Quantifying and minimizing dissipation in a non-equilibrium phase transition

Abstract

In a finite-time continuous phase transition, topological defects emerge as the system undergoes spontaneous symmetry breaking. The Kibble-Zurek mechanism predicts how the defect density scales with the quench rate. During such processes, dissipation also arises as the system fails to adiabatically follow the control protocol near the critical point. Quantifying and minimizing this dissipation is fundamentally relevant to nonequilibrium thermodynamics and practically important for energy-efficient computing and devices. However, there are no prior experimental measurements of dissipation, or the optimization of control protocols to reduce it in many-body systems. In addition, it is an open question to what extent dissipation is correlated with the formation of defects. Here, we directly measure the dissipation generated during the voltage-driven Freedericksz transition of a liquid crystal with a sensitivity equivalent to a ~10 nanokelvin temperature rise. We observe Kibble-Zurek scaling of dissipation and its breakdown, both in quantitative agreement with existing theoretical works. We further implement a fully automated in-situ optimization approach that discovers more optimal driving protocols, reducing dissipation by a factor of three relative to a simple linear protocol.

Figures (4)

• Figure 1: Fréedericksz transition of MBBA under an applied voltage.(a) The LC cell is imaged under a polarized optical microscope (POM) with crossed polarizers $P_1$ and $P_2$. Representative POM images are shown in the image stack. A bipolar voltage ($1$kHz carrier wave) with ramp time $\tau_Q$ is applied to the cell and the current is measured. The LC director is denoted by the vector field $\boldsymbol{\eta}$ with unit norm. (b) Below $V_{\mathrm{th}}$, the vertically aligned configuration minimizes the free energy. Above $V_{\mathrm{th}}$, the director tilts toward the $x-y$ plane and locally chooses a specific azimuthal angle. (c) Colored dotted lines on the voltage trace indicate the acquisition times of the images, with colors matching the borders of the POM frames. The insets in the fourth panel of the third row show magnified views of a topological defect in 2D (view along $z$) and 3D (view along $x$).
• Figure 2: Experimental measurement of dissipated work.(a)$V_{\mathrm{env}}$ and $I_{\mathrm{env}}$ are the envelope amplitudes of voltage $V$ and current $I$. The inset magnifies the boxed region, showing a phase lag of the current behind the voltage close to $\pi/2$. The green curve shows the calculated capacitance of the LC cell. The scale bar of the POM images is 100 $\mu$m. The purple curve shows the relevant input power. The inset highlights the time of negative power input. The brown curve shows the relevant work done $W$. (b) Linear regions in the log-log plot are fitted with power laws (shown in lines), obtaining similar coefficient $\kappa$ between 0.79 and 0.82. The inset (top left) plots the saturation ramp rate $r_Q^{\mathrm{sat}}$ against $\Delta V = V_f - V_{\mathrm{th}}$ in log-log scale. The fitted power law is shown in text.
• Figure 3: Simulated dissipated work as the voltage ramps through the threshold.(a) Simulation counterpart to the experimental data in Fig. \ref{['Fig2']}(a). The scale bar of the POM image represents 50$\mu$m. (b) Simulation counterpart to the experimental data in Fig. \ref{['Fig2']}(b). The power law coefficient $\kappa$ is between 0.81 and 0.82 for different $V_f$.
• Figure 4: Experimental optimization of dissipated work.(a) Top panel: $\tau_Q=1$s and $V_c=6$V, varying $\alpha$. Colormaps (share colorbar): $W_{\mathrm{diss}}$ as a function of $(\alpha, V_c)$ for different $\tau_Q$. All voltages here and below refer to $V_{\mathrm{env}}$ of the protocol. (b) MCMC workflow for protocol optimization. The grid shown in schematics is for illustration (actual size $40\times40$). Red: discretized protocol. Blue: smoothed protocol. The pink arrow shows a random swap. (c) Example MCMC optimization trajectory for $\tau_Q = 1$s and $V_f=10$V. The inset shows the protocol trajectories in faint blue. The darker the curve, the more frequently the protocol appears. (d) Optimal protocols for various $\tau_Q$, plotted with the time axis normalized by $\tau_Q$. (e) Dissipated work under different optimizations. Green: linear ramp. Orange: best protocol from parameter scan in (a). Blue: MCMC-optimized protocol from (d). Fitted power-law exponents are shown in the legend.