department of Physics, Michigan Technological University - Houghton, MI 49931, USA}

received 26 September 2011; accepted in ﬁnal form 28 October 2011 published online 13 December 2011

PACS 87.14.E- – Proteins

PACS 87.15.A- – Theory, modeling, and computer simulation

PACS 87.15.-v – Biomolecules: structure and physical properties

Abstract – The biological activity and functional speciﬁcity of proteins depend on their native three-dimensional structures determined by inter- and intra-molecular interactions. In this paper, we investigate the geometrical factor of protein conformation as a consequence of energy minimization in protein folding. Folding simulations of 10 polypeptides with chain length ranging from 183 to 548 residues manifest that the dimensionless ratio (V/Ar) of the van der Waals volumeV to the surface area A and average atomic radius r of the folded structures, calculated with atomic radii setting used in SMMP (Eisenmenger F. et al., Comput. Phys. Commun., 138 (2001) 192), approach 0.49 quickly during the course of energy minimization. A large scale analysis of protein structures shows that the ratio for real and well-designed proteins is universal and equal to 0.491 ± 0.005. The fractional composition of hydrophobic and hydrophilic residues does not aﬀect the ratio substantially. The ratio also holds for intrinsically disordered proteins, while it ceases to be universal for polypeptides with bad folding properties.

Copyright c EPLA, 2011

Introduction. – In recent decades, physical methods have been widely used to study properties and structures of biopolymers [1–3], including DNA [4,5], RNA [6], and protein [7–10]. Proteins assume specified conformations from their chemical compositions or sequences to develop biological activity and functional specificity. The corre-sponding three-dimensional (3D) structures are a conse-quence of inter- and intra-molecular interactions, in which energy minimization is the principle governing the folding tendency. In spite of various components involved in the interactions, there have been attempts to derive simple geometric factors from a variety of conformations, which can be either considered as a factor for structure validity or used as an effective constraint in folding simulation.

Geometric properties of protein molecules have been studied for more than three decades [11–13]. Among others, the Ramachandran plot [14] is a practical criterion

(a)_{E-mail: [email protected]} (b)_{E-mail: [email protected]}

widely used for improving the quality of NMR or crystal-lographic protein structures. In a polypeptide, the main chain N-C_α and C_α-C bonds are relatively free to rotate, and can be respectively represented by two torsion angles. These angles can only appear in certain combinations due to steric hindrances, which deﬁne the allowed regions of the torsion angles for secondary structures in the plot.

Furthermore, it has been found that the mean volume of an amino acid in the interior of proteins is very close to that of the amino acid in crystals [11,12]. With the help of the Delaunary triangulation method, Liang and Dill [15] have reported that the protein packing is heterogeneous, and in terms of packing density, protein molecules may be either well-packed or loosely packed. Zhang et al. [16] showed that the packing density of single domain proteins decreases with chain length, which shares a generic feature of random polymers satisfying loose constraint in compactness.

Beside the Ramachandran plot and the packing density, which represent conclusions based on observations, there

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non Maxwell-Boltzmann Behavior Induced by Dynamic Anisotropy of Backbone Connectivity over non Equilibrium Collective Relaxation

Processes in Polymer Systems Wen-Jong Ma1,2 ∗ _{and Chin-Kun Hu}2 †

1 _{Graduate Institute of Applied Physics, National Chengchi University, Taipei 11605, Taiwan} 2_{Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan}

Using molecular dynamics simulation, we explore how its steady velocity distribution for monomers deviates from the standard Maxwell-Boltzmann behavior in a system of non-equilibrium polymer chains, as an effect induced by the anisotropy of backbone connectivity along the chains. In our model systems, each individual chain is decorated by weak bending as well as torsion potentials, perturbing to the nearest neighboring (n.n.) bonding along the chain. Additional perturbing effects can also be added by the presence of a small amount of Lennard-Jones fluid and an extra heterogeneity among the monomers along the chains. Over steady non equilibrium relaxation processes, the velocity distributions of monomers of the system are found described by Tsallis q-statistics, with the value of q deviating and larger than unity (the Maxwell-Boltzmann type) in increasing the strength of the n.n. bonding. Most importantly, the state of the system can be characterized by one single q, allowing the instantaneous temperature to vary in time. The feature is identified for the system as reaching a ’quasi-steady’ situation. We found that the latter situation is related to the presence of dynamical balance for the underlying curved frames, which leads to certain symmetries in statistical functions over long range correlations. Under these circumstances, the deviation of

q from unity depends on the degree of dynamic anisotropy along the backbones of the chains.

Our analysis reveals a scenario of dynamically steady non-equilibrium situation, compatible with the inherent anisotropy along the backbones that are useful in tracing the collective relaxation processes in various complex systems of polymers.

KEYWORDS: Maxwell-Boltzmann,Tsallis, polymer,correlation

1. Introduction

The understanding of the statistical mechanical principles underlying the macroscopic properties of systems of complex macromolecules is important for the development of the ability of computation that can provide correct interpretations and predictions for experi-mental observations. This is particularly true for the rapidly advanced studies in systems of biological origins, where phenomena appear at various ranges of time or spatial scales. Aside from the delicately designed native conformations to facilitate biological functioning,

∗_{E-mail: [email protected]} †_{E-mail: [email protected]}

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Fig. 1. Relative direction cosine between the motions at the sites i and i + l along a chain. We record the relationship between the direction of motion (unit vector) ˆvi of site i and the local Cartesian

frame formed by (1) the bond vector bi, (2) the normal (unit) vector ˆuiof the plane extended by

bi−1and bi and (3) the unit vector ˆwiperpendicular to both biand ˆui; and revive the relationship

to obtain the unit vector (ˆvi)l (plotted as a bright arrow) in the local frame at site i + l formed

by the vectors bi+l, ˆui+l and ˆwi+l. The inner product (ˆvi)l· ˆvi+l is denoted brieﬂy by ˆvi· ˆvi+l in

this paper.

