content_image/353833.jpg content_image/353832.jpg content_image/353831.jpg content_image/353830.jpg
where p (0 → 1) = dt/τ$_{c}$ and p (1 → 1) = 1 − p (1 → 0) = 1 − dt/τ$_{e}$ and also P$_{00}$ ( t ) = 1 − P$_{01}$ ( t ). This leads to a simple differencial equation: dP$_{01}$ ( t ) /dt = τ − 1 c − ( τ − 1 e + τ − 1 c ) P$_{01}$ ( t ). If P$_{00}$ (0) = 1, its solution is P$_{01}$ ( t ) = τ$_{e}$ ( τ$_{e}$ + τ$_{c}$ ) − $^{1}$[1 − exp( − t/τ$_{eq}$ )], where 1 /τ$_{eq}$ = 1 /τ$_{e}$ +1 /τ$_{c}$ . Similar results can be performed leading to P$_{11}$ ( t ) = τ$_{e}$ τ$_{e}$ + τ$_{c}$ [ τ$_{e}$ + τ$_{c}$ exp( − t/τ$_{eq}$ )]. Unless fluctuations on the current amplitude, whose average Δ depends on other microscopic factors of the device, this probability corresponds to the autocorrelation of the system A ( t ) = 〈 σ (0) σ ( t ) 〉 , where σ ( t ) corresponds to the state of the trap (occupied σ = 0 or empty σ = 1). In frequency domain this exponential decay of autocorrelation for one trap is described by Lorentzians, since the power spectrum density, i.e., the fourier transform of the autocorrelation is 

<img src='content_image/353833.jpg'>

The known 1/ f noise results from a sum of these Lorentzians (a contribution of the many traps in device). For more details about the origin of 1/ f noise, see for example [7],[4], [13], [14]. 

So, coming back to relaxation phenomena, and starting from n (0) = 0 (all traps empty), we can calculate the average density of occupied traps at time t 

<img src='content_image/353832.jpg'>

where Pr( k | n (0) , t ) is the probability of just k traps are occupied at time t , with k = 0 ...N$_{tr}$ . But the traps have different constants τ$_{e}$ and τ$_{c}$ and from that, we write 

with C$_{k}$ denoting every subset { d$_{1}$, d$_{2}$, ..., d$_{k}$ } from { 1 , 2 , ..., , n } . But { τ ( d$_{i}$ ) c , τ ( d$_{i}$ ) e } N$_{tr}$ i =1 are statistically independent and identically distributed, and we have : 

<img src='content_image/353831.jpg'>

<img src='content_image/353830.jpg'>

