The critical point is a fixed point in finite-size scaling. To quantify the behavior of such a fixed point, we define, at a given temperature and scaling exponent ratio, the width of scaled observables for different sizes. The minimum of the width reveals the position of the fixed point, its corresponding phase transition temperature, and scaling exponent ratio. The value of this ratio tells the nature of the fixed point, which can be a critical point, a point of the first-order phase transition line, or a point of the crossover region. To demonstrate the effectiveness of this method, we apply it to three typical samples produced by the three-dimensional three-state Potts model. Results show the method to be more precise and effective than conventional methods. Finally, we discuss a possible application at the Beam Energy Scan program of the Relativistic Heavy Ion Collider.
A semi-microscopic scaling-field theory is developed for crossover phenomena near critical and tricritical points. The theory is based on a renormalization-group description of a model with two competing fixed points (such as a critical and a tricritical fixed point) in terms of scaling fields. The coupled nonlinear differential equations for scaling fields are truncated such as to preserve the physics essential for crossover phenomena. The approach allows the explicit calculation of thermodynamic functions for (i) tricritical systems and (ii) critical systems with an irrelevant scaling field. We obtain, for example, an explicit expression for the scaling function of the susceptibility, which describes the crossover from the tricritical to the critical region. The idea of "flow diagrams" in the scaling-field space is used to characterize crossover phenomena globally in the whole critical region. The concept of asymptotic critical exponents is generalized and effective critical exponents are introduced as logarithmic derivatives of thermodynamic quantities with respect to experimental fields and scaling fields, respectively. By using the method of effective exponents the size of the crossover region between regions of different asymptotic critical behavior is estimated. For the susceptibility, the width of the crossover region in decades of the effective temperature variable is roughly equal to the inverse of the crossover exponent. In the case of a critical system with a slow transient the asymptotic critical exponent is only reached extremely close to the critical point (unless the amplitude of the transient vanishes). It might then be impossible to determine the asymptotic exponent experimentally or by conventional series-expansion techniques, and an analysis of the data in terms of effective exponents is the alternative. The scaling-field approach is applied to three systems with crossover phenomena: (i) the model for tricritical systems with molecular-field tricritical exponents, (ii) the Ashkin-Teller model in three dimensions, and (iii) a model for phase transitions with Fisher exponent renormalization due to a constraint.
We present a powerful method of identifying the nature of transitions by numerical simulation of finite systems. By studying the finite size scaling properties of free energy barrier between competing states, we can identify unambiguously a weak first order transition even when accessible system sizes are L/{xi} < 0.05 as in the five state Potts model in two dimensions. When studying a continuous phase transition we obtain quite accurate estimates of critical exponents by treating it as a field driven first order transition. The method has been successfully applied to various systems.
We present a powerful method of identifying the nature of transitions by numerical simulation of finite systems. By studying the finite size scaling properties of free energy barrier between competing states, we can identify unambigously a weak first order transition even when accessible system sizes are L/ξ<0.05 as in the five state Potts model in two dimensions. When studying a continuous phase transition we obtain quite accurate estimates of critical exponents by treating it as a field driven first order transition. The method has been successfully applied to various systems.
The finite size scaling method is used to study the critical properties of the spin-1 Baxter-Wu model as function of the fugacity, z, of the vacant (spin zero) sites. For z=0, the thermal exponent converges very quickly to the (exact) value y t =3/2. For z>0, y t monotonically increases beyond the value 2. This increase is interpreted as indicating a first-order transition. Out of several possible renormalisation group flows, the results seem to favour the one in which the critical Baxter-Wu Hamiltonian flows to the fixed point of the 4-state Potts model, with the amplitude of the marginal operator equal to zero.
Finite-size scaling (FSS) for a critical phase transition (
t=0) states that within a window of size
∣t∣∼L−1/ν, the scaling behavior of any observable
Q in a system of linear size
L asymptotically follows a scaling form as
Q(t,L)=LYQQ~(tL1/ν), where
ν is the correlation-length exponent,
YQ is an FSS exponent and
Q~(x) is a function of the scaled distance-to-criticality
x≡tL1/ν. We systematically study the asymptotic scaling behavior of
Q~(∣x∣→∞) for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with
t→0 and
∣x∣→∞. This crossover FSS theory predicts that when the criticality is approached at a slower speed as
∣t∣∼L−λ with
λ<1/ν, the FSS becomes
λ-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality
t=0 can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.
We describe a new approach to the understanding of the thermodynamic properties of a physical system near a critical point. The critical point is characterized by the peculiar dependence of various quantities on the size of the system, rather than by the mathematical singularities which appear in an infinite system. We indicate a very general method of obtaining scaling laws for the critical behavior of any thermodynamic system by considering its thermodynamic fluctuations. As an example, a new scaling law is obtained for the susceptibility at a gas-liquid critical point.