the underlying polymer nature of protein molecules may often render a system to form vari-ous self-assembled collective conformations which cover a long range of time, comparable to human aging phenomena. It is then an important task to track on how the essential polymer feature, such as backbone connectivity, play its role in those long time relaxations, especially those pertaining to aggregation and glass-forming processes. We inquire whether a set of sta-tistical mechanical calculation methods1 can be designed compatibly with the routes of time evolution2 for those long-lived non equilibrium situations in systems of polymer chains.3, 4 To answer this question, we rely on intensive computer experiments with focusing on the roles of backbone connectivity, their eﬀects on the thermal statistical properties over the non equilibrium relaxation processes.5, 6

In this study, we focus first on the equilibration situations reached between non-equilibrium polymer chains and Lennard-Jones fluid. Using the term ‘equilibration’, we refer to a broader scope of situations than those coined by ‘equilibrium’, in that the temperatures of individ-ual species (polymer and fluid) are allowed to be different. Indeed, we find in our previous simulations that the enhanced monomer-monomer bonds along the chains can affect the heat exchange of the latter with the surrounding Lennard-Jones fluid (vapor) and lead the system to relax to the equilibrations with the ratio between the temperatures of the two species dif-ferent from unity.6 Correspondingly, the monomer velocities are found described by the class of statistics proposed more than two decades ago by Tsallis,7 with the Maxwell-Boltzmann distribution as a special case. The statistics has been found to prevail in a host of model poly-mer systems, in presence of or in absence of fluid, with the chain connectivity combined with

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the effects by the bending and torsion angle potentials to play the role.5, 6 In these systems, a ‘quasi-steady’ situation has been reached when the velocity distributions are described by the same value for the parameter q of the Tsallis statistics, where the value of q is affected by the form of angle potentials as well as by the strength of monomer-monomer bonding. Here, we carried out a close examination on the internal statistics over the chains in systems of polymer chains mixed in fluid or systems of chains in absence of fluid and removing of angle potentials.

In the melt, there’s no preferred overall direction in whole system either for the bond orientations or for the directions of motion in monomers. The latter do, however, have local anisotropy governed by the former along the curved backbones. Unlike the static anisotropy prevailed among different chains in liquid crystal systems, such anisotropy is dynamic in origin and, under chain connectivity, induces orientation correlations for the motion of monomers along a chain. A well-defined stable statistics with deviations from the Maxwell-Boltzmann type are observed only when such correlations signal the sustaining of a underlying curved frames. The latter are defined by the intrinsic coordinate systems along the backbones of individual chains, which aredynamically stable. Such a stability is characterized by the spatial symmetry in the long-range monomer-monomer velocity orientation correlations measured over the curved frames along the backbone. Note that, the latter frames are time varying and they can only be described statistically. In contrast to the ordinary Euclidean space, any inner product used to calculate a correlation between two vectors at different locations along a chain should be defined with one vector shifted under a ’parallel transport’ to the local frame of the other vector (see Fig. 1). Such a convention will be adopted without further specifying in this paper.

2. The model system

We consider systems of spatially well-mixed NP = 40 polymer chains and NF = 6000 Lennard-Jones molecules. A polymer chain consists of n = 100 monomers labelled by integer

i = 1, . . . , n from one end to the other, where the monomer at site i for 2 ≤ i ≤ n − 1

is connected by rigid bonds or by harmonic potentials 1₂kspring(rij − r0)2 with two nearest neighbors at sites j = i − 1 and j = i + 1, respectively. We use a strength parameter knn, which is ﬁnite for soft bonding and is ∞ for rigid bonding. In this study, we consider the coupling kspring to be ks = knn × k0, with knn = 10s and s =0, 1, 2, 3, 4 or ∞ for the six types of chains, respectively. The inﬁnite strength is realized as the constraint force assigned numerically to maintain a rigid bonding.

The polymer chains are a modiﬁed ’united- atom’ model. For the convenience of compar-ison, we use the parameters Kb and Kt to denote the strengths of the (bending and torsion) angle potentials, which resume the origin model8 on assigning both parameters with unity values. As in,5, 6 the consecutive monomers along a polymer chain interact via the bending

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Fig. 2. (Color online)Distributions f for the direction cosine ˆv ·ˆb between the velocity direction ˆv of a

monomer site and its bond direction ˆv along a polymer chain in the systems of NP = 40 polymer

chains, each containing n = 100 monomers, mixed with NF = 6000 Lennard-Jones molecules.

The neighboring monomers along a chain are connected by a spring of strength kspring = 10sk0

(s=0,1,2,3 or 4) or by a rigid bond. Each of the distributions is obtained by collecting data of 2000 snapshots, in intervals of 5 steps, over a simulation of period 0.25τ in time step δt = 2.5 × 10−5τ .

