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WORLD SCIENTIFIC eBooks

Finite Size Scaling and Numerical Simulation of Statistical Systems

Vladimir Privman A

Published Jan 1, 1990• 1,285 citations
Publisher

Abstract

The theory of Finite Size Scaling describes a build-up of the bulk properties when a small system is increased in size. This description is particularly important in strongly correlated systems where critical fluctuations develop with increasing system size, including phase transition points, polymer conformations. Since numerical computer simulations are always done with finite samples, they rely on the Finite Size Scaling theory for data extrapolation and analysis. With the advent of large scale computing in recent years, the use of the size-scaling methods has become increasingly important.

DOI
10.1142/1011

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Finite-size scaling at infinite-order phase transitions

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Abstract For systems with infinite-order phase transitions, in which an order parameter smoothly becomes nonzero, a new observable for finite-size scaling analysis is suggested. By construction this new observable has the favourable property of diverging at the critical point. Focussing on the example of the F -model we compare the analysis of this observable with that of another observable, which is also derived from the order parameter but does not diverge, as well as that of the associated susceptibility. We discuss the difficulties that arise in the finite-size scaling analysis of such systems. In particular we show that one may reach incorrect conclusions from large-system size extrapolations of observables that are not known to diverge at the critical point. Our work suggests that one should base finite-size scaling analyses for infinite-order phase transitions only on observables that are guaranteed to diverge.

arXiv:1605.08876•Journal of Statistical Mechanics Theory and Experiment•Sep 1, 2016•4 citations
The theory of Finite Size Scaling describes a build-up of the bulk properties when a small system is increased in size. This description is particularly important in strongly correlated systems where critical fluctuations develop with increasing system size, including phase transition points, polymer conformations. Since numerical computer simulations are always done with finite samples, they rely on the Finite Size Scaling theory for data extrapolation and analysis. With the advent of large scale computing in recent years, the use of the size-scaling methods has become increasingly important.
Both first- and second-order transitions get smeared and shifted due to the finite size of a sample. Recent theoretical developments on this finite size rounding of singularities associated with phase transitions are briefly reviewed, and evidence from Monte Carlo simulations for the validity of these concepts will be discussed. It is also shown how finite size scaling theories in conjunction with simulation data from finite systems are a powerful tool for the quantitative study of bulk critical phenomena. We shall also mention extensions such as finite size tests of hyperscaling, finite size scaling of interfacial properties, finite size effects on critical relaxation, etc. A brief discussion of the experimental situation concludes this review.
The aim of this book is to familiarise the reader with the rich collection of ideas, methods and results available in the theory of critical phenomena in systems with confined geometry. The existence of universal features of the finite-size effects arising due to highly correlated classical or quantum fluctuations is explained by the finite-size scaling theory. This theory (1) offers an interpretation of experimental results on finite-size effects in real systems; (2) gives the most reliable tool for extrapolation to the thermodynamic limit of data obtained by computer simulations; (3) reveals
Using single cluster flip Monte Carlo simulations we accurately determine new finite size scaling functions which are expressed only in terms of the variable x=${\ensuremath{\xi}}_{\mathit{L}}$/L, where ${\ensuremath{\xi}}_{\mathit{L}}$ is the correlation length in a finite system of size L. Data for the d=2 and d=3 Ising models, taken at different temperatures and for different size lattices, show excellent data collapse over the entire range of scaling variable for susceptibility and correlation length. From these finite size scaling functions we can estimate critical temperatures and exponents with rather high accuracy even though data are not obtained extremely close to the critical point. The bulk values of the renormalized four-point coupling constant are accurately measured and show strong evidence for hyperscaling. \textcopyright{} 1996 The American Physical Society.
For systems with infinite-order phase transitions, in which an order parameter smoothly becomes nonzero, a new observable for finite-size scaling analysis is suggested. By construction this new observable has the favourable property of diverging at the critical point. Focussing on the example of the F-model we compare the analysis of this observable with that of another observable, which is also derived from the order parameter but does not diverge, as well as that of the associated susceptibility. We discuss the difficulties that arise in the finite-size scaling analysis of such systems. In particular we show that one may reach incorrect conclusions from large-system size extrapolations of observables that are not known to diverge at the critical point. Our work suggests that one should base finite-size scaling analyses for infinite-order phase transitions only on observables that are guaranteed to diverge.
The implications of the scaling theory for the crossover from one universality class of critical phenomena to another one are worked out for finite-size scaling analyses of numerical simulation data. Near a multicritical point finite-size scaling critical amplitudes have a singular dependence on the distance from the multicritical point. Attention is also paid to the crossover to Landau-like behaviour when the interaction range diverges, and as an application Monte Carlo results for polymer mixtures are discussed.
Abstract For systems with infinite-order phase transitions, in which an order parameter smoothly becomes nonzero, a new observable for finite-size scaling analysis is suggested. By construction this new observable has the favourable property of diverging at the critical point. Focussing on the example of the F -model we compare the analysis of this observable with that of another observable, which is also derived from the order parameter but does not diverge, as well as that of the associated susceptibility. We discuss the difficulties that arise in the finite-size scaling analysis of such systems. In particular we show that one may reach incorrect conclusions from large-system size extrapolations of observables that are not known to diverge at the critical point. Our work suggests that one should base finite-size scaling analyses for infinite-order phase transitions only on observables that are guaranteed to diverge.