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WORLD SCIENTIFIC eBooks

Transfer Matrices, Phase Transitions, and Critical Phenomena: Numerical Methods and Applications

M. P. Nightingale

Published Jan 1, 1990• 7 citations
Publisher
DOI
10.1142/9789814503419_0007

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Abstract This chapter introduces a few typical numerical methods used in modern studies of phase transitions and critical phenomena in spin systems. The first section describes the stochastic dynamics of a generic system with discrete degrees of freedom following the master equation. This section serves as a theoretical basis for the Monte Carlo method that includes the heat bath and Metropolis algorithms of configuration updates. Another useful numerical technique is the transfer matrix method, described in the last section, and which is applied for numerically exact evaluation of the free energy and related physical quantities.

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Our study emphasizes the efficacy of employing matrices resembling Wishart matrices, derived from magnetization time series data within specific dynamics, to elucidate phase transitions and critical phenomena in the Q-state Potts model. Through the application of appropriate statistical methods, we not only identify second-order transitions but also distinguish weaker first-order transitions by carefully analyzing the density of eigenvalues and their fluctuations. Additionally, we investigate the method’s sensitivity to stronger first-order transition points. Notably, we establish a robust correlation between the system’s actual thermodynamics and the spectral thermodynamics encapsulated within the eigenvalues. Our findings are further supported by correlation histograms of the time series data, revealing insightful patterns. Building upon our core results, we provide a didactic analysis that draws parallels between the spectral properties of criticality in a spin system and matrices intentionally imbued with correlations (a toy model). Within this framework, we observe a universal behavior characterized by the distribution of eigenvalues into two distinct groups, separated by a gap dependent on the level of correlation, influenced by temperature-induced changes in the spin system.

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We review recent progress in the understanding of phase transitions and critical phenomena, obtained by means of numerical simulations of the early dynamic evolution of systems prepared at well-specified initial conditions. This field has seen exhaustive scientific research during the last decade, when the renormalization-group (RG) theoretical results obtained at the end of the 1980s were applied to the interpretation of dynamic Monte Carlo simulation results. While the original RG theory is restricted to critical phenomena under equilibrium conditions, numerical simulations have been applied to the study of far-from-equilibrium systems and irreversible phase transitions, the investigation of the behaviour of spinodal points close to first-order phase transitions and the understanding of the early-time evolution of self-organized criticality (SOC) systems when released far form the SOC regime. The present review intends to provide a comprehensive overview of recent applications in those fields, which can give the flavour of the main ideas, methods and results, and to discuss the directions for further studies. All of these numerical results pose new and interesting theoretical challenges that remain as open questions to be addressed by new research in the coming years.

Reports on Progress in Physics•Jan 14, 2011•112 citations
Abstract This chapter introduces a few typical numerical methods used in modern studies of phase transitions and critical phenomena in spin systems. The first section describes the stochastic dynamics of a generic system with discrete degrees of freedom following the master equation. This section serves as a theoretical basis for the Monte Carlo method that includes the heat bath and Metropolis algorithms of configuration updates. Another useful numerical technique is the transfer matrix method, described in the last section, and which is applied for numerically exact evaluation of the free energy and related physical quantities.
Our study emphasizes the efficacy of employing matrices resembling Wishart matrices, derived from magnetization time series data within specific dynamics, to elucidate phase transitions and critical phenomena in the Q-state Potts model. Through the application of appropriate statistical methods, we not only identify second-order transitions but also distinguish weaker first-order transitions by carefully analyzing the density of eigenvalues and their fluctuations. Additionally, we investigate the method’s sensitivity to stronger first-order transition points. Notably, we establish a robust correlation between the system’s actual thermodynamics and the spectral thermodynamics encapsulated within the eigenvalues. Our findings are further supported by correlation histograms of the time series data, revealing insightful patterns. Building upon our core results, we provide a didactic analysis that draws parallels between the spectral properties of criticality in a spin system and matrices intentionally imbued with correlations (a toy model). Within this framework, we observe a universal behavior characterized by the distribution of eigenvalues into two distinct groups, separated by a gap dependent on the level of correlation, influenced by temperature-induced changes in the spin system.
We review recent progress in the understanding of phase transitions and critical phenomena, obtained by means of numerical simulations of the early dynamic evolution of systems prepared at well-specified initial conditions. This field has seen exhaustive scientific research during the last decade, when the renormalization-group (RG) theoretical results obtained at the end of the 1980s were applied to the interpretation of dynamic Monte Carlo simulation results. While the original RG theory is restricted to critical phenomena under equilibrium conditions, numerical simulations have been applied to the study of far-from-equilibrium systems and irreversible phase transitions, the investigation of the behaviour of spinodal points close to first-order phase transitions and the understanding of the early-time evolution of self-organized criticality (SOC) systems when released far form the SOC regime. The present review intends to provide a comprehensive overview of recent applications in those fields, which can give the flavour of the main ideas, methods and results, and to discuss the directions for further studies. All of these numerical results pose new and interesting theoretical challenges that remain as open questions to be addressed by new research in the coming years.