These intervals are (chosen arbitrarily) from the quasi-steady situations starting at t=29.4 (for

s=0), t=42.15 (for s=1), t=41.9 (for s=2), t=41.65 (for s=3), t=59.65 (for s=4) and t=169.2 (for

rigid bonding) respectively, of Fig. 4 in Ref.6). The corresponding value for q can be found in Fig. 4.

and torsional potentials, which are in form

Θ(θijk) = Kbcb(cosθijk− cosθ0)2 and Φ(φijkl) = Kt 3 ι=0 aι(cosφijkl)ι

with bending angle θijk and torsional angle φijkl determined by the consecutive monomers at

sites i, j = i + 1 and k = i + 2 and by i, j = i + 1, k = i + 2 and l = i + 3. The potential Φ has two minima at φ = 0o and at φ = 112.8o, respectively, determined by the parameters

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two diﬀerent degrees of hindrance in the local conformations of peptide chains. In contrast to the united atom model for polyethylene, which corresponds to the case Kb = Kt = 1 and

θ0 = θ(II)0 , we restrict in our models the values of Kb and Kt, no greater than 0.1, and impose

θ0 = θ0(I), so that the potentials Θ and Φ provide eﬀectively local conformation hinderance perturbing to the Lennard-Jones cores along the chains.

Any inter-chain or non-spring-connected (non-rigid-bonded) intra-chain pair of monomers, interact via Lennard-Jones (L-J) potential

V (r) = 4((σ/r)12− (σ/r)6).

We use the energy and size parameters, and σ, of L-J potential and the mass m of a monomer as the basic units for the quantities. The temperature (multiplied by Boltzmann constant) and time are expressed in units of energy and τ = mσ2/. For the springs connecting

monomers, we choose the constant k0 = 1.5552 × 105 and length r0 = 0.357σ. The rest parameters (cb, and aι, ι=0, 1, 2 and 3) have been assigned previously5, 8).

The fluid molecules mutually interact via a Lennard-Jones pair potential, which has the same energy parameter and one-quarter size parameter as those for monomer-monomer pair interactions. Each fluid molecule is assumed to have the same mass m as a monomer. The pair interactions between fluid molecules and monomers are repulsive and take the r−12 part of the L-J interaction. We introduce some minor heterogeneity into the chain by randomly choosing totally five percent (199 sites) “linker sites” from the monomers to have the same interaction parameters as those for fluid molecules.

In controlling the parameters kn.n., Kb and Kt, the system of the collection of polymer chains may be prepared to evolve ’quasi-steadily’5, 6where mechanical balances are maintained dynamically to sustain stable underlying curved frames. To characterize such frames, we collect the statistical distributions of the eﬀective geometric curvature ˜κ and torsion ˜τ ,

˜

κ = 1

2Δbi[(ˆbi− ˆbi−1)· ˆwi− (( ˆwi− ˆwi−1)· ˆbi] (1)

and

˜

τ = 1

2Δbi[( ˆwi− ˆwi−1)· ˆui+ ((ˆui− ˆui−1)· ˆwi] (2)

where ˆbi = bi/|bi| and ˆui, ˆwi are the local basis vectors (see Fig. 1) and Δbi = 1₂|bi−1+ bi| measures the distance variation at the site i along the backbone of a chain. While detailed analysis of local curvature and torsion shows robust statistical distributions as a signature indicating the presence of dynamically steady curved frames along the chains, the appearance of a globally steady situation, where the monomer velocity distributions are characterized by one single q, would require a kind of dynamic balance which we address in this paper.

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Fig. 3. Distributions (probability density functions) of velocity components in parallel to (diamond), or in perpendicular to (ﬁlled circle and open circle, corresponding to the normal direction of the plane extended by the consecutive bond vectors and the direction perpendicular to the bond and the normal, respectively, see Fig. 1) the backbone for (a) the mixed system of polymer chains with rigidly connected bonds (T_P∗ ≈ 103); (b) the mixed system polymer chains bonded by springs with strength kspring= k0 (T_P∗ ≈ 74.5), described in Fig. 2; and those corresponding total velocities (c)

(for the system in (a)) and (d) (for the system in (b)). The data are collected over 40 conﬁgurations in interval of 5× 10−5τ over a period of 0.002 τ . (a) and (b) are ﬁtted by the one dimensional

Maxwell-Boltzmann distribution FMB(v) =

2m

πT∗e− mv2

2T ∗ (black lines) and one-dimension Tsallis distribution Fq(v) = Bqβ1/2e−

βmv2

2

q with q=1.15 (red lines), where the parameter T∗and β−1 are in unit of and T_m∗ = (βm)−1M1(q) give the shared second moment for the two ﬁtted distributions

as well as the raw data, where the one-dimension proportional factor M1(q) = (_1−q2 )[1−_BBq

q q

12_]

and q = _2−q1 . The ﬁtted values for T (dotted line), T⊥1 (dashed line) and T⊥2 (dotted-dashed

line) for the curves of the components v = v, v⊥1 and v⊥2 respectively, shown in the plots. The

curves of Tsallis formula apparently are better fitted to the simulation data, especially near the maximum around zero velocity. (c) and (d) contain the three dimensional data fitted by three-dimensional Tsallis distribution, Eq. (3) (solid line), with the values of parameters q = 1.15 and β shown in the figures. The curve of (three dimensional) Maxwell-Boltzmann having the same second moment < v2 > is also plotted (dashed line) in each figure for comparison. In (c), we plot also

the elliptic Maxwell-Boltzmann Felliptic(v) = m[T1(T1−T)]−1/2ve−mv2/2T1erf (v{m2[T1 −

1

T1]} 1 2) using T1= 1₂(T⊥1+T⊥2) (dotted line) which improves much the discrepancy from the the isotropic

Maxwell-Boltzmann (dashed line) and is very close to, but still not as good as, the ﬁtting by the Tsallis formula (solid line).

