We show how to obtain the instantaneous variance of a nonstationary random process by way of the Wigner spectrum. This is done by transforming the governing differential equation into the time‐frequency domain and by finding the second conditional moment of frequency. We illustrate the method with Brownian motion.
We have previously presented a method to write equations for the Wigner distribution corresponding to the solution of a linear differential equation. We extend this method to arbitrary time-frequency distributions, the Short Time Fourier Transform, and the wavelet transform. Also, we apply the method to stochastic signals and address a number of applications including Brownian motion. We obtain the Wigner distribution for Brownian motion by solving the Langevin equation and also by using our method. We obtain exact solutions.
We use the Wigner distribution to study systems subjected to random forces. We define the instantaneous spectrum as the ensemble average of the Wigner distribution, and we write the differential equation whose solution gives us the time-varying spectrum of the state variable. We consider the cases of both constant and time-varying coefficients. We apply the method to study the instantaneous spectrum of a harmonic oscillator driven by Gaussian noise, with both constant and time-varying coefficients. In the latter case our method clearly reveals the nonstationarity of the power spectrum and we confirm our result by numerical simulations.
We obtain an explicit equation for the time-frequency Wigner spectrum of momentum governed by the Langevin equation when the random driving term is quantum noise. The equation is solved exactly and includes both the transient and the stationary part. The time-dependent Wigner spectrum generalizes the result of Wang and Uhlenbeck wherein they showed that for the white noise driving force the power spectrum in the stationary state regime is Lorenzian. We show that our solution reduces to the classical solution when the parameters of the quantum noise are such that the white noise limit is approached and when the long time limit is taken.
We present a general procedure for obtaining equations of motion for the Wigner distribution of functions that are governed by ordinary and partial differential equations. For the case of fields we show that in general one must consider Wigner distribution of the four variables, position, momentum, time and frequency. We also show that in general one cannot write an equation of motion for position and momentum however it can be done in some cases, the Schrodinger equation being one such case. Our method leads to an equation of motion for the Schrodinger equation with time dependent potentials in contrast to the result obtained by Wigner and Moyal which was for time independent potentials.
We discuss a number of topics relevant to noise propagation in dispersive media. We formulate the problem of pulse propagation with a source term in phase space and show that a four dimensional Wigner distribution is required. The four dimensional Wigner distribution is that of space and time and also wavenumber and frequency. The four dimensional Wigner spectrum is equivalent to the space-time autocorrelation function. We also apply the quantum path method to improve the phase space approximation previously obtained. In addition we discuss motion in a Snell’s law medium.