Definitions

# Persistence

[per-sis-tuhns, -zis-]
Persistence may refer to:

• Persistence (computer science), the characteristic of data that outlives the execution of the program that created it
• Persistence (meteorology), the meteorological phenomenon by which weather remains relatively unchanged over short time intervals, allowing forecasts to become more accurate.
• Persistence of a number, a mathematical quality of numbers
• Image persistence, in LCD monitors
• Inflation persistence, in economics
• Persistence in Physical Sciences, As the name suggests, persistence conveys the meaning of survival and is chiefly the characteristic of stochastic process. There has been a lot of interest in recent years on this particular nature of stochastic process.Standard result exists for a wide range of stochastic process, Random Walk, Diffusion, Surface Growth, Magnetic Systems are few to name. To quantify this meaning of survival we define a probability which is simply the probability that the local field has not changed sign up to time t. The local field can be the displacement x(t) in Random walk, the local surface height h(t) in surface growth, the local magnetization m(t) for magnetic systems. For a wide class of such stochastic models the survival probability goes as $p\left(t\right) sim t^\left\{-theta\right\}$ where $theta$ is a new non trivial exponent called the persistence exponent.

The simplest of all stochastic models that exhibit this typical persistence behavior is the Random walk model. The equation of motion for a random walker is

$frac\left\{mathrm\left\{d\right\} x\right\}\left\{mathrm\left\{d\right\} t\right\}=eta\left(t\right)$
where $eta$ is the stochastic noise with correlation
$langle eta\left(t\right)rangle=0$
$langleeta\left(t\right) eta\left(t\text{'}\right)rangle =2D delta\left(t-t\text{'}\right)$

The two time correlation function $langle x\left(t_1\right)x\left(t_2\right)rangle$ is then given by
$langle x\left(t_1\right)x\left(t_2\right)rangle=2D min\left(t_1,t_2\right)$

The process is not a Gaussian stationary process. To convert it to a G.S.P define a transformation in space $bar\left\{X\right\}\left(t\right)=frac\left\{x\left(t\right)\right\}\left\{langle x^2\left(t\right)rangle\right\}$. In the new transformed variable the two time correlation function takes the form
$langle bar\left\{X\right\}\left(t_1\right)bar\left\{X\right\}\left(t_2\right)rangle=sqrt\left\{frac\left\{t_2\right\}\left\{t_1\right\}\right\}$ with $t_1>t_2$
Now make a transformation in time $t=ln T$ and the correlation function becomes
$langle bar\left\{X\right\}\left(t_1\right)bar\left\{X\right\}\left(t_2\right)rangle=e^\left\{-1/2\left(T_1-T_2\right)\right\}$

The process is now a G.S.P and the correlator is an exponentially decaying. To find out the survival probability we ask the question whether the variable $sigma=sgn\left[x\left(t\right)-langle x\left(t\right) rangle\right]$ has changed sign up to time t. The general result due to Slepian is that if a process is a Gaussian Stationary process and the correlator is exponentially decaying then the survival probability in transformed time variable is
$P\left(T\right)=frac\left\{2\right\}\left\{pi\right\} sin^\left\{-1\right\}\left[e^\left\{-lambda t\right\}\right]$
where $e^\left\{-lambda t\right\}$ is the two time stationary correlation function.

For the Random walk problem the correlation function is $e^\left\{-T/2\right\}$ and for sufficiently long time the survival probability in transformed variable is
$P\left(T\right)sim e^\left\{-T/2\right\}$
and in real time the probability is
$p\left(t\right) sim t^\left\{-1/2\right\}$.

The persistence exponent is therefore $theta=1/2$.

Persistence may also be used for:

• A measure of how long a CRT monitor's phosphors glow after they have been struck by electrons.
• The existence of an infectious agent in a population without causing direct harm to the hosts which carry it, but with the potential to infect and damage other hosts, particularly those who are immunocompromised.