Fisher's static finite-size scaling law is derived on the basis of the renormalization group theory and it is extended to dynamic critical phenomena in a finite system. This dynamic finite-size scaling law yields a cross-over effect with respect to the size and time-region. This effect is useful in analyzing computer simulations and also in studying the scaling property of the Kondo effect near the absolute zero temperature.
Fisher's static finite-size scaling law is derived on the basis of the renormalization group theory and it is extended to dynamic critical phenomena in a finite system. This dynamic finite-size scaling law yields a cross-over effect with respect to the size and time-region. This effect is useful in analyzing computer simulations and also in studying the scaling property of the Kondo effect near the absolute zero temperature.
We propose systematic finite-time scaling and show that it can be used to determine both static and dynamic critical properties. The controllable finite time scale that affects the scaling of a system and that serves as an analogue of the system size in finite-size scaling is achieved simply but generally by manipulating the dynamics with an external field varying linearly with time. An analytic renormalization-group theory is developed to justify the scaling. Monte Carlo simulation results for two- and three-dimensional Ising models agree with existing ones and confirm the scaling and the method.
Originally, the term ‘dynamic critical phenomena’ was coined for time-dependent properties near second-order phase transitions in thermal equilibrium . The kinetics of phase transitions in magnets, at the gas–liquid transition, and at the normal- to superfluid phase transition in helium 4 were among the prominent examples investigated already in the 1960s. The dynamic scaling hypothesis , generalizing the scaling ansatz for the static correlation function and introducing an additional dynamic critical exponent, successfully described a variety of these experiments. Yet only the development of the systematic renormalization group (RG) approach for critical phenomena in the subsequent decade provided a solid conceptual foundation for phenomenological scaling theories. Supplemented with exact solutions for certain idealized model systems, and guided by invaluable input from computer simulations in addition to experimental data, the renormalization group now provides a general framework to explore not only the static and dynamic properties near a critical point, but also the large-scale and low-frequency response in stable thermodynamic phases. Scaling concepts and the renormalization group have also been successfully applied to phase transitions at zero temperature driven by quantum rather than thermal fluctuations. It is to be hoped that RG methods may help to classify the strikingly rich phenomena encountered in far-from-equilibrium systems as well. Recent advances in studies of simple reaction-diffusion systems, active to absorbing state phase transitions, driven lattice gases, and scaling properties of moving interfaces and growing surfaces, among others, appear promising in this respect.
Abstract Time evolution, near a phase transition in the critical domain of critical systems not far from equilibrium, using a Langevin-type evolution is studied. Typical quantities of interest are relaxation rates towards equilibrium, time-dependent correlation functions and transport coefficients. The main motivation for such a study is that, in systems in which the dynamics is local (on short time-scales, a modification of a dynamic variable has an influence only locally in space) when the correlation length becomes large, a large time-scale emerges, which characterizes the rate of time evolution. This phenomenon called critical slowing down leads to universal behaviour and scaling laws for time-dependent quantities. In contrast with the situation in static critical phenomena, there is no clean and systematic derivation of the dynamical equations governing the time evolution in the critical domain, because often the time evolution is influenced by conservation laws involving the order parameter, or other variables like energy, momentum, angular momentum, currents and so on. Indeed, the equilibrium distribution does not determine the driving force in the Langevin equation, but only the dissipative couplings are generated by the derivative of the equilibrium Hamiltonian, and directly related to the static properties. The purely dissipative Langevin equation specifically discussed, corresponding to static models like the f4 field theory and two-dimensional models. Renormalization group (RG) equations are derived, and dynamical scaling relations established.
In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O ( n )-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension d c = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η . The critical exponents ν , η , and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension d c = 4.
Abstract In finite systems singularities, such as discontinuities or divergences, associated with phase transitions are rounded. The extrapolation of finite size calculations to the singular thermodynamic limit makes use of scaling laws, introduced by Fisher, which involve the size of the system. The use and the theoretical basis of these laws are briefly reviewed here; some difficulties associated with the analytic calculation of the scaling functions are discussed. The correct treatment of zero‐momentum modes leads to a modified renormalization group picture; above the upper critical dimension (u.c.d.) finite size scaling does not hold but the properties of the finite system are easily calculable; below the u.c.d. the scaling functions are obtained in a systematic expansion in fractional powers of ϵ = 4–d. The rounding of first order transitions is also briefly considered; in spite of the lack of universality of the bulk transition itself, the rounding is universal because the correlation length of the finite system is large.
The main idea of Wilson's renormalization group theory of critical phenomena is extended to the dynamical case. In the first place, a concept of dynamical fixed point is proposed. The dynamic scaling law is then derived from our new fixed-point hypothesis. Wilson's fixed-point hypothesis is also deduced from ours. Further, a method of renormalizing the master equation by eliminating the short wavelength modes is found. This is performed by means of the projection operator method. As an illustration, the dynamic critical exponents are calculated for a purely dissipative spin system. Related work based on the time-dependent Ginzburg-Landau equation is examined.