$1/f$ 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 way to resolve Gibbs' paradox that avoids 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.

Figure 1

Sketch of possible states in a two-spin region. (a) For distinguishable spins there is one way to have both spins up (${\Omega }_{+2}=1$) or down (${\Omega }_{-2}=1$) but two ways to have zero net alignment (${\Omega }_0$ = 2). (b) During thermal fluctuations the Boltzmann entropy of the spins [$k_B$ln(Ω)] goes up and down (dashed line). To maintain maximum entropy we assume that the entropy of a thermal bath must go down and up (dotted line) so that the combined entropy of the system plus the bath is constant (solid line). (c) When the bath has high entropy each low-entropy state in the region persists twice as long as expected from the Boltzmann factor alone. (d) Alternatively, zero alignment may come from a single state that contains a superposition of spins, consistent with delocalized particles that are indistinguishable in the region.

Figure 2

(a) Time sequence of magnetization per site from simulations on three lattice sizes at two temperatures as given in the legend. Note that $M$($t$) is multiplied by $\sqrt{N}$ to scale the amplitudes, and the data from $N$ = 12${}^3$ and 96${}^3$ are offset for clarity. At $t$ < 0 the spin-flip rate is governed by Boltzmann's factor alone, Eq. (1), yielding white noise. At $t$ ≥ 0 both Eqs. (1) and (2) are used, yielding $1/f$-like noise. (b) Histograms from noise in simulations (symbols) and measurements (lines). Symbols are from the $N$ = 12${}^3$ lattice, similar to (a) at $t$ ≥ 0 but over a much longer time range. The top pair of lines comes from a spin glass at two temperatures [15]. 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].

Figure 3

Frequency dependence of spectral density (in decibels) from simulations at k${}_B$T/$J$ = 50 and 500, similar to those in Fig. 2. Note that $S$($f$) is multiplied by $N$ to scale different lattice sizes (given in the legend) and log${}_{10}$($f$) is multiplied by 10 to match the decibel scale. Also note that the temperature dependence is relatively weak due to effective cancellation of the linear-$T$ dependence of thermal fluctuations and the nearly inverse-$T$ dependence of the magnetic susceptibility at high $T$. The spectra exhibiting white noise (bottom) come from using Eq. (1) alone. Spectra that exhibit $1/f$-like behavior (diagonal) come from the same model using both Eqs. (1) and (2). Over a broad range of frequencies these simulations can be characterized by $S$($f$)∝1/$f^{\alpha \left(T\right)}$ with α($T$) ≈ 1.0 for k${}_B$T/$J$ = 500 (solid line) and α($T$) ≈ 1.15 for k${}_B$T/$J$ = 50 (dotted line). 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.

Figure 4

Spectral-density exponent as a function of normalized temperature $T$/$T_1$, 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.