Eigen microstates and their evolutions in complex systems

Yu Sun,Gaoke Hu,Yongwen Zhang,Bo Lu,Zhenghui Lu,Jingfang Fan,Xiaoteng Li,Qimin Deng and Xiaosong Chen,*

1 Institute of Theoretical Physics,Key Laboratory of Theoretical Physics,Chinese Academy of Sciences,PO Box 2735,Beijing 100190,China

2 School of Physical Sciences,University of Chinese Academy of Science,No.19A Yuquan Road,Beijing 100049,China

3 Beijing Computational Science Research Center,Beijing 100193,China

4 Data Science Research Center,Kunming University of Science and Technology,Kunming 650500,China

5 Laboratory for Climate Studies,National Climate Center,China Meteorological Administration,Beijing 100081,China

6 CAS Key Laboratory of Regional Climate Environment for Temperate East Asia,Institute of Atmospheric Physics,Chinese Academy of Sciences,Beijing 100029,China

7 School of Systems Science,Beijing Normal University,Beijing 100875,China

8 Harvest Fund Management,Beijing 100021,China

9 Lab for Climate and Ocean-Atmosphere Studies,Dept.of Atmospheric and Oceanic Sciences,School of Physics,Peking University,Beijing,100871,China

Abstract Emergence refers to the existence or formation of collective behaviors in complex systems.Here,we develop a theoretical framework based on the eigen microstate theory to analyze the emerging phenomena and dynamic evolution of complex system.In this framework,the statistical ensemble composed of M microstates of a complex system with N agents is defined by the normalized N×M matrix A,whose columns represent microstates and order of row is consist with the time.The ensemble matrix A can be decomposed aswhere r=min(N ,M),eigenvalue σI behaves as the probability amplitude of the eigen microstate UI so thatand UI evolves following VI.In a disorder complex system,there is no dominant eigenvalue and eigen microstate.When a probability amplitude σI becomes finite in the thermodynamic limit,there is a condensation of the eigen microstate UI in analogy to the Bose–Einstein condensation of Bose gases.This indicates the emergence of UI and a phase transition in complex system.Our framework has been applied successfully to equilibrium threedimensional Ising model,climate system and stock markets.We anticipate that our eigen microstate method can be used to study non-equilibrium complex systems with unknown orderparameters,such as phase transitions of collective motion and tipping points in climate systems and ecosystems.

Keywords:complex system,phase transition,critical phenomena,Earth system,statistical ensemble,eigen microstate,dynamic evolution,econophysics

1.Introduction

Statistical physics is a field of physics that studies the macroscopic behaviors of large collection of interacting objects based on their microscopic properties.Traditionally,these objects can be atoms,molecules,magnetic spins,electrons,quanta such as photons and phonons,et al.In recent decades,statistical physics has been extended to study complex systems consisting of many other types of interacting objects,such as individuals in populations,species in an ecosystem,agents in a market,computers in a network,et al.The studies of collective phenomena emerging from the interactions between individuals of social system are reviewed recently[1].With the use of ideals and mathematical methods developed in statistical physics to economic systems,a new interdisciplinary research field is established and called econophysics[2].The criticality and dynamical scaling of statistical physics are found in living systems also[3].The concept of statistical ensemble was introduced by Gibbs[4]in 1902 and serves as a starting point of statistical physics.At a certain time,the states of all individuals in a system can be represented by a point in the high dimensional phase space,which is called a microstate of the system.All microstates under a certain macroscopic condition constitute an ensemble of the system.The ensemble can be described by the probability density of microstate.The macroscopic properties of the system can be obtained by averaging over all microstates.Three important ensembles were defined by Gibbs[4]under different thermodynamic conditions.They are the micro-canonical ensemble,the canonical ensemble,and the grand canonical ensemble,respectively.The probability density of microstate in an ensemble is known for thermodynamic systems in equilibrium.All microstates in a microcanonical ensemble have the same probability.In the canonical ensemble with temperature T,the probability density of a microstate with energy E is proportional towhere kBis the Boltzmann constant.For the grand canonical ensemble with temperature T and chemical potential μ,the probability of a microstate with energy E and particle number N is proportional toIn principle,the thermodynamic properties of equilibrium systems can be calculated from these probability densities of microstate.

