Spectral analysis of current fluctuations in periodically driven stochastic systems

Probability current fluctuations play an important role in nonequilibrium statistical mechanics, and are a key object of interest in both theoretical studies and in practical applications. So far, most of the studies were devoted to the fluctuations of the time-averaged probability current, the zero-frequency Fourier component of the time-dependent current. However, in many practical applications the fluctuations at other frequencies are of equal importance. Here we study the statistics of all the probability current's Fourier component in periodically driven stochastic systems. We restrict our study to "trapped" systems where the degrees of freedom of the system cannot achieve arbitrarily large values as time becomes large, in contrast to, e.g., diffusing systems. First, we discuss possible methods to calculate the current statistics, valid even when the current's Fourier frequency is incommensurate with the driving frequency, breaking the time periodicity of the system. Somewhat surprisingly, we find that the cumulant generating function (CGF), that encodes all the statistics of the current, is composed of a continuous background at any frequency accompanied by either positive or negative discontinuities at current's frequencies commensurate with the driving frequency. We show that cumulants of increasing orders display discontinuities at an increasing number of locations but with decreasing amplitudes that depend on the rational frequency ratio. All these discontinuities are then transcribed in the behavior of the CGF. As the measurement time increases, these discontinuities become sharper but keep the same amplitude and eventually lead to discontinuities of the CGF at all the frequencies that are commensurate with the driving frequency in the limit of infinitely long measurement. We demonstrate our formalism and its consequences on three types of models: an underdamped Brownian particle in a periodically driven harmonic potential; a periodically driven run-and-tumble particle; and a two-state system. Our results show a rich and interesting structure in experimentally accessible and important objects: the fluctuations of alternating currents as a function of their frequency.