1. Introduction
2. Eigen microstates and their evolutions
We define a microstate with fluctuations of all agents, which is represented by an N-dimensional vector [6]
where ${C}_{0}={\sum }_{t=1}^{M}{\sum }_{i=1}^{N}\delta {S}_{i}^{2}(t)$. The column order of A is in accord with the evolution of microstate.
With ${C}_{{tt}^{\prime} }$ as its elements, we can get an M × M correlation matrix of microstate
whose trace ${\rm{Tr}}{\boldsymbol{C}}={\sum }_{t=1}^{M}{C}_{{tt}}={C}_{0}$.
where i = 1, 2, ⋯ ,N.
With Kij as its elements, we can get an N × N correlation matrix of dynamic microstate
whose trace ${\rm{Tr}}({\boldsymbol{K}})={\sum }_{i=1}^{N}{K}_{{ii}}={C}_{0}$.
where Σ is an N × M diagonal matrix with elements
where $r={\rm{\min }}(N,M)$.
where ${{\boldsymbol{A}}}_{I}^{e}\equiv {{\boldsymbol{U}}}_{I}\otimes {{\boldsymbol{V}}}_{I}$ is an N × M matrix with elements ${\left({{\boldsymbol{A}}}_{I}^{e}\right)}_{{it}}={U}_{{iI}}{V}_{{tI}}$. We call the ensemble defined by ${{\boldsymbol{A}}}_{I}^{e}$ an eigen ensemble of system. From statistical ensemble, we can obtain not only eigen microstate UI, which has been introduced in [6], but also their evolution with time VI.
whose components
Therefore, we have
UI is the normalized eigen microstate. Between different eigen microstates, there is no correlation.
whose components
Further, we get
where ${{\boldsymbol{V}}}_{I}^{{\rm{T}}}$ is 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
where the Fourier coefficients are
which will be shown with respect to its period Tn = M/n.
2.2. Condensation of eigen microstate and phase transition
where $\beta$ is the critical exponent of order parameter and ν is the critical exponent of correlation length.
which has a fixed point at critical point t = 0.
3. Application to equilibrium system
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.
Using these microstates, we get the ensemble matrix A with elements
where C0 = M · N.
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. Applications to non-equilibrium systems
4.1. Earth system
The root mean square deviation at the grid i is
We introduce the microstates of the Earth as
where C0 = M · N.
Figure 10. Spatial distributions of the six largest eigen microstates of Earth. |
Figure 11. Evolutions of the six largest eigen microstates of Earth. |
4.1.1. First eigen microstate (EM1)
Figure 12. Fourier power spectrum density of evolution for the six largest eigen microstates of Earth. |
Figure 13. Seasonal cycle of EM1 evolution V1. |
4.1.2. Second eigen microstate (EM2)
4.1.3. Third eigen microstate (EM3)
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. |
4.1.4. Fourth eigen microstate (EM4)
Figure 18. (a) The amplitude of the semiannual oscillation (SAO). (b) The absolute amplitude of U4. |
4.1.5. Fifth eigen microstate (EM5)
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. |
4.1.6. Sixth eigen microstate (EM6)
4.2. Stock markets
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
where ${C}_{0}={\sum }_{t=1}^{M}{\sum }_{i=1}^{N}\delta {S}_{i}^{2}(t)$.
