Measurement of Persistence in 1D Diffusion

Wong, G. P.; Mair, R. W.; Walsworth, Ronald L.; Cory, David G.

doi:10.1103/physrevlett.86.4156

Cited by 52 publications

(68 citation statements)

References 28 publications

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“…The solution is characterized by a single growing length scale ℓ(t) ∝ t 1/z , with z = 2. For a system of linear size L, the persistence p 0 (t, L) is simply the probability that φ(x, t) does not change sign up to time t. It was found [4] that for t ≫ 1, p 0 (t, L) has a finite size scaling formwith h(u) ∼ c st , a constant independent of L and t, for u ≪ 1 andRemarkably, the persistence for d = 1 was observed in experiments on magnetization of spin polarized Xe gas and the exponent θ exp (1) ≃ 0.12 was measured [3], in good agreement with analytical approximations and numerical simulations [4,5]. Another apparently unrelated problem concerns the roots of random polynomials (i.e.…”

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confidence: 72%

“…We further show that the probability that such polynomials have exactly k real roots in [0,1] has an unusual scaling form given by n −φ(k/ log n) whereφ(x) is a universal large deviation function. Persistence properties and related first passage problems have been the subject of intense activities, both theoretically [1] and experimentally [2,3] these last few years. The persistence probability p(t) for a time dependent stochastic process of zero mean, is defined as the probability that it has not changed sign up to time t. In many physical situations, p(t) was found to decay at large time as a power law p(t) ∝ t −θ .…”

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confidence: 99%

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Statistics of the Number of Zero Crossings: From Random Polynomials to the Diffusion Equation

Schehr

Majumdar

2007

Phys. Rev. Lett.

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We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0, 1] decays as a power law n −θ(d) where θ(d) > 0 is the exponent associated to the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n −2(θ(d)+θ (2)) . For d = 1, this connection allows for a physical realization of real random polynomials. We further show that the probability that such polynomials have exactly k real roots in [0,1] has an unusual scaling form given by n −φ(k/ log n) whereφ(x) is a universal large deviation function. Persistence properties and related first passage problems have been the subject of intense activities, both theoretically [1] and experimentally [2,3] these last few years. The persistence probability p(t) for a time dependent stochastic process of zero mean, is defined as the probability that it has not changed sign up to time t. In many physical situations, p(t) was found to decay at large time as a power law p(t) ∝ t −θ . Surprisingly, the persistence exponent θ was found to be highly non trivial even for simple systems. One example is the diffusion equation in space dimension d, ∂ t φ(x, t) = ∇ 2 φ(x, t) with "white noise" initial conditions φ(x, 0)φ(x ′ , 0) = δ d (x − x ′ ). The solution is characterized by a single growing length scale ℓ(t) ∝ t 1/z , with z = 2. For a system of linear size L, the persistence p 0 (t, L) is simply the probability that φ(x, t) does not change sign up to time t. It was found [4] that for t ≫ 1, p 0 (t, L) has a finite size scaling formwith h(u) ∼ c st , a constant independent of L and t, for u ≪ 1 andRemarkably, the persistence for d = 1 was observed in experiments on magnetization of spin polarized Xe gas and the exponent θ exp (1) ≃ 0.12 was measured [3], in good agreement with analytical approximations and numerical simulations [4,5]. Another apparently unrelated problem concerns the roots of random polynomials (i.e. polynomials with random coefficients), which have attracted renewed interest over the last few years [6,7] in the context of probability and number theory. A recent work [8] proposed a physical realization of the complex roots of Weyl complex polynomials in a system of a rotating quasi-ideal atomic Bose gas. Here we focus instead on the real roots of a class of real random polynomials indexed by an integer d f n (x) = a 0 + n−1 i=1

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confidence: 72%

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confidence: 99%

Statistics of the Number of Zero Crossings: From Random Polynomials to the Diffusion Equation

Schehr

Majumdar

2007

Phys. Rev. Lett.

