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重整化群理论通过在实空间可用的方法对微观细节进行形式化平均，从而定义了任意尺度下的有效自由度与相互作用，成为统计力学的重大突破。Kenneth Wilson后续将其发展为连续场论中的重整化方法<ref name="Wilson_renormalization_1974">{{cite journal|author1=Wilson, K. G.|author2=Kogut, J.|title=The renormalization group and the $\epsilon$ expansion|journal=Phys. Rep.|year=1974|volume=12|url=}}</ref><ref name="Castro_scaling">{{cite journal|author1=Castro C|author2=Jona-Lasinio G.|title=On the microscopic foundation of scaling laws|journal=Phys. Lett. A|year=1969|volume=29}}</ref>。Wilson的工作不仅为理解临界现象的标度行为奠定理论基础，更将场论中处理名义发散量的零散方法系统化。这些发散性源于数学上为解决任意小尺度问题的理想化操作，虽然研究者们对此曾提出减除方案<ref name="Gell-Mann_electrodynamics">{{cite journal|author1=Gell-Mann M|author2=Low F. E.|title=Quantum electrodynamics at small distances|journal=Phys. Rev.|year=1954|volume=95}}</ref>，但该方案在重整化群理论建立前缺乏概念基础。此后，重整化群方法催生了“理论之理论”，揭示微观细节迥异的物理系统在临界点附近如何出现普适行为<ref name="Zinn-Justin_field">{{cite journal|author1=Zinn-Justin J.|title=Quantum Field Theory and Critical Phenomena|journal=Oxford Univ. Press|year=2021}}</ref><ref name="Amit_field">{{cite journal|author1=Amit D. J|author2=Martin-Mayor V.|title=Field Theory, the Renormalization Group and Critical Phenomena|journal=World Scientific|year=2005}}</ref><ref name="Täuber_renormalization">{{cite journal|author1=Täuber U. C.|title=Renormalization group: applications in statistical physics|journal=Nucl. Phys. B|year=2012|volume=228}}</ref><ref name="Pelissetto_critical">{{cite journal|author1=Pelissetto A|author2=Vicari E.|title=Critical phenomena and renormalization-group theory|journal=Phys. Rep.|year=2002|volume=368}}</ref><ref name="Efrati_real-space">{{cite journal|author1=Efrati E|author2=Wang Z|author3=Kolan A|author4=Kadanoff L. P.|title=Real-space renormalization in statistical mechanics|journal=Rev. Mod. Phys.|year=2014|volume=86}}</ref>。
重整化群理论通过在实空间可用的方法对微观细节进行形式化平均，从而定义了任意尺度下的有效自由度与相互作用，成为统计力学的重大突破。Kenneth Wilson后续将其发展为连续场论中的重整化方法<ref name="Wilson_renormalization_1974">{{cite journal|author1=Wilson, K. G.|author2=Kogut, J.|title=The renormalization group and the $\epsilon$ expansion|journal=Phys. Rep.|year=1974|volume=12|url=}}</ref><ref name="Castro_scaling">{{cite journal|author1=Castro C|author2=Jona-Lasinio G.|title=On the microscopic foundation of scaling laws|journal=Phys. Lett. A|year=1969|volume=29}}</ref>。Wilson的工作不仅为理解临界现象的标度行为奠定理论基础，更将场论中处理名义发散量的零散方法系统化。这些发散性源于数学上为解决任意小尺度问题的理想化操作，虽然研究者们对此曾提出减除方案<ref name="Gell-Mann_electrodynamics">{{cite journal|author1=Gell-Mann M|author2=Low F. E.|title=Quantum electrodynamics at small distances|journal=Phys. Rev.|year=1954|volume=95}}</ref>，但该方案在重整化群理论建立前缺乏概念基础。此后，重整化群方法催生了“理论之理论”，揭示微观细节迥异的物理系统在临界点附近如何出现普适行为<ref name="Zinn-Justin_field">{{cite journal|author1=Zinn-Justin J.|title=Quantum Field Theory and Critical Phenomena|journal=Oxford Univ. Press|year=2021}}</ref><ref name="Amit_field">{{cite journal|author1=Amit D. J|author2=Martin-Mayor V.|title=Field Theory, the Renormalization Group and Critical Phenomena|journal=World Scientific|year=2005}}</ref><ref name="Täuber_renormalization">{{cite journal|author1=Täuber U. C.|title=Renormalization group: applications in statistical physics|journal=Nucl. Phys. B|year=2012|volume=228}}</ref><ref name="Pelissetto_critical">{{cite journal|author1=Pelissetto A|author2=Vicari E.|title=Critical phenomena and renormalization-group theory|journal=Phys. Rep.|year=2002|volume=368}}</ref><ref name="Efrati_real-space">{{cite journal|author1=Efrati E|author2=Wang Z|author3=Kolan A|author4=Kadanoff L. P.|title=Real-space renormalization in statistical mechanics|journal=Rev. Mod. Phys.|year=2014|volume=86}}</ref>。
