Teng Liu1†, Gao-Ke Hu1†, Jia-Qi Dong2, Jing-Fang Fan1, Mao-Xin Liu3, and Xiao-Song Chen1*
1School of Systems Science/Institute of Nonequilibrium Systems, Beijing Normal University, Beijing 100875, China 2Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou 730000, China 3School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
Abstract：We propose a renormalization group (RG) theory of eigen microstates, which are introduced in the statistical ensemble composed of microstates obtained from experiments or computer simulations. A microstate in the ensemble can be considered as a linear superposition of eigen microstates with probability amplitudes equal to their eigenvalues. Under the renormalization of a factor $b$, the largest eigenvalue $\sigma_1$ has two trivial fixed points at low and high temperature limits and a critical fixed point with the RG relation $\sigma_1^b = b^{\beta/\nu} \sigma_1$, where $\beta$ and $\nu$ are the critical exponents of order parameter and correlation length, respectively. With the Ising model in different dimensions, it has been demonstrated that the RG theory of eigen microstates is able to identify the critical point and to predict critical exponents and the universality class. Our theory can be used in research of critical phenomena both in equilibrium and non-equilibrium systems without considering the Hamiltonian, which is the foundation of Wilson's RG theory and is absent for most complex systems.
Kadanoff L P 1971 Critical Phenomena, in Proceedings of the Enrico Fermi International School of Physics, Course LI, edited by Green M S (New York: Academic) p 100
2022, Vol. 39 