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. |
5. Conclusions
Appendix
Table A1. The 51 stocks, which have negative components in U1, are listed below with sectors in industry and values of component. |
Stock symbol | Industry | Value | |
1 | 600123.SH | Coal mining | −0.0156537 |
2 | 601699.SH | Coal mining | −0.0116389 |
3 | 600546.SH | Coal mining | −0.0116223 |
4 | 600256.SH | Petroleum mining | −0.0115809 |
5 | 600199.SH | Liquor | −0.0109279 |
6 | 600403.SH | Coal mining | −0.0101699 |
7 | 600348.SH | Coal mining | −0.0099700 |
8 | 600971.SH | Coal mining | −0.0097196 |
9 | 000937.SZ | Coal mining | −0.0083262 |
10 | 600691.SH | Pesticide fertilizer | −0.0081774 |
11 | 000157.SZ | Construction machinery | −0.0080799 |
12 | 000799.SZ | Liquor | −0.0079999 |
13 | 000983.SZ | Coal mining | −0.0078971 |
14 | 600508.SH | Coal mining | −0.0077700 |
15 | 601666.SH | Coal mining | −0.0077409 |
16 | 600997.SH | Coal mining | −0.0077055 |
17 | 600844.SH | Chemical raw materials | −0.0072676 |
18 | 600188.SH | Coal mining | −0.0070650 |
19 | 600031.SH | Construction machinery | −0.0067909 |
20 | 601001.SH | Coal mining | −0.0065050 |
21 | 600132.SH | Beer | −0.0058330 |
22 | 000968.SZ | Coal mining | −0.0054416 |
23 | 002069.SZ | Fishery | −0.0053891 |
24 | 000809.SZ | Regional real estate | −0.0053213 |
25 | 002106.SZ | Components | −0.0052831 |
26 | 600395.SH | Coal mining | −0.0048973 |
27 | 600381.SH | Medical care | −0.0044463 |
28 | 000869.SZ | Red and yellow wine | −0.0043872 |
29 | 601958.SH | Small metals | −0.0036315 |
30 | 600809.SH | Liquor | −0.0036107 |
31 | 600111.SH | Small metals | −0.0035444 |
32 | 002128.SZ | Coal mining | −0.0030434 |
33 | 000933.SZ | Coal mining | −0.0029023 |
34 | 600779.SH | Liquor | −0.0027581 |
35 | 600362.SH | Copper | −0.0026821 |
36 | 000960.SZ | Small metals | −0.0025779 |
37 | 600141.SH | Chemical raw materials | −0.0023876 |
38 | 002140.SZ | Construction engineering | −0.0023799 |
39 | 600740.SH | Coke processing | −0.0019607 |
40 | 600121.SH | Coal mining | −0.0019506 |
41 | 600375.SH | Automobiles | −0.0018485 |
42 | 000962.SZ | Small metals | −0.0018200 |
43 | 600547.SH | Gold | −0.0015524 |
44 | 002155.SZ | Gold | −0.0012866 |
45 | 600096.SH | Pesticide fertilizer | −0.0012306 |
46 | 000338.SZ | Auto parts | −0.0011826 |
47 | 600801.SH | Cement | −0.0005575 |
48 | 601088.SH | Coal mining | −0.0005393 |
49 | 600600.SH | Beer | −0.0005185 |
50 | 600875.SH | Electrical equipment | −0.0003222 |
51 | 600489.SH | Gold | −0.0001693 |
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. |
Stock symbol | Industry | Value | |
1 | 002069.sz | Fishery | 0.0550643 |
2 | 600096.sh | Pesticide fertilizer | 0.0528409 |
3 | 600121.sh | Coal mining | 0.0524364 |
4 | 002269.sz | Apparel | 0.0524142 |
5 | 600858.sh | Department stores | 0.0520945 |
6 | 000655.sz | General steel | 0.0518988 |
7 | 000809.sz | Regional real estate | 0.0517208 |
8 | 600509.sh | Thermal power generation | 0.0516920 |
9 | 600331.sh | Lead-zinc | 0.0514581 |
10 | 000962.sz | Small metals | 0.0514527 |
11 | 600644.sh | Hydropower | 0.0513471 |
12 | 002073.sz | Software services | 0.0513363 |
13 | 600397.sh | Coal mining | 0.0512892 |
14 | 300022.sz | Agricultural machinery | 0.0509761 |
15 | 600375.sh | Automobiles | 0.0508973 |
16 | 600333.sh | Gas and heating | 0.0504784 |
17 | 600778.sh | Department stores | 0.0504760 |
18 | 000862.sz | New electric power | 0.0501137 |
19 | 600166.sh | Automobiles | 0.0499687 |
20 | 600078.sh | Chemical raw materials | 0.0499670 |
21 | 600251.sh | Agricultural complex | 0.0499422 |