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3. Simulation procedure and results

A mixed system containing polymer chains of rigid bonding and Lennard-Jones molecules was prepared in the cubic simulation box subject to periodic boundary conditions. Starting from an arbitrary random initial configurations in a large volume, the polymer chains were in a non equilibrium situation under a spontaneous driving toward the formation of dense clusters. We forced the system to relax via a highly biased isobaric environment, allowing a sufficient local adjustment for the chains and the fluid atoms, leaving the global adjustment of the former highly inadequate. In turning off the external controlling, the followed pro-ductive runs were then driven by the tendency of a global adjustment in a constant volume simulation. We are particularly interested in the situations where non equilibrium process pro-ceed in a quasi-steady manner, specifically for the systems with Kb = Kt= 0.1. After some

lengthy preparation procedures,5, 6 including turning the rigid bonds into springs, we obtain six spatially well mixed systems of polymer and fluid, each in a cube of side L = 33.3876σ. They contain chains connected by springs of coupling kspring = 10sk0 for s =0, 1, 2, 3 , 4, or ∞, respectively. We started our productive runs with configurations where velocities for all monomer sites and fluid molecules are virtually zero. The polymer and fluid heat up spontaneously at different rates. We immediately had systems of hot polymer and cold fluid, undergoing relaxation procedure toward reaching equilibration.6

In simulation, the velocity Verlet scheme was used to integrate the equations of motion with the rigid bonds realized via RATTLE scheme,9 which is accurate to the velocity order, to guarantee the relative velocity between two bonded sites along a chain being perpendicular to the direction of their bond. The numerical noises from both schemes render the systems act as extra source or sink of heat that make the system ﬂoating in total energy. Nevertheless, we can track the non-equilibrium situations by identifying the presence of steady single-step velocity distributions. It has been found that, over almost the whole process of relaxations excluding very short transients at beginning, the distributions for monomer velocities are well described by the generalized Maxwell-Boltzmann distribution,5, 6

4πv2F (v) = Aqβ3/24πv2× [e−

β

2mv2

q ] (3)

where the q-exponential function ex_q is deﬁned by ex_q ≡ [1 + (1 − q)x](1−q)1 _{with A}_q ₌

[₀∞4πv2e−_q 21mv2_dv]−1 _{and 1} _{≤ q < 7/5.}5 _{In the limit q → 1, Eq. (3) becomes the}

Maxwell-Boltzmann distribution (MB), in which one can ﬁnd the correspondence of the β factor in Eq. (3) exactly to the inverse temperature factor commonly used in the Gibbs-Boltzmann statistical mechanics and justify the usage of the same notation. According to Eq. (3), the

β−1 and the mean kinetic energy < mv2 >, which deﬁnes temperature, are related by a

q-dependent-only proportional constant, M (q).6

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varying in (inverse) temperature parameter β persists. Second, the ratio Γ = T_P/T_F of the instantaneous temperatures (“T_P∗” for polymer and “T_F∗” for fluid), defined as the (twice of) mean kinetic energy per degree of freedom approaches a constant value. In the five systems of spring-connecting chains, Γ∗ have been found not necessarily to be unity, in contrast to the prediction of the zeroth law of thermodynamics. Indeed, we found that6 the cases among the five systems with Γ∗ < 1, which are the system with kspring = 103k0 and the one with

kspring= 104k0, are exactly those with monomer velocity distribution deviated from Maxwell-Boltzmann (q > 1).10 The ratio Γ∗ can be naively interpreted as the portion of ‘true’ degrees of freedom participating eﬀectively the equilibration between polymer and ﬂuid. In increasing the strength of the springs, Γ∗ becomes smaller, ranging between unity and two-third the latter can be considered as the limit of rigid bonds, where the number degrees of freedom should be enumerated with the number of rigid bonds deducted. In the following, we try to reveal the origin behind such anomalies by investigating the internal statistical properties along the backbones of the polymer chains.

In enhancing the strength of the springs connecting neighboring monomers along the chains, we expect a tendency in the relative velocity between connected sites along a chain to become perpendicular to the direction of their relative position. The latter situation occurs strictly for chains of rigid bonds. The way to arrange the motion of individual monomer sites along a chain all together to accommodate such constriction seems to render their motions preferring the transverse to the backbone direction. Figure 2 shows the probability densities of direction cosine ˆv ·ˆb for the unit vector ˆv = v/|v| of velocity for each monomer, with respect

to the unit vector ˆb of the bond connecting to one of its two neighbors along the chain, for

colorred those six systems, of mixed polymer chains with Lennard-Jones ﬂuid, after reaching equilibration. The increasing of the strengths of the springs do drive the probability density functions deviating from a uniform situation, where the monomers move more transversely to the backbone direction. Once again, the cases with signiﬁcant anisotropy in the distribution of direction cosine ˆv · ˆb are exactly those having q > 1.

The anisotropy is also found among the distributions of the velocity components along backbones. There is a discrepancy (Fig. 3(a)) between the distribution of parallel component

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and those of the perpendicular components for the monomer velocities of the system with chains having rigid bonds. Such a discrepancy can barely be recognized in the system with the weakest bonding (Fig. 3(b)) among the six systems under consideration. For the former system, the monomer velocity distribution deviates from Maxwell-Boltzmann (Fig. 3(c)), in contrast to that of the latter (Fig. 3(d)). Such an analysis inspires the necessity of searching for the geometric origin of these anomalies, beyond the mere statistical eﬀects caused by ﬁnite sampling.11, 12

It is worth to mention that there is no overall anisotropy in the backbone directions, which is evidenced in the distribution of monomer-monomer bond directions. It must be then some kind of correlation along the backbone of the chains that plays the key role. After all, the velocity distributions reflects dynamic properties of chains with ever-chaining conformations, in the first place. Secondly, the enhanced backbone chain connectivity seems to be the most relevant candidate that makes the dynamics different from those in ordinary systems where Maxwell-Boltzmann statistics prevails.

