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Register a new userFluctuation theorems and 1/f noise from a simple matrix
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Here we present a model for a small system combined with an explicit entropy bath that is comparably small. The dynamics of the model is defined by a simple matrix, M. Each row of M corresponds to a macrostate of the system, e.g. net alignment, while the elements in the row represent microstates. The constant number of elements in each row ensures constant entropy, which allows reversible fluctuations, similar to information theory where a constant number of bits allows reversible computations. Many elements in M come from the microstates of the system, but many others come from the bath. Bypassing the bath states yields fluctuations that exhibit standard white noise; whereas with bath states the power spectral density varies as S(f)~1/f over a wide range of frequencies, f. Thus, the explicit entropy bath is the mechanism of 1/f noise in this model. Both forms of the model match Crooks fluctuation theorem exactly, indicating that the theorem applies not only to infinite reservoirs, but also to finite-sized baths. The model is used to analyze measurements of 1/f-like noise from a sub-micron tunnel junction.
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Fluctuation theorems establish deep relations between observables away from thermal equilibrium. Until recently, the research on fluctuation theorems was focused on time-reversal-invariant systems. In this review we address some newly discovered fluctuation relations that hold without time-reversal symmetry, in particular, in the presence of an external magnetic field. One family of relations connects non-linear transport coefficients in the opposite magnetic fields. Another family relates currents and noises at a fixed direction of the magnetic field in chiral systems, such as the edges of some quantum Hall liquids. We review the recent experimental and theoretical research, including the controversy on the microreversibility without time-reversal symmetry, consider the applications of fluctuation theorems to the physics of topological states of matter, and discuss open problems.
The power spectrum of quantum dot fluorescence exhibits $1/f^beta$ noise, related to the intermittency of these nanosystems. As in other systems exhibiting $1/f$ noise, this power spectrum is not integrable at low frequencies, which appears to imply infinite total power. We report measurements of individual quantum dots that address this long-standing paradox. We find that the level of $1/f^beta$ noise decays with the observation time. The change of the spectrum with time places a bound on the total power. These observations are in stark contrast with most measurements of noise in macroscopic systems which do not exhibit any evidence for non-stationarity. We show that the traditional description of the power spectrum with a single exponent $beta$ is incomplete and three additional critical exponents characterize the dependence on experimental time.
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Internal mechanism leading to the emergence of the widely occurring 1/f noise still remains an open issue. In this paper we investigate the distinction between internal time of the system and the physical time as a source of 1/f noise. After demonstrating the appearance of 1/f noise in the earlier proposed point process model, we generalize it starting from a stochastic differential equation which describes a Brownian-like motion in the internal (operational) time. We consider this equation together with an additional equation relating the internal time to the external (physical) time. We show that the relation between the internal time and the physical time that depends on the intensity of the signal can lead to 1/f noise in a wide interval of frequencies. The present model can be useful for the explanation of the appearance of 1/f noise in different systems.
Noise of stochastic processes whose power spectrum scales at low frequencies, $f$, as $1/f$ appears in such diverse systems that it is considered universal. However, there have been a small number of instances from completely unrelated fields, e.g., the fluctuations of the human heartbeat or vortices in superconductors, in which power spectra have been observed to cross over from a $1/f$ to a non-$1/f$ behavior at even lower frequencies. Here, we show that such crossover must be universal, and can be accounted for by the memory of initial conditions and the relaxation processes present in any physical system. When the smallest frequency allowed by the experimental observation time, $omega_{obs}$, is larger than the smallest relaxation frequency, $Omega_{min}$, a $1/f$ power spectral density is obtained. Conversely, when $omega_{obs}<Omega_{min}$ we predict that the power spectrum of any stochastic process should exhibit a crossover from $1/f$ to a different, integrable functional form provided there is enough time for experimental observations. This crossover also provides a convenient tool to measure the lowest relaxation frequency of a physical system.
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