Spectral fingerprints of nonequilibrium dynamics: The case of a Brownian gyrator

The same system can exhibit a completely different dynamical behavior when it evolves in equilibrium conditions or when it is driven out-of-equilibrium by, e.g., connecting some of its components to heat baths kept at different temperatures. Here we concentrate on an analytically solvable and experimentally relevant model of such a system—the so-called Brownian gyrator—a two-dimensional nanomachine that performs a systematic, on average, rotation around the origin under nonequilibrium conditions, while no net rotation takes place under equilibrium ones. On this example, we discuss a question whether it is possible to distinguish between two types of a behavior judging not upon the statistical properties of the trajectories of components but rather upon their respective spectral densities. The latter are widely used to characterize diverse dynamical systems and are routinely calculated from the data using standard built-in packages. From such a perspective, we inquire whether the power spectral densities possess some “fingerprint” properties specific to the behavior in nonequilibrium. We show that indeed one can conclusively distinguish between equilibrium and nonequilibrium dynamics by analyzing the cross-correlations between the spectral densities of both components in the short frequency limit, or from the spectral densities of both components evaluated at zero frequency. Our analytical predictions, corroborated by experimental and numerical results, open a new direction for the analysis of a nonequilibrium dynamics.

Figure 1

A sketch of the circuit in Refs. [54, 55]. The two resistances $R_1$ and $R_2$ are kept in separate screened boxes at the temperatures $T_1$ and $T_2$, respectively, and are coupled only by the capacitance $C$. The signal consists of the voltages $V_1$ and $V_2$, which are measured via the amplifiers ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$, respectively. The noise sources ${\eta }_i$ and the capacitances $C_i$, $i=1,2$, of the two circuits are also sketched. The corresponding mathematical model is defined in Eq. (1).

Figure 2

The Pearson coefficient ${\rho }_{\omega }$ in Eq. (17) (solid curves) as a function of the frequency $f=\omega /2\pi$. The noisy curves present ${\rho }_{\omega }$ evaluated from the experimental data corresponding to the following values of the system parameters: $C=100\mathrm{pF}$, $C_1=680\mathrm{pF}$, and $C_2=430\mathrm{pF}$, and $R_1=R_2=10\mathrm{M}\mathrm{\Omega }$. For nonequilibrium dynamics we have $T_1=88\mathrm{K}$ and $T_2=296\mathrm{K}$, while for equilibrium dynamics the system was maintained at equal temperatures $T_1=T_2=296\mathrm{K}$. Solid line: theoretical prediction, Eq. (17). Symbols: numerical simulations, average over 300 000 histories. Dotted lines: experiment, resolution $1\mathrm{Hz}$. Black dashed: frequency dependence for small $f$.

Figure 3

Probability density function [see Eq. (40)] of the random variable $\chi$ [see Eqs. (14) and (15) for the definition] for nonequilibrium (a) and equilibrium (b) situations. Theoretical prediction (40) (solid thick line), thin line (experiment). Circuit parameters and values of the temperatures for equilibrium and nonequilibrium cases are the same as in Fig. 2.

Figure 4

Ensemble-averaged power-spectral density ${\mu }_2\left(\omega \right)$ (thick solid curve), Eqs. (11), as a function of the frequency $f=\omega /2\pi$, together with realization-dependent random function $S_2\left(\omega \right)$ (thin noisy curves) for five randomly chosen realizations of voltages recorded in experiments in Refs. [54, 55]. (a) Out-of-equilibrium dynamics with $T_1=88\mathrm{K}$ and $T_2=296\mathrm{K}$. (b) Equilibrium dynamics with $T_1=T_2=296\mathrm{K}$. Material parameters are as defined in the caption to Fig. 2.

Figure 5

The univariate probability density functions in Eqs. (26) and (35). (a) Distribution of single trajectory spectra out of equilibrium for $\omega =0$ ($f=0$). Theoretical prediction in Eq. (35) (thick solid curve) vs experiment (same parameters as in Ref. [55] and in the caption to Fig. 2). $S_1$ (light red curve): component $x$, $S_2$ (dark red curve). (b) Distributions of single trajectory spectra at equilibrium. Theoretical predictions (dash-dotted curve) in Eq. (26) (upper curves, $f=1\mathrm{Hz}$) and in Eq. (35) (lower solid curves, $f=0$) vs experiment in Ref. [55] (same parameters). $S_1$ (gray curves): component $x$, $S_2$ (light blue curves): component $y$. Note that a slight change of the value of the frequency ($f=1\mathrm{Hz}$ vs $f=0$) results in a drastic change of the shape of the probability density function, in accord with our theoretical predictions.

Figure 6

The case $\omega =0$ with arbitrary $t$. (a) ${\overline{S}}_1={\mathrm{\Sigma }}_1\left(t\right)$ (${\mathrm{V}}^2\mathrm{s}$) versus the observation time $t$ ($\mathrm{s}$). Solid (blue) curve represents the full exact expression; green dashed curve depicts the asymptotic form in the first line in Eqs. (C21), while the red dotted curve depicts the large-$t$ asymptotic curve in the first line in Eqs. (C22), in which we omit the exponential in $t$ relaxation terms. The horizontal (black) dashed line indicates the asymptotic value ${\overline{S}}_1=\overline{S_1(\omega =0,t=\infty )}=4R_1{k}_{\mathrm{B}}{T}_1$. (b) ${\overline{S}}_2={\mathrm{\Sigma }}_2\left(t\right)$ versus the observation time $t$. Solid (blue) curve represents the full exact expression; green dashed curve depicts the asymptotic form in the second line in Eqs. (C21), while the red dotted curve depicts the large-$t$ asymptotic curve in the second line in Eqs. (C22), in which we omit the exponential in $t$ relaxation terms. The horizontal (black) dashed line indicates the asymptotic value ${\overline{S}}_2=\overline{S_2(\omega =0,t=\infty )}=4R_2{k}_{\mathrm{B}}{T}_2$. (c) The Pearson coefficient ${\rho }_0$ as a function of $t$. Solid (blue) curve presents the full exact expression, while the dotted (red) curve indicates the asymptotic form in Eq. (20). The system parameters are as in Fig. 4.