A large number of complex systems that need to be studied at present are generally not in equilibrium.Their probability densities of microstate are unknown.This is a huge challenge for studying the macroscopic properties of complex systems from the interaction between individuals.As pointed out recently in the commentary[5],the complexity at mesoscales of different levels arises from collective effects.The studies at mesoscales could be an essential step to tackle complex systems.We have proposed a method[6]to study the eigen microstates of complex system,which capture structures of mesoscales.According this method,we can get microstates and the ensemble of a complex system from its experimental investigations or computer simulations.In the original ensemble,microstates are not independent each other.Using the eigenvectors of the correlation matrix between microstates,eigen microstates can be obtained.In the ensemble composed of eigen microstates,there is no correlation between eigen microstates,which have weights proportional to the square of eigenvalue.For a complex systems without localization of microstate,the weight of an eigen microstate approaches zero in the thermodynamic limit.When the weight of an eigen microstate has a finite limit,there is then a condensation of the eigen microstate in the ensemble.This is similar to the Bose–Einstein condensation of Bose gas.The condensation of eigen microstate corresponds to a phase transition with new phase described by the eigen microstate and the eigenvalue as its order parameter.In the studies of phase transitions until now[46],order-parameter is served as the starting point.But the order-parameters of many complex systems are unknown at first.This bottleneck can be overcome by our eigen microstate method.

Complex systems in non-equilibrium are always in the course of evolution.We should study not only eigen microstates but also their evolution.Here we present the method to characterize the evolution of eigen microstate in section 2.The condensation of eigen microstate and its relation to a phase transition are discussed in section 2.2.To demonstrate the effectiveness of our method,we have studied the eigen microstates and their evolutions of the equilibrium threedimensional Ising model,the global temperature fluctuation of Earth,and the price fluctuation in the stock markets of Chinese mainland.Finally,we make some conclusions in section 5.

2.Eigen microstates and their evolutions

For a complex system composed of N agents,we can obtain the states of the agents from experimental measurements or computer simulations.Using the states at times t=1,2,…,M in sequence,we can get the state series Si(t)of agents i=1,2,…,N.

The average state of an agent i is

At a certain time t,the agent i has a fluctuation

We define a microstate with fluctuations of all agents,which is represented by an N-dimensional vector[6]

With the M microstates,we can compose a statistical ensemble of the complex system.This ensemble is described by an N×M matrix A with elements

As in[6],the correlation between the microstates at t andt′is defined by their vector product

WithCtt′as its elements,we can get an M×M correlation matrix of microstate

The statistical ensemble can also be considered as an ensemble of dynamic microstates of agent,which are described by M-dimensional vectors

where i=1,2,…,N.

In addition,we consider here the correlation between the dynamic microstates δSiand δSj

With Kijas its elements,we can get an N×N correlation matrix of dynamic microstate

The correlation matrix C has M eigenvectors VJof J=1,2,…,M.With them we can compose an M×M unitary matrix

The correlation matrix K has N eigenvectors UIof I=1,2,…,N.From these eigenvectors,we can obtain an N×N unitary matrix

According to the singular value decomposition(SVD)[7],the ensemble matrix A can be factorized as

where Σ is an N×M diagonal matrix with elements

wherer=min(N,M).

Furthermore,we can rewrite the ensemble matrix A as

FromTrC=C0orTrK=C0,we have the relation

Therefore,we consider σIas the probability amplitude andas the probability of the eigen ensemblein the statistical ensemble A.This is analogous to quantum mechanics,where a wave function can be written in terms of eigen functions.The square of the absolute value of expansion coefficient is the probability of the corresponding eigen microstate.

Using M eigenvectors of C,we can obtain eigen microstates[6]

whose components

Therefore,we have

UIis the normalized eigen microstate.Between different eigen microstates,there is no correlation.

We can express original microstates by eigen microstates as

Using N eigenvectors of K,we can get eigen dynamic microstates[8]

whose components

Further,we get

whereis the normalized eigen dynamic microstate.Different eigen dynamic microstates have no correlation each other and are independent.

