Finite-size Scaling Effects Research Articles

Fragmentation and the Nuclear Equation of State

Progress on the determination of the order of the fragmentation phase transition, the location of its critical point in the nuclear matter phase diagram, the values of the critical exponents that determine the universality class of the transition, and finite size scaling effects is discussed. Evidence for the presence of Zipf-Mandelbrot-scaling in the relative size of the largest clusters is examined, and the connection to the value of the critical exponent τ is established.

Critical behavior of the Ising model on fractal structures in dimensions between one and two: Finite-size scaling effects

The magnetic critical behavior of Ising spins located at the sites of deterministic Sierpinski carpets is studied within the framework of a ferromagnetic Ising model. A finite-size scaling analysis is performed from Monte Carlo simulations. We investigate four different fractal dimensions between 1.9746 and 1.7227, up to the sixth and eighth iteration step of the fractal structure in one case. It turns out that the finite-size scaling behavior of most thermodynamical quantities is affected by scaling corrections increasing as the fractal dimension decreases, tending towards the lower critical dimension of the Ising model. These corrections are related to the topology of the fractal structure and to the scale invariance. Nevertheless the maxima of the susceptibility follow power laws in a very reliable way, which allows us to calculate the ratio of the exponents $\ensuremath{\gamma}/\ensuremath{\nu}.$ Moreover, the fixed point of the fourth order cumulant at ${T}_{c}$ exhibited by Binder on translation invariant lattices is replaced by a decreasing sequence of intersection points converging towards the critical temperature. The convergence towards the thermodynamical limit as the size of the networks increases is slowed down as the fractal dimension decreases. At last, the evolution of the discrepancies between Monte Carlo simulations and $\ensuremath{\epsilon}$ expansions with the fractal dimension is set out.

Finite size effects at the Yang-Lee edge singularity and branched polymers in a plate geometry

Finite size scaling effects are investigated for Ising-like systems in a hypercube L d near the Yang-Lee edge singularity. Besides exact results for d = 1, we present series in ε 1 3 for some universal quantities in dimensions d = 6 − ε gained by field-theoretic techniques. Using the supersymmetric connection between the Yang-Lee theory in dimension d and the statistics of branched polymers in D = d + 2, we find the animal number in a periodic plate geometry with 2 infinite and d compactified dimensions. In particular, we exactly calculate the full cross-over between three-dimensional and two-dimensional animals.

Relaxation times in a finite Ising system with random impurities

Finite-size scaling effects of the Ising model with quenched random impurities are studied, focusing on critical dynamics. In contrast to the pure Ising model, disordered systems are characterized by continuous relaxation time spectra. Dynamic field theory is applied to compute the spectral densities of the magnetizationM(t) and ofM2(t). In addition, universal cumulant ratios are calculated to second order in e1/4, where e=4−d andd<4 denotes the spatial dimension.

Scaling and dynamical effects on the superfluid transition in helium films

The results of a series of measurements of superfluid transition behavior in helium films are presented. Variations of the superfluid density and dissipation in the vicinity of the transition were measured. Dynamical effects were probed by comparing transitions at two different frequencies. Results are compared with a comprehensive numerical calculation in which finite-size scaling effects are explicitly introduced. A consistent picture for the dependence of critical thickness on temperature, of the scaling of the transition shape with temperature and of dynamical effects is obtained.

Kosterlitz-Thouless behavior and scaling in superfluid helium films

The results of a series of measurements of Kosterlitz-Thouless behavior in helium films are presented. Use is made of high Q piezoelectric crystals so as to probe the phenomena in a single flat substrate geometry at high frequencies. Variations of the superfluid density and dissipation in the vicinity of the transition were measured. Results are compared with a comprehensive numerical calculation in which finite size scaling effects are explicitly introduced. A consistent picture for the dependence of critical thickness on temperature is obtained.

Dynamical and scaling effects in two-dimensional superfluid helium.

The results of a series of measurements of Kosterlitz-Thouless behavior in helium films are presented. Use is made of high-Q piezoelectric crystals so as to probe the phenomena in a single flat substrate geometry at high frequencies. Variations of the superfluid density and dissipation in the vicinity of the transition were measured. Dynamical effects were probed by comparing transitions at two different frequencies. Results are compared with a comprehensive numerical calculation in which finite-size scaling effects are explicitly introduced. A consistent picture for the dependence of critical thickness on temperature, of the scaling of the transition shape with temperature, and of dynamical effects is obtained.

Finite size scaling for directed percolation and related stochastic evolution processes

Finite size scaling effects are investigated for certain evolution processes modelled by a one-component reaction-diffusion system with an absorbing state. The model possesses a non-equilibrium critical point, and the associated universality class includes directed bond percolation, cellular automata, Reggeon field theory and a stochastic version of Schlogl's first autocatalytic reaction scheme. Using renormalisation group techniques, we calculate the linear relaxation time in a cubic geometry of finite sizeL, with periodic boundary conditions imposed. The corresponding scaling behaviour toO(ɛ) (ɛ=4−d,d being the spatial dimension) is presented in universal form.

Finite size scaling effects in dynamics

We calculate the linear relaxation time for a finite size system with a cubic geometry and “model A” dynamics both above T c in a 4 - ϵ expansion and below T c in a 2 + ϵ expansion, and express the results in a scaling form. The universal scaling functions are obtained to one-loop order. We use the method of the effective hamiltonian for the homogeneous modes. The quantum mechanical hamiltoninian is supersymmetric. The large n limit (infinitely many spin components) is also considered both above and below T c.