22 | 600307.sh | General steel | 0.0498540 |
23 | 600844.sh | Chemical raw materials | 0.0497993 |
24 | 000969.sz | Steel processing | 0.0496595 |
25 | 601005.sh | General steel | 0.0494886 |
26 | 600875.sh | Electrical equipment | 0.0493948 |
27 | 002140.sz | Construction engineering | 0.0492951 |
28 | 600403.sh | Coal mining | 0.0492887 |
29 | 600425.sh | Cement | 0.0487832 |
30 | 002225.sz | Other building materials | 0.0487225 |
31 | 600724.sh | Cement | 0.0484018 |
32 | 600395.sh | Coal mining | 0.0483786 |
33 | 600792.sh | Coke processing | 0.0483285 |
34 | 600381.sh | Medical care | 0.0482970 |
35 | 601001.sh | Coal mining | 0.0482963 |
36 | 600467.sh | Fishery | 0.0482299 |
37 | 600582.sh | Construction machinery | 0.0477340 |
38 | 000960.sz | Small metals | 0.0476099 |
39 | 000893.sz | Food | 0.0475348 |
40 | 601918.sh | Coal mining | 0.0474391 |
41 | 600616.sh | Red and yellow wine | 0.0469954 |
42 | 600058.sh | Business agency | 0.0468971 |
43 | 600537.sh | Semiconductor | 0.0467404 |
44 | 002029.sz | Apparel | 0.0467173 |
45 | 601958.sh | Small metals | 0.0467052 |
46 | 000557.sz | Railway | 0.0465262 |
47 | 000937.sz | Coal mining | 0.0464698 |
48 | 601898.sh | Coal mining | 0.0464407 |
49 | 600339.sh | Petroleum processing | 0.0461237 |
50 | 300029.sz | Special machinery | 0.0460201 |
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. |
Stock symbol | Industry | Value | |
1 | 000401.SZ | Cement | 0.0745595 |
2 | 000425.SZ | Construction machinery | 0.0733337 |
3 | 600231.SH | General steel | 0.0732387 |
4 | 600581.SH | General steel | 0.0711848 |
5 | 000877.SZ | Cement | 0.0710412 |
6 | 600456.SH | Small metals | 0.0709966 |
7 | 600729.SH | Department stores | 0.0704835 |
8 | 601899.SH | Gold | 0.0703957 |
9 | 601088.SH | Coal mining | 0.0698539 |
10 | 600132.SH | Beer | 0.0688951 |
11 | 000968.SZ | Coal mining | 0.0688016 |
12 | 002083.SZ | Textile | 0.0668737 |
13 | 600547.SH | Gold | 0.0668275 |
14 | 600859.SH | Department stores | 0.0659839 |
15 | 600295.SH | Small metals | 0.0656441 |
16 | 000898.SZ | General steel | 0.0650409 |
17 | 000528.SZ | Construction machinery | 0.0649777 |
18 | 601111.SH | Air transport | 0.0648821 |
19 | 600326.SH | Cement | 0.0642478 |
20 | 600449.SH | Cement | 0.0638847 |
21 | 600810.SH | Chemical fiber | 0.0633524 |
22 | 000597.SZ | Chemical pharmacy | 0.0631210 |
23 | 600720.SH | Cement | 0.0630507 |
24 | 000546.SZ | Cement | 0.0624129 |
25 | 000703.SZ | Chemical fiber | 0.0619837 |
26 | 000918.SZ | National real estate | 0.0611608 |
27 | 002128.SZ | Coal mining | 0.0610831 |
28 | 000933.SZ | Coal mining | 0.0609055 |
29 | 600362.SH | Copper | 0.0603090 |
30 | 000338.SZ | Auto parts | 0.0602836 |
31 | 600216.SH | Chemical pharmacy | 0.0602699 |
32 | 002110.SZ | General steel | 0.0601954 |
33 | 002092.SZ | Chemical raw materials | 0.0601682 |
34 | 600970.SH | Construction engineering | 0.0596964 |
35 | 000951.SZ | Automobiles | 0.0593858 |
36 | 600239.SH | Regional real estate | 0.0590865 |
37 | 601168.SH | Copper | 0.0585304 |
38 | 000825.SZ | Special steel | 0.0581011 |
39 | 600188.SH | Coal mining | 0.0578562 |
40 | 002088.SZ | Mineral products | 0.0578412 |
41 | 600308.SH | Papermaking | 0.0575344 |
42 | 600888.SH | Aluminum | 0.0573318 |
43 | 000617.SZ | Diversified finance | 0.0571679 |
44 | 600508.SH | Coal mining | 0.0571411 |
45 | 600997.SH | Coal mining | 0.0571142 |
46 | 601600.SH | Aluminum | 0.0567971 |
47 | 000822.SZ | Chemical raw materials | 0.0565928 |
48 | 601003.SH | General steel | 0.0560530 |
49 | 600068.SH | Construction engineering | 0.0560270 |
50 | 600569.SH | General steel | 0.0555972 |