2.1.Fourier spectrum analysis of eigen microstate evolution

The time evolution of the Ith eigen microstate is described byTo get more information about the evolution of eigen microstate,we expand VtIby a Fourier series

where the Fourier coefficients are

To characterize the proportion of different frequences,we introduce the Fourier power spectrum density

which will be shown with respect to its period Tn=M/n.

2.2.Condensation of eigen microstate and phase transition

In a statistical ensemble without localization of microstate,the weights of all eigen microstates are of the same order.In the limits M→∞and N→∞,all probability amplitudes σI→0.In this case,the system is in a random and disorder state.

If a probability amplitude σIbecomes dominant so that it has a finite limit at M→∞and N→∞,there is a condensation of eigen microstaten the statistical ensemble.This is similar to the Bose–Einstein condensation of Bose gases,where a finite proportion of infinite bosons share the lowest quantum state.This condensation indicates the appearance of a new phase in the Bose gas and there is a phase transition.The condensation of an eigen microstate manifests that a finite proportion of infinite eigen microstates share this eigen microstate.A new phase appears in the system.The new phase is characterized by the eigen microstate UI.The evolution of new phase is described by VI.

If σIincreases to finite continuously,the complex system experiences a continuous phase transition.With a jump of σIat transition point,there is a discontinuous phase transition.

For system with finite N,the limit of σIfor M→∞can be obtained with large enough M[6].For a continuous phase transition,the distance from critical point Tcis defined as t=(T-Tc)/Tc.In the asymptotic region with|t|?1,we propose the finite-size form of σI[6]as

where β is the critical exponent of order parameter and ν is the critical exponent of correlation length.

Correspondingly,the ratio σ2/σ1follows the finite-size scaling form[9]

which has a fixed point at critical point t=0.

In the bulk limit N→∞,we obtain the bulk order parameter

The eigen microstates and their condensation in one-and two-dimensional equilibrium Ising models have been investigated[8,6].Near and below critical point,the largest eigen microstate(EM1)U1becomes a cluster with spins of the same orientation.For the second largest eigen microstate(EM2)U2,there are two clusters which have opposite orientation.The finite-size scaling of probability amplitude in equation(26)has been confirmed by Monte Carlo simulation data of oneand two-dimensional Ising models[6].

In the next section,we will study the condensation of eigen microstate and their evolution in equilibrium threedimensional Ising model by Monte Carlo simulations.To demonstrate that our method presented above can be used to general complex systems,the eigen microstates of the Earth system and stock markets will be explored in section 4.

3.Application to equilibrium system

We consider a three-dimensional Ising model with square geometry and periodic boundary conditions.The Hamiltonian of N=L×L×L spins is

where Si=±1 and〈i,j〉means a summation over all spins in the nearest neighbourhood.The model is characterized by the system size L and the reduced temperature T*=kBT/J.We can obtain microstates of the system using the Wolff algorithm[10],which flips a cluster of spins rather than a single spin.

In a finite Ising system,there is no symmetry breaking.Therefore〈Si〉=0 and δSi(t)=Si(t).The microstate at time t is

Using these microstates,we get the ensemble matrix A with elements

where C0=M·N.

In figure 1(a),the largest probabilityw1=is plotted with respect to T*.At higher temperatures,w1is very small and comparable to other probabilities.With the decrease of temperature,w1increases faster and becomes dominant.This indicates the emergence of a new phase and there is a phase transition.

The finite-size scaling form of w1is presented in figure 1(b).To calculate the scaling variable,the critical temperature?4.5114[11]and the critical exponent of correlation length ν=0.6335[12]are used.Our simulation data confirm the finite-size scaling form of w1.

The second largest probability w2and its finite-size scaling form are shown in figure 2.Near the critical pointw2becomes finite and its finite-size scaling form is confirmed also.

In figure 3(a),the ratiois plotted with respect to T*.A fixed point is found at the critical temperatureThe finite-size scaling form of w2/w1is presented in figure 3(b).The finite-size scaling form of eigenvalues has been verified completely in three-dimensional Ising model also.This is in accord with the finite-size scaling of correlation function[13].