过去五十年间，重整化群技术已扩展至从最大尺度的宇宙学到最小尺度的粒子物理等广泛领域。在统计力学范畴内，重整化群方法已推广至同质均匀的分形格点上的非平衡系统<ref name="Gefen_fractal">{{cite journal|author1=Gefen Y|author2=Mandelbrot B. B|author3=Aharony A.|title=Critical phenomena on fractal lattices|journal=Phys. Rev. Lett.|year=1980|volume=45}}</ref><ref name="Gefen_backbone">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B|author4=Kirkpatrick S.|title=Solvable fractal family, and its possible relation to the backbone at percolation|journal=Phys. Rev. Lett.|year=1981|volume=47}}</ref><ref name="Phani_percolation">{{cite journal|author1=Phani M. K|author2=Dhar D.|title=Real-space renormalisation group: application to directed percolation|journal=J. Phys. C|year=1982|volume=15}}</ref><ref name="Gefen_phase1">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. I. Quasi-linear lattices|journal=J. Phys. A|year=1983|volume=16}}</ref><ref name="Gefen_phase2">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Shapir Y|author4=Mandelbrot B. B.|title=Phase transitions on fractals. II. Sierpinski gaskets|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Gefen_phase3">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. III. Infinitely ramified lattices|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Das_critical">{{cite journal|author1=Das D|author2=Dey S|author3=Jacobsen J. L|author4=Dhar D.|title=Critical behavior of loops and biconnected clusters on fractals of dimension d \< 2|journal=J. Phys. A|year=2008|volume=41}}</ref>，其局部平移不变性能够简化计算。超越初始应用的重要范例包括：流行病动力学模型<ref name="Della_SIR">{{cite journal|author1=Della Morte M|author2=Sannino F.|title=Renormalization group approach to pandemics as a time-dependent SIR model|journal=Front. Phys.|year=2021}}</ref>、生物群体行为动力学<ref name="Cavagna_scaling">{{cite journal|author1=Cavagna A.|title=Dynamic scaling in natural swarms|journal=Nat. Phys.|year=2017|volume=13}}</ref><ref name="Cavagna_swarms">{{cite journal|author1=Cavagna A.|title=Dynamical renormalization group approach to the collective behavior of swarms|journal=Phys. Rev. Lett.|year=2019}}</ref><ref name="Cavagna_dimensions">{{cite journal|author1=Cavagna A.|title=Natural swarms in 3.99 dimensions|journal=Nat. Phys.|year=2023|volume=19}}</ref>、神经网络<ref name="Koch_information">{{cite journal|author1=Koch-Janusz M|author2=Ringel Z.|title=Mutual information, neural networks and the renormalization group|journal=Nat. Phys.|year=2018|volume=14}}</ref><ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Hu_machine">{{cite journal|author1=Hu H.-Y|author2=Li S.-H|author3=Wang L|author4=You Y.-Z.|title=Machine learning holographic mapping by neural network renormalization group|journal=Phys. Rev. Res.|year=2020|volume=2}}</ref><ref name="Caso_graph">{{cite journal|author1=Caso F|author2=Trappolini G|author3=Bacciu A|author4=Liò P|author5=Silvestri F.|title=Renormalized graph neural networks|journal=Preprint at arXiv|year=2023}}</ref>、动态与有向逾渗<ref name="Young_percolation">{{cite journal|author1=Young A. P|author2=Stinchcombe R. B.|title=A renormalization group theory for percolation problems|journal=J. Phys. C|year=1975|volume=8}}</ref><ref name="Reynolds_percolation">{{cite journal|author1=Reynolds P. J|author2=Stanley H. E|author3=Klein W.|title=A real-space renormalization group for site and bond percolation|journal=J. Phys. C|year=1977|volume=10}}</ref>、观点动力学的选民模型<ref name="Galam_voting">{{cite journal|author1=Galam S.|title=Real space renormalization group and totalitarian paradox of majority rule voting|journal=Physica A|year=2000|volume=285}}</ref>，以及一维链<ref name="Kogan_synchronization">{{cite journal|author1=Kogan O|author2=Rogers J. L|author3=Cross M. C|author4=Refael G.|title=Renormalization group approach to oscillator synchronization|journal=Phys. Rev. E|year=2009|volume=80}}</ref>、格点<ref name="Östborn_disorder">{{cite journal|author1=Östborn P.