Application of the Real-Space Renormalization Group to Dynamic Critical Phenomena

Mazenko, Gene F.; Nolan, Michael J.; Valls, Oriol T.

doi:10.1103/physrevlett.41.500

Cited by 42 publications

(13 citation statements)

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“…Here we use a real space RG scheme [9][10][11] which renormalizes the transition probability W. The succession of RG transformations corresponds to a trajectory in the space spanned by the parameters that defines W. The scheme we use is an implementation of the DDRG [1][2][3] and is accomplished by transforming cells of b sites into a cell of just one site. To treat the vacuum state properly any cell with at least one particle renormalizes into an occupied site.…”

Section: Renormalization Schemementioning

confidence: 99%

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Renormalization group of probabilistic cellular automata with one absorbing state

Oliveira

Satulovsky

1997

Phys. Rev. E

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We apply a recently proposed dynamically driven renormalization group scheme to probabilistic cellular automata having one absorbing state. We have found just one unstable fixed point with one relevant direction. In the limit of small transition probability one of the cellular automata reduces to the contact process revealing that the cellular automata are in the same universality class as that process, as expected. Better numerical results are obtained as the approximations for the stationary distribution are improved.

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Section: Renormalization Schemementioning

confidence: 99%

“…Similarly, for the renormalized system, let P (τ, τ ′ ) be the probability of occurrence of state τ ′ at a given time and state τ at one time step later. The RG transformation are obtained by demanding that [9] …”

Section: Renormalization Schemementioning

confidence: 99%

Renormalization group of probabilistic cellular automata with one absorbing state

Oliveira

Satulovsky

1997

Phys. Rev. E

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“…It is worth noting that (8) gives the exact critical coupling u c = V"2 -1 and the exact thermal exponent v m l =y T = 1. The recursion relation (8) does not come directly out of our previous analysis but it gives a (7) 1 large hint as to how to develop a systematic theory. If we define e(n) as the two-spin static correlation function for spins separated by n lattice sites along the x ory directions, then we have from (7) that e(2n) =^1 2 €'(w) for n> 1.…”

Section: The Intracell Part Is Of the Formmentioning

confidence: 98%

Dynamical Correlation Functions in the Two-Dimensional Kinetic Ising Model: A Real-Space Renormalization-Group Approach

et al. 1980

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A real-space dynamic renormalization-group scheme is used to evaluate static and dynamic correlation functions for a kinetic Ising model on a two-dimensional square lattice. The critical exponents obtained from the correlation functions calculated using this method satisfy the proper static and dynamic scaling relations and are in excellent agreement with known values.PACS numbers: 64.60.-i The calculation of static and dynamic correlation functions in systems that undergo a phase transition is a problem of great interest. These functions can be measured in scattering experiments and exhibit a wide variety of phenomena in the region near the phase transition. Recently, 1 ' 2 a real-space dynamic renormalization-group (RSDRG) method was introduced for the investigation of lattice dynamical models. In this Letter we apply this method to the kinetic Ising (Kl) model on a square lattice and show how, within the formalism, one can calculate space-and timedependent correlation fucntions.The KI model 3 describes the stochastic time evolution of a set of Ising spins {cr}. The equilibrium properties are governed by the nearestneighbor Ising-model Hamiltonian, H a , characterized by a coupling K. The time evolution of this model is assumed to be generated by a single-spin-flip operator (SFO):o> ln W i [v]a i +J 2 cr

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“…14). 17) at Therefore, the transformation function R(,ua) should satisfy the orthogonality condition (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) in addition to the normalization condition (2)(3)(4)(5)(6)(7). This orthogonality condition also ensures the positivity of P;t(fl) through P;t(fl) = 8(fllfl)P;t(fl) =(R(fliJ»2pstUn because (R(fl(J»2 and Pst«J) are positive.…”

mentioning

confidence: 97%

“…The difficulty of this approach is that the coarse grained time-evolution operator obtained by the Markov approximation has a nonlocal form in space. 14 ) Thus, this approach has not been studied systematically up to now. However, it is shown in the present paper that this convolution-type approach yields a promissing systematic method to obtain the dynamic critical exponent.…”

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confidence: 98%

Real-Space Renormalization Group Approach to the Kinetic Ising Model

Takano

Suzuki

1982

Progress of Theoretical Physics

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A systematic real·space dynamic renormalization group method is proposed and it is applied to the kinetic Ising model on a triangular lattice. The coarse grained master equation is constructed by using the memory function formalism and the Markov approximation. The high temperature series renormalization group method proposed by Betts, Cuthiell and Plischke is extended for the present purpose to make a perturbation expansion in a self·consistent way with respect to all orders of interactions. Terms nonlocal in space appear in the coarse grained time-evolution operator in our approach, because of the memory effect accompanied with the coarse graining procedure. It is shown within the second order calculation that these nonlocal terms are irrelevant around the nontrivial fixed point. The values of the static exponent v and dynamic exponent z are given by 1/v=0.99 and z=2.23 from the present second order calculation. These results are quite reasonable. § 1. IntroductionThe renormalization group approach first introduced by Wilson!) to study critical phenomena has turned out a very useful and powerful method in this field. 2 ) This approach is classified roughly into two categories, namely, the momentum-space and real-space approaches. The latter one first proposed by Niemeijer and van Leeuwen,3) in contrast with the former, deals with the microscopic Hamiltonian directly in real space and it can be used in any dimension. In fact, the real-space approach has given very successful results concerning the static critical properties and the global thermodynamic properties of the twodimensional Ising-like spin systems. 4) Therefore, it is of great interest to generalize the real-space renormalization group method to the dynamic case. In recent years, many attempts 5H7 ) at this generalization have been made in the kinetic Ising model. 18)-20)There are two problems in the generalization of the real-space renormalization group approach to the dynamic case. The first one is how to deal with the memory effect accompanied with the coarse graining procedure. The second one is how to construct a consistent perturbation theory.Concerning the memory effect, one possible approach is the convolutionlesstype approach, in which a time-convolutionless master equation for the coarse grained system is constructed. The coarse grained time-evolution operator thus obtained has the time dependence, which corresponds to the memory effect. We usually take the long-time limit to make the time-evolution operator time-inde-*) Present address:

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Application of the Real-Space Renormalization Group to Dynamic Critical Phenomena

Cited by 42 publications

References 7 publications

Renormalization group of probabilistic cellular automata with one absorbing state

Renormalization group of probabilistic cellular automata with one absorbing state

Dynamical Correlation Functions in the Two-Dimensional Kinetic Ising Model: A Real-Space Renormalization-Group Approach

Real-Space Renormalization Group Approach to the Kinetic Ising Model

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