We identify the character of phase transition by studying the spatial distribution of eigen microstates UI.For an eigen microstate in three-dimensional space,four sectional drawings with equal distance are presented.In figure 4,we show three eigen microstates U1,U2,and U3at reduced temperature T*=5.5116,which is above the critical temperature.The clusters of spins in these eigen microstates are of micro size and distributed randomly in space.

At temperatures near and below critical point,the largest eigen microstate(EM1)becomes a single cluster with the system size.The results are presented in figures 5 and 6.A ferromagnetic phase appears when the probability of the eigen microstate becomes finite.

The evolutions of eigen microstates around critical point are shown in figures 7–9.In relation to the dynamics of the Wolff algorithm[10],VtIfluctuates around zero with time t.

4.Applications to non-equilibrium systems

4.1.Earth system

Because of nonlinear interactions and feedback loops between and within geosphere,atmosphere,hydrosphere,cryosphere,and biosphere,the Earth operates as a complex system.The statistical physics,that studies the behaviors of large collections of interacting objects,has been applied to the complex Earth system[14].Until now,many concepts and methods in statistical physics such as correlation,synchronization,network and percolation theory,tipping points analysis et al have been used.Here we use the statistical ensemble,which is the foundation of statistical physics,to study complex Earth system.It is hoped that the statistical physics of complex Earth system will be established gradually.

As one of the most important climate phenomena on Earth,the El Ni?o and Southern Oscillation(ENSO)has been studied with network and its percolation[15,16].We consider here the Earth as a non-equilibrium complex system composed of many latitude-longitude grids.In this complex system,there are N=94×192=18 048 grids with 1.9°×1.875° latitude-longitude.Using a part of these grids,their principal modes are investigated recently[17].Here the microstates of the Earth as a whole are studied based on daily surface air temperature(SAT)(2 m)provided by the National Centers for Environmental Prediction-National Center for Atmospheric Research(NCEP-NCAR)Reanalysis[18].The dataset spans the time period between 01-January-1950 and 31-December-2018.Both the El Ni?o and the La Ni?a events are defined by the Oceanic Ni?o Index(ONI)from the National Oceanic and Atmospheric Administration.The ONI is defined as 3 month running mean of ERSST.v5 SST anomalies in the Ni?o 3.4 region(5°S–5°N,170°W–120°W).

At a certain time t,the SAT at a grid i is Ti(t).The average SAT at the grid i during a time period M can be calculated as

The fluctuations of the SAT at the grid i are

The root mean square deviation at the grid i is

It is better to characterize the fluctuations of SAT by

We introduce the microstates of the Earth as

Using the dataset spanning from 01-January-1950 to 31-December-2018,we can get an N×M ensemble matrix A with elements

where C0=M·N.

We present the eigen microstates UIof I=1,2,…,6 in figure 10.Their evolutions VtIare shown in figure 11.In the following,we will discuss these eigen microstates in detail and their relevance to climate phenomena especially.

4.1.1.First eigen microstate(EM1).As presented in figure 10(a),EM1 reveals a classic solstitial mode and contributes 54.38% of the total variance.Its spatial pattern exhibits an inter-hemisphere contrast.This asymmetry is roughly a mirror image along the equator,although the slight shift is evident over the African,South American and Eastern Pacific sections.The spectrum analysis(figure 12(a))of the corresponding time series(figure 11(a))shows a robust peak at 1 year,following the annual variation of the solar zenith angle.Further results(figure 13)show that the maximum of EM1 often occurs in August and the minimum of EM1 occurs in Late January,indicating 1 month delay of air temperature after the solar forcing.

