Thermodynamic uncertainty relation of interacting oscillators in synchrony

The thermodynamic uncertainty relation sets the minimal bound of the cost-precision tradeoff relation for dissipative processes. Examining the dynamics of an internally coupled system that is driven by a constant thermodynamic force, we, however, find that the tradeoff relation of a subsystem is not constrained by the minimal bound of conventional uncertainty relation. We made our point explicit by using an exactly solvable model of interacting oscillators. As the number ($N$) of interacting oscillators increases, the uncertainty bound of individual oscillators is reduced to $2k_{B}T/N$ upon full synchronization under strong coupling. The cost-precision tradeoff for interacting subsystems is particularly relevant for subcellular processes where interactions among multiple energy-expending components lead to emergence of collective dynamics.

Figure 1

Two interacting oscillators embedded in a thermal reservoir at temperature $T$. The motion of each oscillator, described by the phase variable ${\theta }_{1,2}$, is powered by the inherent frequency ${\omega }_{1,2}$, corresponding to the nonconservative driving force. The parameter $K$ characterizes the interaction strength between the oscillators, which elicits the synchronization of the phases. The heat dissipated from each oscillator is denoted by $q_{1,2}$.

Figure 2

Thermodynamic uncertainty relations for two coupled oscillators. (a) Heat dissipation rates of two oscillators (blue squares and red circles) and their total heat dissipation rate (black triangles). The two oscillators have intrinsic frequencies of ${\omega }_1=10$ and ${\omega }_2=5$, respectively, and the noise strength is set to $D=1$. (b) Mean phase velocities of two oscillators. (c) Relative diffusion constants that represent phase fluctuations. (d) The uncertainty measures for the cost-precision tradeoffs calculated for the total systems $\left[\mathcal{Q}(q,{\theta }_i)\right]$ and subsystems $\left[\mathcal{Q}(q_i,{\theta }_i)\right]$. The dotted black line represents the usual minimal bound of TUR, $2k_{B}T$. To compute $\sigma$, $v$, and $\mathcal{D}$, an ensemble of ${10}^3$ realizations of stochastic process are used. The lines depict analytical expressions of $\sigma$, $v$, $\mathcal{D}$, and $\mathcal{Q}$ (see Appendix pp2).

Figure 3

Thermodynamic uncertainty relation for interaction strengths of the form of $Ksin\left[m({\theta }_j-{\theta }_i)\right]$. ${\mathcal{Q}}^{\text{sub}}/k_{B}T$ for various $m\in \{1,2,3,4\}$ for two coupled oscillators ($N=2$) with intrinsic frequencies, ${\omega }_1=10$ and ${\omega }_2=5$. The noise strength is set to $D=1$. To compute the cost-precision tradeoff, we took an average over an ensemble of ${10}^3$ realizations.

Figure 4

Thermodynamic uncertainty relations for subsystems of interacting oscillators. (a) ${\mathcal{Q}}^{\text{sub}}/k_{B}T$ for various numbers ($N=2,4,8,16,32$) of oscillators as a function of interaction strength $K$. Plotted are ${\mathcal{Q}}^{\text{sub}}/k_{B}T$ averaged over $N$ oscillators. Individual oscillators have intrinsic frequencies sampled from a normal distribution $\mathcal{N}(\overline{\omega },\mathrm{\Delta }\omega )$ with a mean $\overline{\omega }=10$ and a standard deviation $\mathrm{\Delta }\omega =5$. The noise strength is set to $D=1$. (b) The minimal bound of ${\mathcal{Q}}^{\text{sub}}$ for large $K\gg \mathrm{\Delta }\omega$ (${\mathcal{Q}}_{\text{min}}^{\text{sub}}$) for varying $N$ (data point) is obtained from the simulations of $N$-oscillator system. The red line depicts $2/N$.

Figure 5

Subsystem cost-precision tradeoffs for various collective dynamics. (a) Two coupled oscillators ($N=2$) with intrinsic frequencies ${\omega }_1=10$ and ${\omega }_2=5$ under attractive ($K>0$) and repulsive ($K<0$) interactions. (b) Three coupled oscillators ($N=3$) with intrinsic frequencies ${\omega }_1=15$, ${\omega }_2=10$, and ${\omega }_3=5$ under attractive and repulsive interactions. (c) Thirty-two coupled oscillators ($N=32$) whose intrinsic frequencies are sampled from a normal distribution $\mathcal{N}(\overline{\omega },\mathrm{\Delta }\omega )$ with a mean $\overline{\omega }=10$ and a standard deviation $\mathrm{\Delta }\omega =5$. Among them, 24 oscillators (“conformists”) have attractive coupling strengths ($K_{+}=K$), whereas eight oscillators (“contrarians”) have negative coupling strengths ($K_{-}=-K/2$). The noise strength is set to $D=1$. To compute the tradeoff, an ensemble of ${10}^3$ realizations of stochastic process are used. Insets represent phase snapshots of attractive oscillators (blue circles) and repulsive oscillators (red circles) at strong coupling regimes.