I am a theoretical quantum physicist trying to find probability distributions for heat transfer in an open quantum system. I am testing an approximate method with an analytically exact one, so I know what the characteristic function should look like. The probability distribution is a function of Q, and the characteristic function is a function of X. I have defined the probability distribution as the inverse Fourier transform of the characteristic function.
The real and imaginary parts of the analytic characteristic function have a feature at around X=0, but quickly decays to zero afterwards.
The real and imaginary parts of the characteristic function I calculate using my approximate method does not decay to zero like they do in the analytically exact case. Instead, they repeat the main features found at around X=0 for the analytic case, indefinitely.
If I limit the range of X to only include this first feature, the probability distributions I get out are almost identical. But if I include the repeated features by sampling over a wider range of X, then my approximate method gives a probability distribution that has fast oscillations in it compared to the analytic case.
I have the following questions about what it means for a characteristic function to decay to zero:
(i) How does decay of the characteristic function effect the probability distribution?
(ii) Do we expect the characteristic function to always decay to zero?
(ii) What are the differences we expect to see in the probability distributions for a characteristic function that does decay, versus one that doesn't?
p.s. please forgive a physicist if any wording or phrases are not correct.