Role of interactions in nonequilibrium transformations

For arbitrary nonequilibrium transformations in complex systems, we show that the distance between the current state and a target state can be decomposed into two terms: one corresponding to an independent estimate of the distance, and another corresponding to interactions, quantified using the relative mutual information between the variables. This decomposition is a special case of a more general decomposition involving successive orders of correlation or interactions among the degrees of freedom of the system. To illustrate its practical significance, we study the thermal relaxation of two interacting, optically trapped colloidal particles, where increasing pairwise interaction strength is shown to prolong the longevity of the time-dependent nonequilibrium state. Additionally, we study a system with both pairwise and triplet interactions, where our approach identifies their distinct contributions to the transformation. In more general setups where it is possible to control the strength of different orders of interactions, our findings provide a way to disentangle their effects and identify interactions that facilitate the transformation.

Figure 1

(a) Schematics of two identical, hydrodynamically coupled colloidal particles in two spatially separated quadratic potential wells of stiffnesses $k_1$ and $k_2$. (b) The distance between the initial equilibrium system at temperature $T_0$ and the final equilibrium system at temperature $T$. We consider a particular parameter choice $T_0/T=2.5$, as marked. The other parameter choices are $k_1=1,k_2=2,\gamma =1,\eta =1,k_B=1.$

Figure 2

(a) The total distance function $D\equiv D_{\mathrm{KL}}(P\left({\mathit{x}}_t\right)\left|\right|P_{\mathrm{eq}}\left({\mathit{x}}_t\right))$ as well as the independent distance $D_{\mathrm{ind}}$ for different values of $R$ and $t$, for fixed values of initial and final temperatures as well as other model parameters. We find that $D>D_{\mathrm{ind}}$ for all values of $R$ and $t$. (b) The total distance function $D\equiv D_{\mathrm{KL}}(P\left({\mathit{x}}_t\right)\left|\right|P_{\mathrm{eq}}\left({\mathit{x}}_t\right))$ for two values of the separation $R$. (top: $R=0.1$; bottom: $R=0.5$) between the two optical traps. We find that the system at small separation takes longer to thermalize.

Figure 3

(a) The interaction distance $D_{\mathrm{int}}$ for different values of time $t$ and separation $R$. We find that as we decrease $R$ and bring the two particles closer to each other, the interaction distance $D_{\mathrm{int}}$ increases. (b) The total distance $D$ compared with the distance computed for the noninteracting case $D_{\mathrm{nonint}}\equiv {lim}_{R\to \infty }D$, for different values of $R$. As expected, we find that $D_{\mathrm{nonint}}\le D$ for all values of $R$ and $t$, saturating the bound in the $R\to \infty$ limit.

Figure 4

(a) The total distance function $D$ as well as the independent distance $D_{\mathrm{ind}}$ for different values of strength of the external driving, $\alpha$ and $t$, for a fixed value of $R$ and other model parameters. (b) The interaction distance $D_{\mathrm{int}}$ for different values of time $t$ and $\alpha$ for the parameter choice in (a). We find that all the distance functions decrease in value for any $t$ with increasing $\alpha$. The other parameter choices are $k_1=1,k_2=2,\gamma =1,\eta =1,k_B=1.$

Figure 5

(a) The confining potential in Eqs. (14) for $a,\cdots ,d=1$ and $\theta =\frac{\pi }{4}$. (b) The total distance function $D\equiv D_{\mathrm{KL}}(P\left({\mathit{x}}_t\right)\left|\right|P_{\mathrm{eq}}\left({\mathit{x}}_t\right))$ for two values of the parameter $\alpha$ ($\alpha =0$ and $\alpha =10$). We find that the system with larger values of $\alpha$ takes longer to relax to the stationary state. The distance functions are computed by numerically integrating the Langevin equation in Eq. (13) with time-step $dt=0.01$, and constructing histograms at different times using ${10}^5$ copies of trajectories.

Figure 6

(a) The distance functions for different orders of interaction: $D^{\left(1\right)}=D_{\mathrm{ind}}, D^{\left(2\right)}$, and $D^{\left(3\right)}=D$, as well as (b) the contributions to total distance arising solely from pairwise and triplet contributions, $D_{\mathrm{int}}^{\left(2\right)}$ and $D_{\mathrm{int}}^{\left(3\right)}$, for $\alpha =0$ (dot-dashed lines) and $\alpha =10$ (solid lines). We find that the contributions from interactions, especially triplet interactions, are significantly higher (note the logarithmic scale used for the $y$ axis) when $\alpha =10$ as compared to the case with $\alpha =0$. The distance functions are computed by numerically integrating the Langevin equation in Eq. (13) with time step $dt=0.01$, and constructing histograms at different times using ${10}^5$ copies of trajectories.

Figure 7

Dependence of the $x, y$ correlations on $\alpha$ as a function of time $t$ for a fixed value of $R$. The other parameter choices are $k_1=1,k_2=2,\gamma =1,\eta =1,k_B=1.$ The values of $\alpha$ considered are $\alpha =0,0.05,0.1,0.15$.

Figure 8

Various correlations of the $x, y$, and $z$ variables for the dynamical system in Eq. (13) as a function of time, $t$, for two different values of $\alpha$: (a) $\alpha =0$ and (b) $\alpha =10$. The other parameters are kept fixed ($a=1,b=1,c=1,\theta =\frac{\pi }{4},d=1,\gamma =1,\eta =1,k_B=1,T_0=1,T=\frac{1}{10}$). The correlation functions are computed by numerically integrating the Langevin equation in Eq. (13) with time step $dt=0.01$, and using ${10}^5$ copies of trajectories.