The motions of monomers are damped oscillations under the constraints of a dynamically stable, time-varying curved frames. We consider the velocity orientation correlation along a chain, measured by the probability density fl of relative (Fig. 1) direction cosine ˆvi · ˆvi+l

observed along the curved frame, where we record the relation between ˆviand the local orien-tations at site i and revive ˆvi by resuming such a relation in the new local frames at site i + l

(Fig. 1). The distribution measured in such curved frame (Fig. 4) has a better balanced densi-ties between the two ends near ˆvi· ˆvi+l =−1 and ˆvi· ˆvi+l= +1 over the horizontal axis of the

plot, than that observed in laboratory frame. (The latter distribution would have a dominant higher population near the end of ˆvi· ˆvi+l = 1.) The data (Fig. 4) shows the observations in

the curved frames for various cases of the mixed systems. For systems with spring-connected monomers (Fig. 4(a)-(e)), the distributions change from isotropy (ﬂat distribution) toward anisotropy, in increasing the strength of the springs. The system with monomers connected by rigid bonds has the strongest anisotropy in its distribution (Fig. 4(f))among the six cases. In each case described in Fig. 4, there is the same trend in the distribution that that the tendency of the antiparallel situation (ˆvi· ˆvi+l =−1) and of the parallel situation (ˆvi· ˆvi+l= 1)

are equal probable, in increasing the distance index l. While the enhanced monomer-monomer connection reduce the longitudinal ﬂuctuations and forces the directions of motions of the monomer sites regulated by the backbones, the orientations of motions between two distantly separated sites along a chain are uncorrelated relative to their local curved frame. In other words, a long-range correlation is established by such a backbone regulated long distance isotropy along the chains. converging to some symmetric curve. A measure of degree of such a symmetry along the chains is carried out in Appendix.

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index distance l, between the side of parallel and that of antiparallel (having values 1 and -1 respectively) situations (see Appendix for such symmetries in reﬁned correlation distribution functions). To test how robust such observation, we collect the data for additional six systems, corresponding to those in Figs. 4 and A·2, with the angle parameters Kb and Kt replaced by 10−4. We take the functions fl with l = 16 as an approximation of the functions for long

range correlation and plot the data of f16 for all cases of Fig. 4, together with those for the six new systems, in one plot in Fig 5(a). It shows that the distributions are indeed form consistently a family of curves governed by the ﬁtted q, possessing the symmetry between parallel and antiparallel situations. We measure the deviation of the function f16 from the isotropic situation, which is fisotropic(x) = 0.5, for −1 ≤ x ≤ 1, by the quantity

f16− fisotropic2 ≡ 1

−1(f16(x) − fisotropic(x))

2_dx shown in Fig. 5(b).

The importance of the symmetry between parallel and antiparallel in fl to maintain the

stability of the ﬂoating curved frame is due to the fact that they signal the correlation property between the motions perpendicular to the backbone directions at distant location along a chain. The property suggests the chain reaches a kind of stochastic mechanical balance that the transverse motion at one position does not aﬀect the motion along the same transverse direction at a distant site. Indeed, whenever a major change is externally imposed, the system would intend to eliminate any unbalance and, after a transient period, reach a situation of dynamical balance signalled by the stochastic symmetry between the parallel and the antiparallel. In the Appendix, we show that the violation of this kind of symmetry does occur, for example, in presence of persistent external interference.

5. Discussion

We have revealed the dynamic correlations that are induced by the anisotropy of the back-bones of the chains with simulations on several well-deﬁned model systems. Using computer experiments we cannot exhaust all possible implicit factors of various systems or locations in phase space for one system, that may change the shape of the velocity distributions. Our analysis do help to propose the examination of the presence two families of symmetric

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pro-ﬁles in the probability density functions describing such correlations as a system-independent condition for the emerging of ‘quasi-steady’ state, where the monomer velocities are described by Tsallis statistics with only one single value for the parameter q.

In the studies of mixed systems with non-zero angle potentials,5, 6 we have observed that, in increasing the strengths of the springs, the correspondence between the increasing deviation of q from 1 and the trend of reduction of effective degrees of freedom in reaching equilibration with fluid suggests the dynamic anisotropy reduces the ability to exchange heat with fluid. It should be related to the restricted collective motion constrained by the orientation correlations. The increasing in the monomer-monomer bonding strength render larger effective body involve in the interaction with fluid molecules. The role played by the backbone connectivity is then the limitation on the the effective dynamic modes for heat exchanging between the chains and the fluid. The fast propagation of energy along the backbone of a strongly connected chain weakens its ability to exchange kinetic energy with the fluid.

It has been proposed that the statistical behavior over curved space would be modified by the underlying geodesics, leading to a deviation from Gaussian properties.15 The evaluation of the effective dimensions over the local frames in a stochastic manner may help to reveal exactly how the backbone curvature affect the statistics along a chain, and deserves further investigation.

In summary, the deviation from Maxwell-Boltzmann type is closely related to the dynamic correlations maintaining a balance under the anisotropy induced by the backbone curvature. The study may inspire some insight concerning the microscopic basis of unconventional sta-tistical mechanics.1, 7 The designing of well-controlled ‘quasi-steady’ relaxation processes in simulations may enable to reveal useful coarse-grained parameters that facilitate the eﬃcient studies of various complex collective properties that are relevant for biological or material applications.16

Acknowledgment

This work was supported by the National Science Council of the Republic of China (Tai-wan) under Grants No. NSC 96-2911-M 001-003-MY3 and No. NSC 101-2112-M-004 -001, National Center for Theoretical Sciences, and Academia Sinica (Taiwan) under Grant No. AS-95-TP-A07. We thank M. Doi and M. Parrinello for stimulating discussion.

