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1/f Noise from the Laws of Thermodynamics for Finite-Size Fluctuations

Computer simulations of the Ising model exhibit white noise if thermal fluctuations are governed by Boltzmann's factor alone; whereas we find that the same model exhibits 1/f noise if Boltzmann's factor is extended to include local alignment entropy to all orders. We show that this nonlinear correction maintains maximum entropy during equilibrium fluctuations. Indeed, as with the usual resolution of Gibbs' paradox that avoids net entropy reduction during reversible processes, the correction yields the statistics of indistinguishable particles. The correction also ensures conservation of energy if an instantaneous contribution from local entropy is included. Thus, a common mechanism for 1/f noise comes from assuming that finite-size fluctuations strictly obey the laws of thermodynamics, even in small parts of a large system. Empirical evidence for the model comes from its ability to match the measured temperature dependence of the spectral-density exponents in several metals, and to show non-Gaussian fluctuations characteristic of nanoscale systems.


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. Here we assume explanation 3): changes in entropy of the region are compensated by changes in entropy of the bath, thereby maintaining maximum entropy and retaining Boltzmann's definition. Thus, we assume that the entropy of the bath can be changed in two ways, from changes in energy and alignment of the region, so that successful inversion of a spin involves two criteria using random numbers between 0 and 1. The first criterion yields the which yields 1/f noise in our model.
A similar (but not identical) nonlinear correction has been found to improve agreement between the Ising model and measured critical scaling in high-purity crystals [28]. One difference is that here we calculate the nonlinear correction in Eq. (2) to all orders using the exact expression for S m -S 0 , not just the lowest-order (quadratic) term. The other difference is that Eq. (2) was bypassed in [28] when δE = 0. Indeed, one reason for bypassing Eq. (2) was to avoid low-frequency fluctuations, whereas here we focus on these fluctuations. Another reason is that, unlike high-purity crystals 1/f noise may require defects [42], which reduce the likelihood of δE = 0 between states. 1/f noise can also be enhanced by non-equilibrium effects [43]. Here we present equilibrium fluctuations of the standard Ising model on simple-cubic lattices using either Eq. (1) or both Eq. (1) and Eq. (2) in every region, only briefly describing simulations where Eq.
(2) is bypassed when δE = 0 in a subset of regions.
Our assumption that Eq. (2) comes from maintaining maximum entropy can be justified in other ways. One mechanism involves an additional change in energy of the bath from the offset in alignment entropy, as in adiabatic magnetization or demagnetization [44,45]. For n noninteracting spins with magnetic moment μ B in an external field B, the internal energy is [46] Here, the free energy (logarithmic term) comes from the thermal average over both states of each spin. Letting B0 the free energy becomes constant, so that the offset in energy from its maximum is ΔE m = T ΔS m . This ΔE m enhances the energy reduction when a region fluctuates into its low-entropy state, increasing the energy of the bath and furthering the fluctuation, consistent with Eq. (2). Another mechanism uses S m as a local bath of alignment entropy [28]. Specifically, high-entropy regions (S m ≈ S 0 ) have many states available facilitating fast spin flips; whereas low-entropy regions (S m ≈ 0) have few states available inhibiting spin dynamics, consistent with Eq. (2). Thus this mechanism is a type of entropic force, similar to Boltzmann's factor where the low entropy of a low-temperature bath inhibits transitions to higher energy. In any case, thermal fluctuations in small regions should obey smallsystem thermodynamics [33], which includes nonextensive thermal properties to all orders.
Additional justification for Eq. (2) comes from the statistics of indistinguishable particles, as described below.   Boltzmann's factor [28]. The central figure depicts how two configurations might be combined into a single m = 0 state, with each site containing a superposition of up and down spins, so that this unaligned state is as likely as each aligned state. For n = 2, these three alignments correspond to the triplet state of spin-½ particles, with the singlet state missing from this basic picture. Thus the nonlinear correction may be a simplistic way to simulate quantum-like statistics in an otherwise classical model. Indeed, a related nonlinear correction has been shown to restore extensive entropy and remove Gibbs' paradox from computer simulations of the Ising model [27,29], similar to how indistinguishable particles remove Gibbs' paradox in an ideal gas.
We simulate the Ising model on simple-cubic lattices of N spins, with interaction energy J between nearest-neighbor spins, and periodic boundary conditions on all external surfaces.
Large lattices are often subdivided into smaller regions, each containing n ≤ N lattice sites. sharp jumps if the sample contains a single region (lower two sets of data). Similar on-off intermittency with varying time duration is known to yield 1/f noise in mathematical models [47,48]. In our model the jumps are due to entropic trapping from Eq. (2). Specifically, as the entropy of the region decreases (due to increased alignment) the entropy of the bath increases, enhancing the lifetime of these highly-aligned states. When the region eventually fluctuates back to high entropy, the time spent near m=0 is brief because the bath has low entropy. nanopore [23] also yield significant probability at the center between the endpoints, unlike the purely bimodal behavior of a double-well potential [49].    Fig. 3 shows no saturation over the full frequency range of our simulations, providing an explanation for the fact that saturation in 1/f noise at low frequencies is rarely observed in real systems. Thus, our model predicts a low-frequency limit to 1/f noise in every finite system, but because of the factorials in the entropy this limit can be at extremely low frequencies.
We have shown that a nonlinear correction to Boltzmann's factor yields 1/f noise in simulations of a simple model, but most materials also show white noise at higher frequencies.
One explanation is that many regions in real systems may have sufficient symmetry to yield nearly-degenerate states that cause Eq. (2) to be bypassed, as found for critical scaling in highpurity crystals [28]. Evidence that 1/f noise involves defects comes from many measurements, including the dependence on electron irradiation [42]. The diamond-shaped symbols that lie below the other 1/f spectra in Fig. 3 show simulations from a heterogeneous system, with one region using Eq. (1) and Eq. (2), while the other 26 regions have Eq. (2) bypassed when δE = 0.
The combination of 1/f noise at low frequencies and white noise at higher frequencies is similar to equilibrium measurements showing that both types of noise usually coexist [50]. Thus, both white noise and 1/f noise may come from thermal fluctuations, with 1/f noise requiring a nonlinear correction to Boltzmann's factor from the local entropy.    The next line comes from ionic conduction through a nanopore [23]. The bottom pair of lines comes from a colloidal particle in two different double-well potentials [49]. Diamond-shaped symbols, which show 1/f noise at low frequencies and white noise at higher frequencies, come from a heterogeneous system described in the text. where α(T)1 as TT 1 . Solid symbols are from measurements [9] on four metallic films. Open symbols connected by solid lines are from simulations using Eqs. (1) and (2). Specifically α(T) is the magnitude of the slope from simulations similar to those that lie along the diagonal in Fig. 3.
The dot-dashed line shows a linear least-squares fit to α(T) from the simulations at T/T 1 ≤1.