Ornstein-Uhlenbeck noise

The outlay of this chapter is as follows In Section 9.2, we introduce the system/ spin-bath model and derive the operator Langevin equation for the particle. This is followed by a discussion on stochastic dynamics in the presence of c-number noise, highlighting the role of the spectral density function in the high- and low-temperature regimes. A scheme for the generation of spin-bath noise as a superposition of several Ornstein-Uhlenbeck noise processes and its implementation in numerical simulation of the quantum Langevin equation are described in Section 9.3. Two examples have been worked out in Section 9.4 to illustrate the basic theoretical issues. This chapter is concluded in Section 9.5. 

Equation 9.33 expresses the FD relation, which is the key element for the generation of c-number noise. n(0n(O)s is the correlation function, which is classical in form but quantum mechanical in its content. We now show that c-number noise q(t) can be generated as a superposition of several Ornstein-Uhlenbeck noise processes (Banerjee et al. 2004). To this end, we begin by expressing (n(t)Ti(t ))s = C t - t ) in the continuum limit as follows ... 

The c-number noise q(0 due to spin-bath is therefore given by a superposition of several Ornstein-Uhlenbeck noise processes as follows ... 

Exercise. Verify that the linear noise approximation always leads to an Ornstein-Uhlenbeck process for the fluctuations in a stable stationary state. 

For multiplicative noise the determination of these moments requires a more detailed consideration of the stochastic integral since white noise is too irregular for Riemann integrals to be applied. Application of Stieltjes integration yields a dependence of the moments on how the limit to white noise is taken. If t) is the limit of the Ornstein-Uhlenbeck -process with r —> 0 (Stratonovich sense) the coefficients read [50]... 

For colored noise sources the derivation of evolution equations for the probability densities is more difficult. In a Markovian embedding, i.e. if the Ornstein-Uhlenbeck process is defined via white noise (cf. chapter 1.3.2) and v t) is part of the phase space one again gets a Fokker-Planck equation for the density P x,y, Similarly, one finds in case of the telegraph... 

An often-used coloured noise is the Ornstein-Uhlenbeck process its expectation E[ j] = 0 vanishes, and its correlation function is exponential ... 

V is the volume of the system. To include finite correlation time the noise was described by the following Ornstein-Uhlenbeck process ... 

The influence of the external noise is taken into account as another Ornstein-Uhlenbeck process given by... 

Such a noise is easy to generate electronically, iii) Since we are interested in macroscopic systems, we will observe the system usually only on macroscopic time scales. It is then reasonable to assume that Z is a Markov process. Furthermore, it has been argued that a non-Markovian noise will not introduce any essentially new physics into the problem [5]. Properties i) - iii) uniquely specify the noise process. In the case of ii)a) we find, in light of DOOB s theorem [6], that Z is given by a stationary Ornstein-Uhlenbeck process, i.e. it obeys.the following Lan-gevin equation ... 

The assumption that f(t) is purely random may actually be a highly restrictive and unnecessary limitation. In fact, many physical processes cannot be described by Equation 1.4 simply because they exhibit memory effects. Thus, we must introduce an extended Ornstein-Uhlenbeck process, deflned as the solution of the most general linear stochastic equation with additive noise, which we write as... 

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