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Fig. 4. (Color online) Distributions flof the cosine ˆvi· ˆvi+lfor l = 1, 2, 4, 8 and 16, observed in the

curved frame, for mixed systems described in Fig.1, with monomers along a chain connected by springs of strength (a) kspring= k0, (b) kspring= 10×k0, (c) kspring= 102×k0, (d) kspring= 103×k0

, (d) kspring= 104× k0; or by (f) a rigid bond. Each of the distributions is obtained over the same

snapshots as those used in Fig. 1. The dashed horizontal line marks the uniform distribution

f (ˆvi· ˆvi+l) = 0.5 for the isotropic situation. We also list the estimated value for the corresponding

q-parameter in the ﬁtting of velocity distributions. In (a), (b) and (c), we mark ‘MB’ for ‘

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Fig. 5. (Color online) (a) function fl, l = 16, and (b) its deviation fl− fisotropic2from the

distribu-tion for random uncorrelated situadistribu-tion versus q-parameter, for the 6 systems considered in Fig. 4 (solid lines in (a) and ﬁlled circles in (b)) and the corresponding mixed systems with the angle potential parameters Kb and Kt, both set to 10−4 (dashed lines in (a) and open circles in (b)),

obtained in the simulations described in Ref.13, 14For the latter systems with Kb=Kt=10−4, the

velocity distributions are Maxwell-Boltzmann type for kspring= k0 and kspring= 10k0); the ﬁtted

values in q for the remaining cases are 1.05 (for kspring = 102k0), 1.19 (for kspring= 103k0), 1.24 (for kspring= 104k0) and 1.25 (for rigidly bond), respectively.

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Mechancis, edited by S. Abe, M. Sakagami and N. Suzuki, Proceedings of International Workshop,

Prog. Theor. Phys. Suppl. 162 (2006).

8) D. Steele: J. Chem. Soc. Faraday Trans. 2 81 (1985) 1077. 9) H. C. Andersen: J. Comput. Phys. 52 (1983) 24.

10) In the cases the fitted q values less than 1.05, the difference between Eq. (3) and standard MB are within the numerical fluctuation of the simulation data. They are considered as Maxwell-Boltzmann type.6

11) H. J. Hilhorst and G. Schehr, J. Stat. Mech. (2007) P06003.

12) C. Vignat, A. Plastino, Phys. Lett. A 365 (2007) 370; C. Vignat, A. Plastino, Phys. Lett. A 343 (2007) 411; V.E. Bening, V.Yu. Korolev, Theory Probab. Appl. 49 (2005) 377.

13) W.-J. Ma and C.-K. Hu, J. Phys. Soc. Japan 79 (2010) 054001. 14) W.-J. Ma and C.-K. Hu, J. Phys. Soc. Japan 79 (2010) 104002.

15) P. H. Roberts and H. D. Ursell: Phil. Trans. Roy. Soc. London A 252 (1960) 317.

16) T. P. J. Knowles, J. F. Smith, A. Craig, C. M. Dobson and M. E. Welland: Phys. Rev. Lett. 96 (2006) 238301; T. P. J. Knowles, A. W. Fitzpatrick, S. Meehan, H. R. Mptt, M. Vendruscolo, C. M. Dobson and M. E. Welland: Science 318 (2007) 1900; H. D. Nguyen and C. K. Hall: Proc. Natl. Acad. Sci. 101 (2004) 16180; B. Urbanic, L. Cruz, S. Yun, S. V. Buldyrev, G. Bitan, D. B. Teplow and H. E. Stanley: Proc. Natl. Acad. Sci. 101 (2004) 17345.

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Fig. A·1. Quantities for the curves of Fig. 4, (a) f_l− f_l+12, measuring the diﬀerence between the curves of consecutive index distances, l and l + 1; (b) fl(x) − fl(−x)2, measuring the degree

of asymmetry in fl. The vertical axes are in log scales. While fls are insensitive to l for weakly

connected cases kspring ≤ 102k0, (Fig. 4(a)-(c)) and the quantities fl− fl+12 in (a) reﬂects

only the level of numerical noises, the diﬀerence between fl and fl+1 are prominent for strongly connected cases kspring≥ 103k0 (Fig. 4(d)-(f)) and fl− fl+12s are well above this level in the regime l < 8, where we can see the exponentially decreasing of fl− fl+12 in increasing l. Such

tendency of convergence is accompanied by the decreasing in the degree of asymmetry evident in (b), as l becomes larger.

Appendix A: Further analysis of anisotropy and correlation

The symmetry between the parallel and the antiparallel can be quantitatively described by Fig. A·1, where the diﬀerence between f_land fl+1, measured by the quantity

fl− fl+12≡

1

−1(fl(x) − fl+1(x))

2

dx,

(Fig. A·1(a)) decreases exponentially with increasing l (for the systems with kspring ≥ 103k0),

down to the regime where the numerical ﬂuctuations dominate (indicated by the data for the systems with kspring ≤ 102k0). The degree of asymmetry in the curve fl(x) between the two

opposite sides centered at x = 0, which is measured by

fl(x) − fl(−x)2≡

1

−1(fl(x) − fl

(−x))2dx,

decays with l to the level masked by numerical noises (Fig. A·1(b)).

