Experimental measurement of the persistence exponent of the planar Ising model

Abstract

Using a twisted nematic liquid crystal system exhibiting planar Ising model dynamics, we have measured the scaling exponent θ which characterizes the time evolution, p(t) ∼ t −θ , of the probability p(t) that the local order parameter has not switched its state by the time t. For 0.4 seconds to 200 seconds following the phase quench, the system exhibits scaling behavior and, measured over this interval, θ = 0.19 ± 0.031, in good agreement with theoretical analysis and numerical simulations. PACS: 82.20.Fd, 02.50.Ey, 05.40.+j, 05.50.+q There has been a recent surge of interest in determining the so-called nontrivial "persistence" exponent θ [1-11] associated with the dynamics of a phase ordering system following a quench from the high-temperature phase to zero temperature. This exponent describes the asymptotic power-law decay, p(t) ∼ t −θ , of the probability that the local order parameter φ(x, t) has not changed sign up to time t after the quench. For the Ising model, p(t) is simply the fraction of spins that have not flipped up to time t, or, equivalently, the probability that no interface has ever crossed a given spin. While there have been considerable theoretical and numerical efforts directed towards calculating θ, so far there has been no experimental measurement of this exponent in spin-like systems. A kind of persistence exponent was first introduced and measured in a breath figure experiment [1], and recently for soap bubbles [12] (in this latter example θ takes a rather trivial value, which can be inferred on simple physical grounds [12]). In this Letter, we present the first experimental measurement of the nontrivial persistence exponent for an Ising-like system, that we find in good agreement with a recent theoretical calculation.This power-law decay of persistence is quite ubiquitous and not just restricted to the phase ordering dynamics of the Ising or Potts models. For instance, a similar question arises in the study of Gaussian processes. Considering such a process m(t), the calculation of the probability that this Gaussian walker never crosses the origin (or changes sign) is a particularly difficult problem when this process is not Markovian [10,11]. For example, even for the simple diffusion equation, ∂ t φ = ∇ 2 φ, the probability that the Gaussian variable φ(x, t) does not change sign up to time t, decays algebraically with t with a nontrivial exponent θ [11]. This exponent θ also appears in other contexts such as reaction-diffusion systems [13][14][15] and driven diffusive systems [16]. For a quench to the critical point of spin systems, the persistence exponent θ associated with the total magnetization (another nonMarkovian Gaussian variable) has recently been argued to be a new non-equilibrium critical exponent [17].For the purpose of the experimental results presented below, we will restrict ourselves to the phase ordering dynamics of the Ising model at T = 0. The Ising model at T = 0 has two ordered states: either all spins are up or they are all down. Following a quench from the hig...

Random walks in logarithmic and power-law potentials, nonuniversal persistence, and vortex dynamics in the two-dimensionalXYmodel

“…Consider separately the two terms in the general solution (9). The term involving J −ν behaves as r for r → 0, while the term in J ν behaves as r −b .…”

“…Persistence phenomena have been widely studied in recent years [1][2][3][4][5][6][7][8][9][10]. Theoretical and computational studies include spin systems in one [1] and higher [2] dimensions, diffusion fields [3], fluctuating interfaces [4], phaseordering dynamics [5], and reaction-diffusion systems [6].…”

“…Theoretical and computational studies include spin systems in one [1] and higher [2] dimensions, diffusion fields [3], fluctuating interfaces [4], phaseordering dynamics [5], and reaction-diffusion systems [6]. Experimental studies include the coarsening dynamics of breath figures [7], soap froths [8], and twisted nematic liquid crystals [9]. Persistence in nonequilibrium critical phenomena has been studied in the context of the global order parameter, M (t), (e.g.…”

Persistence of a continuous stochastic process with discrete-time sampling

“…Persistence is simply the probability P (t) that a stochastic process x(t) does not change sign up to time t. In most of the systems mentioned above, P (t) ∼ t −θ for large t, where the persistence exponent θ is nontrivial. Apart from various analytical and numerical results, this exponent has also been measured experimentally in systems such as breath figures [4], liquid crystals [5], soap bubbles [6], and more recently in laser-polarized Xe gas using NMR techniques [7].…”

“…Persistence is simply the probability P (t) that a stochastic process x(t) does not change sign up to time t. In most of the systems mentioned above, P (t) ∼ t −θ for large t, where the persistence exponent θ is nontrivial. Apart from various analytical and numerical results, this exponent has also been measured experimentally in systems such as breath figures [4], liquid crystals [5], soap bubbles [6], and more recently in laser-polarized Xe gas using NMR techniques [7].Persistence has also remained a popular subject among applied mathematicians for many decades [8]. They are most interested in the probability of 'no zero crossing' of a Gaussian stationary process (GSP) between times T 1 and T 2 [9].…”