Delta-correlated fluctuating force

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

Here f is the friction coefficient, m is the monomer mass, LT(Ro...Rn) is the intrachain interaction potential, and fj( ) are random forces from the media. The simplest assumption about the random forces is that they are independent for each particle and delta-correlated in time. In this case, eqn [11] is often called the Langevin equation. The fluctuation-dissipation theorem establishes the following relationship between the friction f and the random forces ... 

The statistical properties of the random force f(0 are modeled with an extreme economy of assumptions f(t) is assumed to be a stationary and Gaussian stochastic process, with zero mean (f(0 = 0), uncorrelated with the initial value v(t = 0) of the velocity fluctuations, and delta-correlated with itself, f(0f(t ) = f25(t -1 ) (i.e it is a purely random, or white, noise). The stationarity condition is in reality equivalent to the fluctuation-dissipation relation between the random and the dissipative forces in Equation 1.1, which essentially fixes the value of y. In fact, from Equation 1.1 and the assumed properties of f(t), we can derive the expression y(f)v(t) = exp [v(0)v(0) -+ ylM °, where Xg = In equilibrium, the long-time asymptotic value y/M must coincide with the equilibrium average (vv) = (k TIM)t given by the equipartition theorem (with I being the 3 X 3 Cartesian unit tensor), and this fixes the value of y to y=... 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

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