To collect more information on how such correlation is induced by the backbone anisotropy, we carry out the analysis of the correlation between the two types of random variables ui= ˆvi·biand

ui,l= ˆvi· ˆvi+l, which are themselves variables for correlations and have been the focus of analysis so far. We intend to quantify the anisotropy, measuring the separation between the longitudinal

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Let hl and hl be the probability density functions for the two products uiui,l and (1− |ui|) ui,l,

respectively. We consider the quantities

sl(wl) = uiui,l=wl hl(uiui,l) f (ui)fl(ui,l)duidui,l (A·1) and s_l(w_l) =

(1−|ui|)ui,l=wl

h_l((1− |u_i|)u_i,l)

f (ui)fl(ui,l) duidui,l, (A·2) which contain the information of true correlations between ui and ui,l in removing the masks

of weights f (ui) and fl(ui,l) (Fig. 2 and Fig. 4). In increasing the bond strength, we found the

diﬀerence between the distributions sl(wl) and sl(wl) become pronounced. While the peak at

wl= 0 for sl(wl) grows, there are two emerging shoulders next to wl= 1 and wl=−1 in sl(wl),

respectively. The information suggest the correlation occurring mainly from the region surround 0 in variable ui, coupled to the neighborhoods of +1 and -1 in variable ui,l. Each of such correlations

contribute to one of the shoulders in s_l(w_l). The parallel-like ˆvi· ˆvi+l ≈ 1 and anti-parallel-like

ˆ

vi· ˆvi+l≈ −1 situations between the orientations of motions at site i and i + l are thus veriﬁed to

occur when their motions are transverse-bound with respect to backbone, i.e. ˆvi·bi≈ 0.

We can do the similar analysis that has been done in Fig. 5 for the distributions s16, by putting the

data of for all cases in Fig. A·2 and those for the six new systems, in one plot in Fig. A·3(a). The distributions are re-weighed in dividing by the corresponding distribution sisotropic of the totally

uncorrelated situation, so that the true magnitude of correlations can be displayed. We see again that a family of symmetric curves, regardless of the values of Kb and Kt, are governed by the

parameter q. The large amplitudes of the re-weighted curves around the ends of -1 and 1 along the horizontal axes reaﬃrm the coupling between the parallel and antiparallel situations with the backbone anisotropy revealed in the previous section. The deviation of s16from sisotropic for each

case is measured by s 16/sisotropic− 12≡ 1 −1(s 16(x)/sisotropic(x) − 1)2dx shown in Fig. A·3(b).

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Appendix B: Failure in reaching quasi-steady and asymmetry between par-allel and antiparpar-allel

In this section, we show that the features in the functions fl and sl are crucial to maintain the

systems to have a single q-values, i.e. in true ‘quasi-steady’ states. Figures B·1 and B·2 contain the time evolutions of instantaneous temperatures and the fitted parameters β and q for two systems with rigid bonds undergoing aggregation in the slow quenching processes13 and show the the difference in reaching the ‘quasi-steady’ (Fig. B·1) or not (Fig. B·2). One system (Fig. B·1) with Kb= Kt= 10−4 is mixed with L-J fluid, for which the analysis of fland slhas been carried

out in Figs. 5 and A·3 and the other (Fig. B·2) contains pure polymer chains with K_b= Kt= 0.0.

While the former system is in the ‘quasi-steady’, the latter does not have steady q and T∗β values.

Correspondingly, the functions fl and sl are highly asymmetric between the -1 and +1 ends for

the latter system (Figs. B·3(a) and (b)), in contrast to those for the former system (Figs. 4(f) and A·2(f)).

Such a kind of delicate correspondence between the shapes of the functions fl and sl, and the

presence of quasi-steady situations characterized by having steady values in q and T∗β is found

vulnerable to external temperature controlling. Indeed, in applying Gaussian thermostat14to the system of chains with rigid bonds (having their fl and sl described by Figs. 4(f) and A·2(f))

causes the value of q subject to large variations (Fig. B·4), upon the symmetric features in the two functions being destroyed (Figs. B·5(a) and (b)).

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Fig. A·2. (Color online) Functions s_land s_ldeﬁned by Eqs. (A·1) and (A·2), for l = 1, 2, 4, 8 and 16, for mixed systems described in Figs. 1 and 2, with monomers along a chain connected by springs of strength (a) kspring = k0, (b) kspring= 10× k0, (c) kspring = 102× k0, (d) kspring = 103× k0 ,

(d) kspring= 104× k0; or by (f) a rigid bond. Each of the distributions is obtained over the same

snapshots as those used in Fig. 1. The curves of sl and those of sl are virtually identical for the

MB cases, (a)-(c). In deviating from MB with increasing q, from (d) to (f), the functions slbecome

sharper around wl= 0 and those of sl are more evenly distributed to have expanding shoulders

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Fig. A·3. (Color online) (a) s_l, l = 16 divided by the distribution sisotropic and (b) its deviation

s

l− sisotropic2 from the distribution for random uncorrelated situation versus q-parameter, for

the 12 systems considered in Fig. 5, for the situation of no correlation. The latter is the distribution for the product between two independent random variables, uniformly distributed over the interval between -1 and 1. We again use diﬀerent line styles or symbols for those in Fig. A·2 (solid lines in (a) and ﬁlled circles in (b) ) and their counterparts with Kb= Kt= 10−4 (dashed lines in (a)

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Fig. B·1. Time evolution of the instantaneous temperature T∗, its product with the ﬁtted β parameter and the ﬁtted q parameter, for the mixed system of polymer chains, with Kb = Kt= 10−4 and

monomer-monomer bonds being rigid, for which the analysis of fl and sl has been carried out in

Figs. 5 and A·3.

