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I am studying a system subject to random noise, or a system driven by some noise, for example, heat flow or wave propagation perturbed by noise. I would like to know if there is a real system where the noise take effects slowly instead from the very beginning.

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Are you looking for a noise driven system where the noise has varying amplitude? Or maybe something else? It is difficult to tell from the question, a bit more detail may be appropriate. – tmac May 17 '12 at 7:11
Dear tmac, I am looking for a noise driven system where in the long run, the noise has fixed strength or amplitude. But in the beginning, the amplitude is small. Are there such physical scenarios? Thanks a lot. :-) – Anand May 18 '12 at 9:49

There are many possible examples of this, and you may need to be more specific in what you want. Here are two that immediately come to mind:

1) A bead in a harmonic trap (or a bending cantilever) that is undergoing thermal kicks from Brownian motion. The strength of these fluctuations depends on temperature; if the temperature of the system changes over time, the noise amplitude will change. (The simplest case of the bead in a harmonic potential with a fluctuating thermal noise is the Ornstein-Uhlenbeck process: - I do not guarantee that generalizing this to a time-dependent temperature is easy.)

2) "Demographic noise" - suppose you look at the dynamics of a population of animals that can spawn new individuals or die (a birth/death process). You may have seen this in the context of ODEs - . However, if you actually track the population, there will be stochastic fluctuations (intrinsic noise) arising because there happened to be more births or deaths than you would expect on average. These fluctuations often scale as $\sqrt{N}$, where $N$ is the population size. So if the population size changes, so does the fluctuation around the mean. There's some nice treatment of this problem in Plischke and Bergerson's Equilibrium Statistical Physics, which can be found online, I believe.

The difference between these two cases is that in 1), the noise is arising from the thermal bath (the air/water outside the bead or cantilever, which can be controlled), but in 2) the noise arises from the inherently discrete nature of the dynamics.

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It's not clear what you mean by "noise" but if you accept black-body radiation as electromagnetic noise then the average peak and total power output ramps up slowly for many systems including something as simple as a light bulb.

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