Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExact derivation of a finite-size-scaling law and corrections to scaling in the geometric Galton-Watson process
121
0
0.0
(
0
)
Added by
Rosalba Garc\\'ia-Mill\\'an
Publication date
2016
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary.
rate research
Read More
Corrections to scaling in the two-dimensional scalar phi^4 model are studied based on non-perturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes L (from 4 to 1536) and different values of the phi^4 coupling constant lambda, i.~e., lambda = 0.1, 1, 10. According to our analysis, amplitudes of the nontrivial correction terms with the correction-to-scaling exponents omega_l < 1 become small when approaching the Ising limit (lambda --> infinity), but such corrections generally exist in the 2D phi^4 model. Analytical arguments show the existence of corrections with the exponent 3/4. The numerical analysis suggests that there exist also corrections with the exponent 1/2 and, very likely, also corrections with the exponent about 1/4, which are detectable at lambda = 0.1. The numerical tests clearly show that the structure of corrections to scaling in the 2D phi^4 model differs from the usually expected one in the 2D Ising model.
Universal scaling laws form one of the central issues in physics. A non-standard scaling law or a breakdown of a standard scaling law, on the other hand, can often lead to the finding of a new universality class in physical systems. Recently, we found that a statistical quantity related to fluctuations follows a non-standard scaling law with respect to system size in a synchronized state of globally coupled non-identical phase oscillators [Nishikawa et al., Chaos $boldsymbol{22}$, 013133 (2012)]. However, it is still unclear how widely this non-standard scaling law is observed. In the present paper, we discuss the conditions required for the unusual scaling law in globally coupled oscillator systems, and we validate the conditions by numerical simulations of several different models.
The present review is devoted to the problems of finite-size scaling due to the presence of long-range interaction decaying at large distance as $1/r^{d+sigma}$, where $d$ is the spatial dimension and the long-range parameter $sigma>0$. Classical and quantum systems are considered.
We consider scaling of the entanglement entropy across a topological quantum phase transition in one dimension. The change of the topology manifests itself in a sub-leading term, which scales as $L^{-1/alpha}$ with the size of the subsystem $L$, here $alpha$ is the R{e}nyi index. This term reveals the universal scaling function $h_alpha(L/xi)$, where $xi$ is the correlation length, which is sensitive to the topological index.
Corrections to scaling in the 3D Ising model are studied based on non-perturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes L. Analytical arguments show the existence of corrections with the exponent (gamma-1)/nu (approximately 0.38), the leading correction-to-scaling exponent being omega =< (gamma-1)/nu. A numerical estimation of omega from the susceptibility data within 40 =< L =< 2048 yields omega=0.25(33). It is consistent with the statement omega =< (gamma-1)/nu, as well as with the value omega = 1/8 of the GFD theory. We reconsider the MC estimation of omega from smaller lattice sizes to show that it does not lead to conclusive results, since the obtained values of omega depend on the particular method chosen. In particular, estimates ranging from omega =1.274(72) to omega=0.18(37) are obtained by four different finite-size scaling methods, using MC data for thermodynamic average quantities, as well as for partition function zeros. We discuss the influence of omega on the estimation of exponents eta and nu.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