|title=Renormalization of oscillator lattices with disorder|journal=Phys. Rev. E|year=2009|volume=79}}</ref>和层级树<ref name="Garlaschelli_synchronization">{{cite journal|author1=Garlaschelli D|author2=Hollander F|author3=Meylahn J|author4=Zeegers B.|title=Synchronization of phase oscillators on the hierarchical lattice|journal=J. Stat. Phys.|year=2019|volume=174}}</ref>上非线性振子模型（如Kuramoto模型<ref name="Hinrichsen_non-equilibrium_2000">{{cite journal|author1=Hinrichsen, H.|title=Non-equilibrium critical phenomena and phase transitions into absorbing states|journal=Adv. Phys.|year=2000|volume=49|url=}}</ref><ref name="Dornic_Langevin_2005">{{cite journal|author1=Dornic, I.|author2=Chaté, H.|author3=Muñoz, M. A.|title=Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions|journal=Phys. Rev. Lett.|year=2005|volume=94|url=}}</ref>）的同步模型。此外，重整化群方法在张量网络[192,201–203]及信息论框架<ref name="Lenggenhager_optimal">{{cite journal|author1=Lenggenhager P. M|author2=Gökmen D. E|author3=Ringel Z|author4=Huber S. D|author5=Koch-Janusz M.|title=Optimal renormalization group transformation from information theory|journal=Phys. Rev. X|year=2020|volume=10}}</ref><ref name="Gökmen_statistical">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Statistical physics through the lens of real-space mutual information|journal=Phys. Rev. Lett.|year=2021|volume=127}}</ref><ref name="Gordon_relevance">{{cite journal|author1=Gordon A|author2=Banerjee A|author3=Koch-Janusz M|author4=Ringel Z.|title=Relevance in the renormalization group and in information theory|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Sarra_mutual">{{cite journal|author1=Sarra L|author2=Aiello A|author3=Marquardt F.|title=Renormalized mutual information for artificial scientific discovery|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Gökmen_symmetries">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Symmetries and phase diagrams with real-space mutual information neural estimation|journal=Phys. Rev. E|year=2021|volume=104}}</ref>中亦获成功应用。
过去五十年间，重整化群技术已扩展至从最大尺度的宇宙学到最小尺度的粒子物理等广泛领域。在统计力学范畴内，重整化群方法已推广至同质均匀的分形格点上的非平衡系统<ref name="Gefen_fractal">{{cite journal|author1=Gefen Y|author2=Mandelbrot B. B|author3=Aharony A.|title=Critical phenomena on fractal lattices|journal=Phys. Rev. Lett.|year=1980|volume=45}}</ref><ref name="Gefen_backbone">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B|author4=Kirkpatrick S.|title=Solvable fractal family, and its possible relation to the backbone at percolation|journal=Phys. Rev. Lett.|year=1981|volume=47}}</ref><ref name="Phani_percolation">{{cite journal|author1=Phani M. K|author2=Dhar D.|title=Real-space renormalisation group: application to directed percolation|journal=J. Phys. C|year=1982|volume=15}}</ref><ref name="Gefen_phase1">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. I. Quasi-linear lattices|journal=J. Phys. A|year=1983|volume=16}}</ref><ref name="Gefen_phase2">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Shapir Y|author4=Mandelbrot B. B.|title=Phase transitions on fractals. II. Sierpinski gaskets|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Gefen_phase3">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. III. Infinitely ramified lattices|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Das_critical">{{cite journal|author1=Das D|author2=Dey S|author3=Jacobsen J. L|author4=Dhar D.|title=Critical behavior of loops and biconnected clusters on fractals of dimension d \< 2|journal=J. Phys. A|year=2008|volume=41}}</ref>，其局部平移不变性能够简化计算。超越初始应用的重要范例包括：流行病动力学模型<ref name="Della_SIR">{{cite journal|author1=Della Morte M|author2=Sannino F.|title=Renormalization group approach to pandemics as a time-dependent SIR model|journal=Front. Phys.