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“…The purpose of the present paper is twofold : (i) we will give detailed derivations of the results announced in Ref. [14] together with some new results, like the distribution of the largest real root, for generalized Kac polynomials K n (x) (3), (ii) we extend these results (4,5) to two other classes of random polynomials which were recently considered in the literature. First we will study Weyl polynomials W n (x) defined as…”

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confidence: 94%

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

Schehr

Majumdar

2008

J Stat Phys

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We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0, 1] decays as a power law n −θ(d) where θ(d) > 0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n ≫ 1 even, the probability that they have no real root on the full real axis decays like n −2(θ(2)+θ(d)) . For Weyl polynomials and Binomial polynomials, this probability decays respectively like exp (−2θ∞ √ n) and exp (−πθ∞ √ n) where θ∞ is such thatWe also show that the probability that such polynomials have exactly k roots on a given interval [a, b] has a scaling form given by exp (−N abφ (k/N ab )) where N ab is the mean number of real roots in [a, b] andφ(x) a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent −2. These analytical results are confirmed by detailed numerical computations. Some of these results were announced in a recent letter [G. Schehr and S. N. Majumdar, Phys. Rev. Lett. 99, 060603 (2007)].

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“…A uniform spacing, ∆T , between measurements in the latter systems, therefore, corresponds to measurements uniformly spaced in log time in the former. Such a measurement regime has indeed been used in a recent experimental study of diffusive persistence [7], with a spacing in log-time equivalent to ∆T ≈ 0.24. The present paper is the first step in understanding how such discretization affects the measured result.…”

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confidence: 99%

“…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].…”

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confidence: 99%

Persistence of a continuous stochastic process with discrete-time sampling

2001

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We introduce the concept of 'discrete-time persistence', which deals with zero-crossings of a continuous stochastic process, X(T ), measured at discrete times, T = n∆T . For a Gaussian Markov process with relaxation rate µ, we show that the persistence (no crossing) probability decays as [ρ(a)] n for large n, where a = exp(−µ∆T ), and we compute ρ(a) to high precision. We also define the concept of 'alternating persistence', which corresponds to a < 0. For a > 1, corresponding to motion in an unstable potential (µ < 0), there is a nonzero probability of having no zero-crossings in infinite time, and we show how to calculate it.PACS numbers: 05.70. Ln, 05.40.+j, 81.10.Aj Persistence of a continuous stochastic process has generated much recent interest in a wide variety of nonequilibrium systems including various models of phase ordering kinetics, diffusion, fluctuating interfaces and reactiondiffusion processes [1]. Persistence has also been recently used in fields as diverse as ecology [2] and seismology [3]. 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]. It is well known that this probability usually decays as ∼ exp(−θT ) for large T = |T 2 − T 1 | where θ is nontrivial [9,8]. The persistence of some of the nonstationary processes mentioned in the previous paragraph such as the diffusion processes, can be mapped to that of a corresponding GSP [10]. This makes the two sets of problems related to each other and the power law exponent in the former problem becomes the inverse decay rate in the latter. Even though θ is, in general, hard to compute analytically, it is very easy to evaluate numerically in most cases. Given this fact, and the combined interest of both statistical physicists and applied mathematicians, much recent effort has been devoted to computing θ numerically to extremely high precision.This raises a natural question: How accurately can one measure θ? Is there a natural limitation and if so, can it be overcome? This issue arises from the following simple observation. All the stochastic processes mentioned above occur in continuous time. However, when one performs numerical simulations or experiments on persistence, one has to discretize time in some way and sample the data only at these discrete time points to check if the process has retained its sign. Due to this discretization, some information is lost. For example, the process may have cro...

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Measurement of Persistence in 1D Diffusion

Cited by 52 publications

References 28 publications

Statistics of the Number of Zero Crossings: From Random Polynomials to the Diffusion Equation

Statistics of the Number of Zero Crossings: From Random Polynomials to the Diffusion Equation

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

Persistence of a continuous stochastic process with discrete-time sampling

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