4.1.2.Second eigen microstate(EM2).Figure 10(b)demonstrates the spatial pattern of EM2,which contributes to 11.36% of the total variance.Similar to EM1,the obvious period of 1 year is detected,implying the regular annual variation of this mode.A remarkable land-sea contrast is observed in EM2.In the southern(northern)hemisphere,the SAT warms up(cools down)faster over the continents than the oceans due to the land-sea thermal capacity difference.It should be noted that this thermal contrast in the tropicalsubtropical regions drives the local monsoons over the South Asia[20],Australia[21],West Africa[22],North America[23],and South America[24].Wang et al[19]proposed the concept of global monsoon system by considering the seasonal contrast of precipitation globally.It is interesting that the domain with the spatial absolute intensity above 40%of the maximum aboslute intensity in EM2 resembles the global monsoon area as presented in figure 14,with the spatial matching degree reaching 70%.(Since the land-sea thermal contrast is not the only cause of East Asia monsoon,the monsoon area near East Asia is not very matched[25]).We further compare the difference between summer and winter averages of V2with the global monsoon index(figure 15).Again,a significant correlation with R=0.52 is obtained.The above evidences reveal that EM2 represents the land-sea contrasts of SAT which is regarded as the primary driver for monsoon.

Figure 1.(a)Weight w1 of the largest eigen ensemble matrix with respect to reduced temperature T*=kBT/J.(b)Finite-size scaling form of w1 with respect to tL1/ν,where t=(T-Tc)/Tc.

Figure 2.(a)Weight w2 with respect to reduced temperature T*=kBT/J.(b)Finite-size scaling form of w2 with respect to tL1/ν,where t=(T-Tc)/Tc.

Figure 3.(a)Ratio w2/w1 with respect to reduced temperature T*=kBT/J.(b)Finite-size scaling form of w2/w1 with respect to tL1/ν,where t=(T-Tc)/Tc.

Figure 4.Four sectional drawings with equal distance of eigen microstates U1,U2,and U3 at reduced temperature T*=5.5116.

Figure 5.Four sectional drawings with equal distance of U1,U2,and U3 at T*=4.5116.

Figure 6.Four sectional drawings with equal distance of U1,U2,and U3 at T*=3.5116.

Figure 7.Evolutions of the largest three eigen microstates at T*=5.5116.

Figure 8.Evolutions of the largest three eigen microstates at T*=4.5116.

Figure 9.Evolutions of the largest three eigen microstates at T*=3.5116.

4.1.3.Third eigen microstate(EM3).Figure 10(c)demonstrates the spatial pattern of EM3,which accounts for 3.08% of the total variance.It is featured as the climate fluctuation within the tropics.Specifically,the pronounced variations are observed over the tropical Indian Ocean,tropical Pacific Ocean and northern tropical Atlantic Ocean.As shown in figure 16,This pattern is anchored by the sea surface temperature(SST).For example,the climatology SST stays north of the equator over the Atlantic and eastern Pacific Oceans due to the wind-evaporation-SST loop feedback[26],which is well captured by the SAT pattern in EM3(figure 10(c)).Given the fact that the tropical convection is largely coupled with the local SST,EM3 pattern is also able to reveal the tropical rainfall.For example,the intertropical convergence zone north of equator over the eastern Pacfic and Atlantic Oceans[27],as well as the wide South Pacific convergence zone[28];also see in figure 17),are represented by EM3.The spectrum of EM3 is shown in figure 12(c).The periods of both 0.5 year and 1 year are evident.The 0.5 year period of EM3 follows the maximum solar forcing due to the Sun crossing the equator twice a year,which is evident over the Indian Ocean.Meanwhile,due to the surface wind adjustment[29],the SAT and SST in the equatorial eastern Pacific and Atlantic exhibit a distinct annual cycle despite of the occurrences of the two equinoxes,which explains the 1 year period of EM3.Figure 11(c)shows the time series of EM3.A robust upward trend is evident,implying the potential impact of global warming on the tropical SST and convective precipitation.It is interesting that a tipping point in mid-1970s is detected by EM3.This is consistent with the well-known 1976–1977 climate shift[30].