|year=2021}}</ref>、生物群体行为动力学<ref name="Cavagna_scaling">{{cite journal|author1=Cavagna A.|title=Dynamic scaling in natural swarms|journal=Nat. Phys.|year=2017|volume=13}}</ref><ref name="Cavagna_swarms">{{cite journal|author1=Cavagna A.|title=Dynamical renormalization group approach to the collective behavior of swarms|journal=Phys. Rev. Lett.|year=2019}}</ref><ref name="Cavagna_dimensions">{{cite journal|author1=Cavagna A.|title=Natural swarms in 3.99 dimensions|journal=Nat. Phys.|year=2023|volume=19}}</ref>、神经网络<ref name="Koch_information">{{cite journal|author1=Koch-Janusz M|author2=Ringel Z.|title=Mutual information, neural networks and the renormalization group|journal=Nat. Phys.|year=2018|volume=14}}</ref><ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Hu_machine">{{cite journal|author1=Hu H.-Y|author2=Li S.-H|author3=Wang L|author4=You Y.-Z.|title=Machine learning holographic mapping by neural network renormalization group|journal=Phys. Rev. Res.|year=2020|volume=2}}</ref><ref name="Caso_graph">{{cite journal|author1=Caso F|author2=Trappolini G|author3=Bacciu A|author4=Liò P|author5=Silvestri F.|title=Renormalized graph neural networks|journal=Preprint at arXiv|year=2023}}</ref>、动态与有向逾渗<ref name="Young_percolation">{{cite journal|author1=Young A. P|author2=Stinchcombe R. B.|title=A renormalization group theory for percolation problems|journal=J. Phys. C|year=1975|volume=8}}</ref><ref name="Reynolds_percolation">{{cite journal|author1=Reynolds P. J|author2=Stanley H. E|author3=Klein W.|title=A real-space renormalization group for site and bond percolation|journal=J. Phys. C|year=1977|volume=10}}</ref>、观点动力学的选民模型<ref name="Galam_voting">{{cite journal|author1=Galam S.|title=Real space renormalization group and totalitarian paradox of majority rule voting|journal=Physica A|year=2000|volume=285}}</ref>，以及一维链<ref name="Kogan_synchronization">{{cite journal|author1=Kogan O|author2=Rogers J. L|author3=Cross M. C|author4=Refael G.|title=Renormalization group approach to oscillator synchronization|journal=Phys. Rev. E|year=2009|volume=80}}</ref>、格点<ref name="Östborn_disorder">{{cite journal|author1=Östborn P.|title=Renormalization of oscillator lattices with disorder|journal=Phys. Rev. E|year=2009|volume=79}}</ref>和层级树<ref name="Garlaschelli_synchronization">{{cite journal|author1=Garlaschelli D|author2=Hollander F|author3=Meylahn J|author4=Zeegers B.|title=Synchronization of phase oscillators on the hierarchical lattice|journal=J. Stat. Phys.|year=2019|volume=174}}</ref>上非线性振子模型（如Kuramoto模型<ref name="Hinrichsen_non-equilibrium_2000">{{cite journal|author1=Hinrichsen, H.|title=Non-equilibrium critical phenomena and phase transitions into absorbing states|journal=Adv. Phys.|year=2000|volume=49|url=}}</ref><ref name="Dornic_Langevin_2005">{{cite journal|author1=Dornic, I.|author2=Chaté, H.|author3=Muñoz, M. A.|title=Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions|journal=Phys. Rev. Lett.|year=2005|volume=94|url=}}</ref>）的同步模型。此外，重整化群方法在张量网络<ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Levin_tensor">{{cite journal|author1=Levin M|author2=Nave C. P.|title=Tensor renormalization group approach to two-dimensional classical lattice models|journal=Phys. Rev. Lett.|year=2007}}</ref><ref name="Evenbly_tensor">{{cite journal|author1=Evenbly G|author2=Vidal G.|title=Tensor network renormalization|journal=Phys. Rev. Lett.|year=2015}}</ref><ref name="Bal_flows">{{cite journal|author1=Bal M|author2=Mariën M|author3=Haegeman J|author4=Verstraete F.|title=Renormalization group flows of Hamiltonians using tensor networks|journal=Phys. Rev. Lett.