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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Does measurement noise increase as a phase transition is approached?

Zhi Chen, Clare C. Yu

Published Jun 7, 2007• 0 citations
Publisher

Abstract

We study the energy and magnetization noise spectra associated with first and second order phase transitions by using Monte Carlo simulations of the Ising model and 5-state Potts model in 2D. For a finite size system, the total noise power and the low frequency white noise S(f < fknee) increase as Tc is approached. In the thermodynamic limit S(f < fknee) diverges but fknee → 0 and the total noise power vanishes. f-1knee is approximately the equilibration time. At high frequencies S(f > fknee) ~ f-μ. For the Ising model, we relate μ to the critical exponents.

DOI
10.1117/12.725521

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We study the energy and magnetization noise spectra associated with first and second order phase transitions by using Monte Carlo simulations of the Ising model and 5-state Potts model in 2D. For a finite size system, the total noise power and the low frequency white noise S(f < fknee) increase as Tc is approached. In the thermodynamic limit S(f < fknee) diverges but fknee → 0 and the total noise power vanishes. f-1knee is approximately the equilibration time. At high frequencies S(f > fknee) ~ f-μ. For the Ising model, we relate μ to the critical exponents.
We provide a general scheme to predict and derive the contribution to the noise spectrum of a stochastic sequence of pulses from the distribution of pulse parameters. An example is the magnetization noise spectra of a two-dimensional Ising system near its phase transition. At $T\ensuremath{\le}{T}_{c}$, the low-frequency spectra is dominated by magnetization flips of nearly the entire system. We find that both the predicted and the analytically derived spectra fit those produced from simulations. Subtracting this contribution at Tc{T}_{c}Tc​ leaves the high-frequency spectra which follow a power law determined by the critical exponents.
First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussians corresponding to the different phases; all finite-size effects then vary as N or 1/N. The latter case involves the Gaussian ensemble where the entropy of the bath has a convenient form; for small N′, first-order transitions are characterized by van der Waals' loops in (for example) the energy vs. temperature curves. Results from extensive Monte Carlo simulations of Ising and Potts models in d = 2 are presented to confirm the predictions. Comparison is made with data from second-order transitions to show that the order of a transition can be distinguished through such studies, although problems still occur for first-order transitions very close to critical points.
Multifractal properties of the two-dimensional Ising model are studied by means of the $f(\ensuremath{\alpha})$ spectra of singularities of probability measures supported by energy spectra. These measures are determined through calculations of energy degeneracies for finite-size systems up to $12\ifmmode\times\else\texttimes\fi{}12$ spins in cases of square and triangular lattices and up to $22\ifmmode\times\else\texttimes\fi{}22$ spins in the case of the hexagonal lattice. The calculations are performed in an exact manner using the transfer-matrix method. It is argued that, in the thermodynamic limit, the scaling exponent ${\ensuremath{\alpha}}_{\mathrm{max}}$ associated with the most probable energy of the system takes at the critical temperature a minimum value and, consequently, it is argued that a given system reveals a phase transition at a finite temperature if ${\ensuremath{\alpha}}_{\mathrm{max}}$ possesses a minimum at the finite temperature. It is also shown that, in the thermodynamic limit, the spectrum $f(\ensuremath{\alpha})$ shrinks when the critical temperature is approached.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
Using the ``multihit'' Swendsen-Wang cluster flipping method, we performed extensive Monte Carlo simulations to investigate the critical behavior of the two-dimensional (2D) eight-state random-bond Potts model. We applied finite-size-scaling techniques to extract the critical exponents for two different sets of bond strengths, from which we concluded that the transition is second order with critical exponents for both sets falling into same universality class, that of a 2D Ising model. A variation of the Lee-Kosterlitz method for determining the order of a phase transition was also applied. The double-peaked structure of the specific heat, which was found in some of the bond configurations, was also studied by simulation on periodic arrangements of strong and weak bonds.