4.1.4.Fourth eigen microstate(EM4).Figure 10(d)shows the spatial pattern of EM4,which explains 2.13%of the total variance.As seen in figure 10(d),the distribution of significant temperature anomaly is mainly located in the tropics and high latitudes.The EM4 evolution series(figure 11(d))is dominated by semiannual signal as the power spectrum showing(figure 12(d)).In meteorology,this semiannual signal is called semiannual oscillation(SAO),which is component of the annual cycle of a variable that consists of a sinusoidal oscillation with a period of six months[31].Strong semiannual signals in thermal and momentum fields of the troposphere are found in both the Tropics and in Southern Hemisphere middle and high latitudes.The SAO is detectable as a significant contribution of the second harmonic to the annual march of temperature,SAT,wind speed et al[32–35];here,we applied a harmonic analysis method called nonlinear mode decomposition to extract its second harmonic wave and showed its amplitude as seen in the figure 18(a),which is highly consistent with the results of figure 18(b),especially in Antarctic region.Previous studies have pointed out that the mechanism of SAO in Antarctic is mainly attributed to different annual cycles of temperature in the mid-latitude ocean and Antarctic regions with the complex thermodynamics and dynamics of the oceanatmosphere system[36–38].

Figure 10.Spatial distributions of the six largest eigen microstates of Earth.

4.1.5.Fifth eigen microstate(EM5).The spatial pattern of EM5 is given in figure 10(e),accounting for 1.34% of the total variance.The most remarkable feature is evident over the tropical Pacific Ocean,with the warming over the central to eastern Pacific and cooling over the western Pacific.This pattern resembles the ENSO,which is the strongest climate fluctuation at interannual time scale[39].Spectrum analysis is conducted(figure 12(e)).Other than the semi-annual cycle,the broad spectrum is evident at interannual time scale from 2 to 7 years in EM5,which is consistent with ENSO spectrum.In figure 19 the time series of EM5 is compared with the observed ONI.Significant correlation of R=0.89 is obtained after averaging U5over 90 days.Three super El Ni?o events in 1982/1983,1997/1998 and 2015/2016 are all detectable by EM5,further confirming the linkage between EM5 and ENSO.Further results(figure 20)show that the maximum of EM5 often occurs in December and its minimum occurs in June after removing the semi-annual and higher frequency component.

4.1.6.Sixth eigen microstate(EM6).Figure 10(f)illustrates the spatial pattern of EM6,explaining 1.06% of total variance.In the tropical Pacific Ocean,a cooling is observed in the central basin,with the warming in the western and southeastern basin.This pattern resemble the El Ni?o Modoki[40],which is distinct from the conventional El Ni?o as detected by EM5 in terms of the dynamics and teleconnections[41].In the tropical Indian Ocean,a dipole structure is evident,with warming off Sumatra coast and cooling over tropical northwestern Indian Ocean.This pattern reflects the well-known Indian Ocean Dipole mode[42].In the North Atlantic Ocean,a positive-negative-positive pattern is obtained from the equator to the mid-latitude,indicating the so-called North Atlantic SST tripole mode[43].In addition,the warm SAT is evident over mid-latitude Oceans.

Figure 11.Evolutions of the six largest eigen microstates of Earth.

Figure 12.Fourier power spectrum density of evolution for the six largest eigen microstates of Earth.

Figure 13.Seasonal cycle of EM1 evolution V1.

Figure 14.(a)The global monsoon area derived by Wang et al[19].(b)The domain with the spatial absolute intensity above 40%of the maximum absolute intensity in U2.

Figure 15.(a)The GMI calculated by Wang et al[19].(b)The difference between summer and winter averages of V2.

Figure 16.Annual mean sea surface temperature(SST)pattern during 1950–2018.Note that the SST below 26C is removed for clarity.

Figure 17.Annual mean precipitation pattern during 1979–2018.

Figure 18.(a)The amplitude of the semiannual oscillation(SAO).(b)The absolute amplitude of U4.

Figure 19.Comparison between the ONI(red)and the average of V5 over 90 d and with a running step of 30 d(black).

Figure 20.Seasonal cycle of the evolution V5 with semi-annual and higher frequency components filtered.

Figure 21.The largest three eigen microstates U1,U2 and U3 with respect to stocks.

Figure 22.Evolutions of the largest three eigen microstates V1,V2 and V3.

Figure 23.Comparison between the evolutions of V1(black line)and the SSE100 index(red line).

