Ornstein-Uhlenbeck process equation

At this point it has to be emphasized the links of the Langevin description with the diffusion processes. By comparing the transition density functions (4.121) and (4.130), it is clear that the Langevin equation (4.126) is equivalent to the Ornstein-Uhlenbeck process. Equation (4.130) satisfies the following one-dimensional Fokker-Planck... 

These are the so-called forward and backward Kolmogorov equations for the Ornstein-Uhlenbeck process. Their paramount importance will appear in VIII.4 under the more familiar name of Fokker-Planck equation. 

If A < 0 the stationary solution (1.4) is Gaussian. In fact, in that case it is possible by shifting y and rescaling, to reduce (1.5) to (IV.3.20), so that one may conclude the stationary Markov process determined by the linear Fokker-Planck equation is the Ornstein-Uhlenbeck process. For Al 0 there is no stationary probability distribution. 

This is a linear Fokker-Planck equation. Apart from constants which can be scaled away, it is identical with the equation (IV.3.20) obeyed by the transition probability of the Ornstein-Uhlenbeck process. The stationary solution of (4.6) is the same as the Pl given in (IV.3.10). Thus, in equilibrium V(t) is the Ornstein-Uhlenbeck process. 

Thus our additional approximation for the neighborhood of rf leads to a linear Fokker-Planck equation of the same form as (4.6). The fluctuations in the stationary state are therefore again an Ornstein-Uhlenbeck process. It will be shown in X.4 that (5.6) is a consistent approximation.510... 

Exercise. The Ornstein-Uhlenbeck process (IV.3.10), (IV.3.11) satisfies the generalized Langevin equation with memory kernel ... 

This is the master equation for the Ornstein-Uhlenbeck process. This result is readily extended to more variables, but we emphasize that the equations must be linear. 

In particular let us take for the stationary solution of (1.10) = s = 1. Then (1.11) reduces to a time-independent Fokker-Planck equation whose solution is the Ornstein-Uhlenbeck process. More directly one finds from (1.12b)... 

Equation (58) is equivalent to the fractional Rayleigh equation [75, 77], and therefore we refer to Eq. (58) as the fractional Ornstein-Uhlenbeck process. For the sharp initial condition Wo(x) = <5(x — xo), the solution to this process is, according to Eq. (46), given by... 

The flawed time evolution tells us that, starting from some proper quantum states, we may obtain improper ones. However, Eq. (35) is the correct evolution equation for the Ornstein-Uhlenbeck process. Thus, starting from allowed classical states, it will always produce allowed classical ones. This strange situation indicates that irreversible time evolution is more intricate in quantum theory than we may guess from the classical counterpart. 

The study of a Brownian particle suspended in a fluid lead also to the introduction of the exponentially correlated Ornstein-Uhlenbeck process [48], the only Markovian Gaussian non-white stochastic process [19, 22]. We present here the Langevin approach to this problem, hence we analyze the forces that act on a single Brownian particle. We suppose the particle having a mass m equal to unity, and we assume the force due to the hits with thermal activated molecules of the fluid to be a stochastic variable. Moreover, due to the viscosity of the fluid, a friction force proportional to the velocity of the particle has to be considered. All this yields the following equation... 

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... 

Consider a variant of Langevin dynamics coupling the Ornstein-Uhlenbeck process to only one degree of freedom. Such equations of motion for a planar system with potential energy U(x,y) are... 

Equation (4.4) represents an Ornstein-Uhlenbeck process. When r is above or below b, it will be pulled toward b, although random shocks generated by dz may delay this process. 

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 ... 

We take it to be the Ornstein-Uhlenbeck process given by the stochastic differential 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... 

We have shown in Appendix A how these results can be obtained directly from the Langevin equation). Thus the conditional probability distribution of the velocities in the Ornstein-Uhlenbeck [11] process is... 

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 ... 

---