|year=2017}}</ref>及信息论框架<ref name="Lenggenhager_optimal">{{cite journal|author1=Lenggenhager P. M|author2=Gökmen D. E|author3=Ringel Z|author4=Huber S. D|author5=Koch-Janusz M.|title=Optimal renormalization group transformation from information theory|journal=Phys. Rev. X|year=2020|volume=10}}</ref><ref name="Gökmen_statistical">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Statistical physics through the lens of real-space mutual information|journal=Phys. Rev. Lett.|year=2021|volume=127}}</ref><ref name="Gordon_relevance">{{cite journal|author1=Gordon A|author2=Banerjee A|author3=Koch-Janusz M|author4=Ringel Z.|title=Relevance in the renormalization group and in information theory|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Sarra_mutual">{{cite journal|author1=Sarra L|author2=Aiello A|author3=Marquardt F.|title=Renormalized mutual information for artificial scientific discovery|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Gökmen_symmetries">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Symmetries and phase diagrams with real-space mutual information neural estimation|journal=Phys. Rev. E|year=2021|volume=104}}</ref>中亦获成功应用。
与平衡态不同，非平衡系统中决定普适类的要素往往未知，且微扰处理中高阶重整化群计算通常不可行。非微扰重整化群作为有潜力的替代方案<ref name="Canet_reaction">{{cite journal|author1=Canet L|author2=Delamotte B|author3=Deloubrière O|author4=Wschebor N.|title=Nonperturbative renormalization-group study of reaction--diffusion processes|journal=Phys. Rev. Lett.|year=2004|volume=92}}</ref><ref name="Canet_branching">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B.|title=Quantitative phase diagrams of branching and annihilating random walks|journal=Phys. Rev. Lett.|year=2004|volume=92}}</ref><ref name="Canet_fixed">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B|author4=Dornic I|author5=Muñoz M. A.|title=Nonperturbative fixed point in a nonequilibrium phase transition|journal=Phys. Rev. Lett.|year=2005|volume=95}}</ref><ref name="Canet_framework">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B.|title=General framework of the non-perturbative renormalization group for non-equilibrium steady states|journal=J. Phys. A|year=2011|volume=44}}</ref>，已应用于格点上的流行病扩散动力学模型等研究<ref name="Tarpin_epidemic">{{cite journal|author1=Tarpin M|author2=Benitez F|author3=Canet L|author4=Wschebor N.|title=Nonperturbative renormalization group for the diffusive epidemic process|journal=Phys. Rev. E|year=2017|volume=96}}</ref>。统计物理和场论中的重整化总结了传统应用于物理系统的重整化和重整化群变换的基本概念。
与平衡态不同，非平衡系统中决定普适类的要素往往未知，且微扰处理中高阶重整化群计算通常不可行。非微扰重整化群作为有潜力的替代方案<ref name="Canet_reaction">{{cite journal|author1=Canet L|author2=Delamotte B|author3=Deloubrière O|author4=Wschebor N.|title=Nonperturbative renormalization-group study of reaction--diffusion processes|journal=Phys. Rev. Lett.|year=2004|volume=92}}</ref><ref name="Canet_branching">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B.|title=Quantitative phase diagrams of branching and annihilating random walks|journal=Phys. Rev. Lett.|year=2004|volume=92}}</ref><ref name="Canet_fixed">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B|author4=Dornic I|author5=Muñoz M. A.|title=Nonperturbative fixed point in a nonequilibrium phase transition|journal=Phys. Rev. Lett.|year=2005|volume=95}}</ref><ref name="Canet_framework">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B.|title=General framework of the non-perturbative renormalization group for non-equilibrium steady states|journal=J. Phys. A|year=2011|volume=44}}</ref>，已应用于格点上的流行病扩散动力学模型等研究<ref name="Tarpin_epidemic">{{cite journal|author1=Tarpin M|author2=Benitez F|author3=Canet L|author4=Wschebor N.|title=Nonperturbative renormalization group for the diffusive epidemic process|journal=Phys. Rev. E|year=2017|volume=96}}</ref>。统计物理和场论中的重整化总结了传统应用于物理系统的重整化和重整化群变换的基本概念。