Figure 24.Comparison between the evolutions of V2(black line)and the SSE Energy Sector Index(red line).

Figure 25.U2 with respect to stocks in the order of components.The stocks of the SSE Energy Index are marked by red dots.

Figure 26.Comparison between the evolutions of V3(black line)and the SSE materials sector index(red line).

Figure 27.U3 with respect to stocks in the order of components.The stocks of the SSE Material Index are marked by red dots.

The rest of eigen microstates take the proportions of the total variance less than 1%.Their spatial distributions vary in even smaller length scales.

4.2.Stock markets

The mutual fertilization between economics and physics has existed for a very long time[47].As an example,the random matrix theory in physics has been used to investigate financial price fluctuations[48].The global stock market is considered as a complex system and its principal fluctuation modes were studied[49].

The fluctuations of stock prices are results of interactions of multiple human and non-human factors in real life.Stock price is a direct indicator of the average expectation of an asset’s value asserted by all its investors.Since the limitation of information,judgements to stock prices are insufficient.Consequently,value distributions of stock formed naturally enable trade process to match up bid orders and ask orders.Once an investor reallocates his/her portfolio,cashes are transferred from other bearish assets to some bullish assets.Under such reallocation actions,different assets are correlated each other.Collective behaviors of investors can result in fluctuation patterns of stock prices.To some degree,these patterns reflect developments of related industries and companies.

Here we study the eigen microstates of stock markets to analyze the fluctuation patterns of stock prices.We take the stock price data from 04-January-2010 to 26-May-2020 in Chinese mainland for investigations.The stock price data comes from the public data published by Shanghai Stock Exchange and Shenzhen Stock Exchange within the period from 04-January-2010 to 26-May-2020.In the dataset,there are 1600 stocks with prices of 2525 trading days.After removing the impaired data of 140 stocks,there are 1460 stocks left.

For financial markets,composite indexes are constructed to describe their global trends.The SSE 100 Index is a broadbased index that reflects the performance of the entire Shanghai Stock Exchange market.It is composed of 100 sample stocks which have rapid operating income growth and high return on equity in the market.The SSE sector indices reflect the performances of 10 industry sector:the energy sector,the materials sector,the industrials sector and etc.Two important indices of them are the SSE energy sector index and SSE materials sector index.

The price of a stock i at a time t is denoted by Pi(t).The average price of the stock i during a specified period can be calculated as

Here,M refers to the total number of trading days.At the time t,the stock i has the price fluctuation

The root mean square fluctuation of the stock i is

The state of a stock i at a time t is characterized by the reduced fluctuation

The microstate of the stock markets with N stock prices is described by the vector

From these vectors,we can get the ensemble matrix A with elements

This ensemble matrix can be decomposed into a set of eigen microstates by the SVD.The largest three eigen microstates EM1,EM2,and EM3 are responsible for 48.78%,19.2% and 8.54% of the total variance respectively.Their evolutions are shown in figure 22.The components of an eigen microstate indicate the involvements of the corresponding stocks in the evolution of this eigen microstate.In figure 21,the components of U1,U2,and U3are shown with respect to the stocks in the order of their components.

The EM1 evolution V1resembles the SSE100 Index perfectly with a Pearson correlation coefficient R=0.95,as shown in figure 23.The sharp volatility during the stock market crash in 2015 can be observed both in V1and the SSE100 Index.As shown in figure 21(a),there are overwhelming positive components in V1and only 51 stocks have negative components.V1characterizes the main trend of the stock markets.There are 51 stocks that evolute against this trend.It can be found in the appendix table A1 that the 51 stocks come mostly from the section of coal mining.

The evolution of EM2 is shown in figure 24,together with the SSE energy sector index.The Pearson correlation coefficient between them is R=0.84.Except for the stock market crash in 2015,V2declines from 2010 to 2020.In U2,About half of stocks have positive components.The top 50 stocks with positive components are listed in the appendix table A2.They are closely related to the energy sector,such as petroleum,coal and related industries.With the stocks of the SSE energy sector index marked by red dots,U2is presented in figure 25.The red dots have the largest positive components.