Fig. B·2. Same as Fig. B·1, for a pure system of polymer chains, with K_b = Kt= 0 and

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Fig. B·3. (Color online) (a) f_l and (b) s_l, normalized as in Fig. A·2 (l=1,2,4,8,16), for the system described in Fig. B·2. Same as in Figs. 4 and A·2, each of the distributions is obtained by collecting data of 2000 snapshots, in intervals of 5 steps, over a simulation of period 0.25τ in time step

δt = 2.5 × 10−5τ . The interval starts at t=48.25.

Fig. B·4. Same as Fig. B·1, for the mixed system of polymer chains, with Kb = Kt = 0.1 and monomer-monomer bonds being rigid, which the same as the system in Fig. B·2, under the control of a Gaussian thermostat to impose T0∗ to a ﬁxed value.

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Fig. B·5. (Color online) Same as Fig. B·3, for the system described in Fig. B·4, collected over an interval starting at t =56.0.

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Abstract

The presence of the “market mode” in the collective movements of fluctuations of a large number of stocks suggests a way to describe the state of a stock market by macroscopic parameters. The concepts and the methods employed for the identification of macroscopic parameters in many-particle physics systems provide useful guideline for such issues. In this study, we show that the differentiation of the microscopic from the macroscopic time scales enable us to characterize the systems of collections of stocks by similar effective kinetic parameters, as their counterparts in material systems. We have embedded the correlation information revealed from market data by a stochastic model, that the fitted parameters be realized as the quantities for macroscopic state. Under the scenario, the cross correlation among stocks is realized as relating the characteristic time required to relax any unbalance in prices caused by external information flux. In the analysis of pools of stocks in US and Taiwan markets over the years 1996-1999, such characteristic times are found to depend on the time scale of observation. The underlying cause of such dependence seems quite robust and deserves further investigation.

PACS numbers: 89.65.Gh

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There are two different and complimentary ways to describe the state of a physical sys-tem consisting of a large number of molecules: microscopic or macroscopic. For example, to describe the the state of a system of N gas molecules in a box of volume V , one can use microscopic quantity such as distribution of the velocities of the molecules and the macroscopic quantities, such as volume V , pressure P , and temperature T . The connection between microscopic and macroscopic quantities of a system has been established via sta-tistical mechanics. For financial markets, such a connection is an issue of current interest, when more and more data bases are available providing microscopic details of the markets. It is essentially a non equilibrium problem, that dynamic properties play important roles. The comparison in the collective properties between stocks and particles is always inspiring. The concept of random walks, for example, had been raised to describe the time dependence of financial prices at a time before Einstein introduced the idea for the Brownian motion of particles. While the later scenario has been routinely used to describe motions of particles over time scales allowing coarsening of microscopic details, it is still a question how time scale dependence should be addressed in the analysis of financial time series. The purpose of the present paper is to show that the analogy between stocks and particles in their collective behaviors indeed allows us to reveal macroscopic information from pools of empirical data. Fluctuations in financial markets are important quantities of practical as well as academic interests. In 1900, Bachelier [1] proposed that fluctuations in financial market follow random walks. However, later studies indicate that fluctuations in stocks are not totally random. In 1966, King found that changes in prices of different stocks during time intervals of a day or longer are often highly correlated and the correlation is higher for firms in the same industry [2]. In 1977-1979, Epps studied correlations in log price for four major automakers in the United States-AMC (American Motors Corporation, 1954-1987), Chrysler, Ford, and GM during intervals of 10 minutes to three days [3]. Epps found that AMC has less correlation with other companies and the correlations in other three companies increases with the length of the time intervals used to calculate changes in log price [3]. This has been called “Epps effect”. Such dependence is understood as the time-scale dependence of the correlations [4, 5]. It contains the information on the degree of transaction synchronicity [4, 5], the lead-lag phenomena between pairs of stocks [6] and other important but less recognized

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and is approximately exponential, ensuring that (as one would expect for a price diﬀerence distribution) the variance of the distribution is ﬁnite, The scaling exponent is remarkably constant over the six-year period (1984-89).

In 1999, Laloux, et al. [8] calculated correlation matrix for 406 stocks in S & P 500 during 1991-1996 with time interval of one day and Plerou, et al. [9, 10] calculated correlation matrix for 1000 stocks in USA during 1994-1995 with time interval of 30 minutes. Both group found that in the eigenvalue distribution of the correlation matrix, there are some discrete large eigenvalues above the continuous component predicted by the random walk model for stocks. In the eigenvector corresponding to the largest eigenvalue of the correlation

matrix (λM), all stocks in the market move (deviate from the average value) in the same

direction. The mode corresponding to λM is called market mode.

In 2004, Ma, Hu and Amritkar [11] proposed a model of coupled random walks for stock-stock correlations [14]; the walks are coupled via a mechanism that the displacement (price change) of each walk (stock) is activated by the price gradients over some underlying network. They assumed that the network has two underlying structures: one for the correlations among the stocks of the whole market and another for those within individual groups; each with a coupling parameter controlling the degree of correlation. The model provides the interpretation of the features displayed in the distribution of the eigenvalues for the correlation matrix of real market on the level of time sequences. They veriﬁed that such modelling indeed gives good ﬁtting for the market data of US stocks.

In this paper, we extend the analysis to include two additional parameters, the diﬀusivity and the mobility, by the analogous comparison in their collective properties to a many-particle system.