We present the evolution of EM3 in comparison with the SSE materials sector index in figure 26.They are correlated with a Pearson correlation coefficient R=0.75.The top 50 stocks of U3are listed in the appendix table A3.They come mostly from the SSE materials sector,such as steel,gold,copper,cement,etc.In figure 27 of U3,the stocks of the SSE materials sector index are marked by red dots.It can be seen that the EM3 is dominated by the stocks of the SSE materials sector index.

In conclusion,the eigen fluctuation patterns in stock markets have been captured by the eigen microstates above.EM1 describes the general trend of the stock market.EM2 and EM3 are related to the evolutions of the energy and material sectors in the stock market.The hierarchical features of stock market have been exhibited by the eigen microstates.

5.Conclusions

We provide a framework to study emergence behaviors in equilibrium and non-equilibrium complex systems.For a complex system with N agents,we can compose a statistical ensemble with M microstates described by N-dimensional vectors with components defined by fluctuations of agents.The ensemble is described by an N×M normalized matrix A.We can decompose the matrix aswhereand UIis the Ith eigen microstate with evolution described by VI.For a complex system in disorder phase,all eigenvalues σI→0 in the limits N→∞,M→∞.If the limit of σIbecomes finite,there is a condensation of the eigen microstate UIin analogy to the Bose–Einstein condensation of Bose gases.This indicates a phase transition of the complex system with the order parameter given σIand the new phase described by UI.Our method can be applied to both equilibrium and non-equilibrium complex systems.

For the three-dimensional Ising model in equilibrium,we have studied its eigen microstates and the corresponding eigenvalues in different temperatures.With the decrease of temperature,the largest microstate U1of the Ising model changes from disorder to ferromagnetic.The eigenvalue σ1becomes finite also.This indicates that there is a condensation of ferromagnetic microstate and a ferromagnetic phase transition.Our Monte Carlo simulation data have confirmed that the eigenvalues σIfollow a finite-size scaling for temperatures near the critical point.

The eigen microstates of non-equilibrium complex systems including the Earth system and stock markets are studied.For the Earth system,the temperature microstates between 01-January-1950 and 31-December-2018 have been investigated.U1contributes about 54% of the total temperature variance.It reveals a classic solstitial mode,which reflects the impact of anti symmetric annual solar forcing with a one-to-two-month phase delay in the atmospheric response.Its spatial distribution exhibits an inter-hemisphere contrast.U2contributes about 11%of the total temperature variance.It reveals a remarkable land-sea temperature contrast resulted by the land-sea thermal capacity difference.We have shown that U2is correlated with monsoons.U3is featured as the climate fluctuation within the tropics.It has a dominant period of 0.5 year,which is resulted by the maximum solar forcing due to the Sun crossing the equator twice a year.We find a tipping point of V3in consistence with the well-known 1976–1977 climate shift.U4is mainly located in the tropics and high latitudes.V4is dominated by a semiannual signal,which is related to the so-called SAO in meteorology.U5is related to the ENSO,and its evolution V5resembles the ONI.In addition,we have studied the microstates of stock markets in Chinese mainland with the stock data from 04-January-2010 to 26-May-2020.The first eigen microstate resembles the SSE100 Index with a Pearson correlation coefficient R=0.95.The second and third eigen microstates are correlated with the SSE Sector Indices of energy and materials,respectively.

This eigen microstate method can be applied to other non-equilibrium complex systems[44,45].It is expected that phase transitions of collective motion[50,51]and tipping points in climate systems[52]and ecosystems[53]can be studied by the eigen microstate method.

Acknowledgments

This work was supported by the Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.QYZD-SSW-SYS019)and the HPC Cluster of ITP-CAS.We also acknowledge Naiming Yuan and Jiaqi Dong for discussions.

Appendix

Table A1.The 51 stocks,which have negative components in U1,are listed below with sectors in industry and values of component.

Table A2.List of the top 50 stocks with positive components of U2 and sectors in industry.They are closely related to the energy sector of industry.

Table A3.List of the top 50 stocks with positive components of U3 and sectors in industry.They are closely related to the raw materials sector of industry.