Measurement-Induced Phase Transitions in Informational Active Matter

Abstract

Various biological and synthetic media out of equilibrium can be viewed as many-ratchet systems that rectify environmental noise through local measurements and information processing, like in Maxwell's prototypical demon. These systems pose a challenge to standard coarse-graining approaches because they are better described in terms of decision-making protocols similar to computer programs rather than force laws. Here, we study a many-body generalization of the Maxwell demon problem: a fluid composed of adaptive particles that achieve collective behavior by biasing noise-driven scattering events subject to measurements. Using a combination of information-theoretic, kinetic, and hydrodynamic tools, we elucidate how microscopic decision-making protocols, rather than microscopic forces, generate macroscopic active states sustained by continuous measurements. These include an informational version of flocking whose order parameter is bounded by the information measured, and the onset of which may be viewed as a measurement-induced phase transition. We find that the signature of such microscopic choices is an `informational activity' that selectively compresses phase space, without work, and causes deviations from equilibrium scaling with the magnitude of environmental noise. We envision applications to noise-induced patterning performed by collections of microrobots guided by reinforcement learning or programmable phoretic colloids in turbulent flows that exploit local measurements and control actions to counteract the scrambling of information by chaos.

Figures (10)

• Figure 1: The demon gas uses measurement to break detailed balance without work.a. A gas of feedback controllers, hard disks capable of changing their diameter in response to a measurement. Here we consider gases where measurements are instantaneous and synchronized, with a delay of $t_m$ between subsequent measurements. b. Controllers maintain energy conservation during collision events by disallowing diameter changes that would introduce particle overlaps. c. Diameter as a function of particle velocity breaks detailed balance - a 'reverse' collision that begins where a 'forward' collision ends does not recover the initial configuration of the forward collision. d. A demon gas with periodic boundary conditions, simulated via molecular dynamics. e-f. Diameter changing conditions for a nematically-aligning demon gas. e. When particle velocity is aligned (or anti-aligned) with the majority of nearby neighbors, a small diameter is chosen. f. When particle velocity is perpendicular to the majority of nearby neighbors, a large diameter is chosen. g. Snapshots of binned orientation fields for an initially passive gas at short times after diameter rescaling is enabled. $\langle \tau_{col}\rangle$ is the mean collision time assuming an even split of large and small diameter particles. Nematic defects form and annihilate leading to a nearly uniform orientation. h. Long-time distribution of particle orientations for the gas in part (g). The times corresponding to snapshots in (g) are indicated with dashed lines. i The anisotropic pressure of a demon gas trapped within a passive flexible container inflates it into an ellipsoidal shape (colored by radial deformation $r$ from circular initial condition $r_0$).
• Figure 2: The demon gas is driven by information. a. Particles instantaneously measure their velocity and update their diameters (subject to constraints due to proximity) at an interval of $t_m$. b. The entropy (relative to the Maxwell-Boltzmann distribution) of a demon gas measured from its velocity distribution function. Here the measurement interval is several times longer than the mean collision time ($t_m/\tau_{\text{col}}=[3.6,4.3,5]$ for $\Delta D=[1,2,3]$). c. Relative entropy of demon gases with a range of measurement intervals, up to approximately one collision time. d. The nonequilibrium free energy injected at each measurement, as estimated by fitting curves similar to (c) as a function of the information per measurement of a Markov process describing the diameter of particles (see extended Fig. \ref{['efig:parameter_fitting']}). Note that $I$ increases with the interval between measurements $t_m$.e. The average nonequilibrium free energy as a function of the information rate of the diameter Markov process. Note that for the values studied here $dI/dt$ increases as $t_m$ is reduced.f-j. The result of various diameter functions on the gas velocity distribution. f. The Maxwell-Boltzmann distribution in the velocity plane (constant diameter). g. A zero-centered Gaussian diameter function. h. A quadratic diameter function of one velocity component. i. A diameter step function that breaks reflection symmetry. j. A diameter step function in angular velocity coordinates that breaks pressure symmetry. k. The entropy production rate of various demon gases as a function of non-equilibrium free energy. l. Growth of the nematic order parameter with $Q_0$ for simulations of demon gases with explicitly (blue) and spontaneously (orange) broken symmetry. The system-wide average of the order parameter does not approach zero as $Q_0\to 0$ due to thermal fluctuations. m. Nematic order parameter as temperature is increased 50-fold. Nematic ordering is approximately independent of temperature.
• Figure 3: Noise-driven active patterning.a. A collection of variable-diameter particles are dropped into a hard-sided box. As particles undergo elastic collisions, their velocities are reduced by a linear drag ($\gamma=0.01$) term and they settle onto the bottom of the box. The two species (red and blue) have opposite $\bm{\xi}(\bm{x})$ fields (inset), driving separation into a designed pattern. b. Self-propelled particles that exert a constant magnitude force in the direction specified by $\bm{\xi}$ (i.e. blue (red) particles push themselves away from (towards) the nearest segment of the pattern), while experiencing constant agitation (linear drag $\gamma=0.01$, random forces with mean magnitude $6k_bT_\textrm{eff}\gamma/\delta t$, see methods for details). As agitation is increased ($k_bT_{\text{eff}}$), pattern resolution degrades, since particles can only push with a fixed amount of force. Frames are instantaneous snapshots of simulations at the time and effective temperature indicated by the position of the inset lower left corner. c. Controller particles following the diameter rule of eq. \ref{['eq:step']} and the same $\bm \xi$ field and non-thermal agitation as (b). The time required to obtain a given resolution decreases with increasing agitation. Black curves are $t = sL/\langle |\bm{v}| \rangle$ for $s=0.15\to 2$ where $L$ is the size of the simulation domain and $\langle | \bm{v}|\rangle = \sqrt{2k_bT_{\text{eff}}/m}$ is the mean speed of a particle in the gas. d. Velocity distribution function of demon gas particles that adopt small (large) diameters when traveling right (left) immersed in a fixed-diameter isothermal gas. Exchange of linear momentum with the passive gas allows the demon gas to concentrate in the positive half of the velocity plane, as predicted by kinetic theory. e. Mean bulk velocity $u_x$ of the demon gas in (d) as a function of temperature, with fixed $\Delta D=1$. Mean bulk flow due to collisional biasing is a fraction of the thermal speed. f. Density of demon gas particles that adopt small (large) diameters when moving towards (away) from the origin. Selective collisions with each other and a surrounding passive isothermal gas concentrate them near the target location. g. Depth of an effective potential consistent with the increased density of demon gas particles near the target in (f), as a function of temperature.
• Figure E1: Parameter fitting of the free energy of the demon gas. a-c Estimated parameters of the free energy cycles of demon gases with various measurement intervals ($t_m$) and diameter differences ($\Delta D$). These parameters summarize the data presented in d-l. a. Value of free energy at the start and end of a measurement cycle. b. Free energy impulse delivered to the demon gas from measurement and diameter change. c. Relaxation parameter for free energy dynamics. d-l. Free energy data, fit by the functional form presented in methods to extract the parameters in a-c. Rows increase measurement time top to bottom, columns increase $\Delta D$ left to right. Note that the smallest measurement time that can be reliably fit is a function of $\Delta D$.
• Figure E2: Markov chain model and simulation of a demon particle immersed in an isothermal gas. a. Transition diagram for the states of a demon particle following a binary diameter rule: $D_L$ when $v_x<0$, $D_S$ when $v_x>0$. Collisions occur in the time between measurements ($t_m$) with probability $P_\text{col}^L$ (for large particles or $P_\text{col}^S$ for small) and randomize velocities (scattering into negative or positive velocities with probabilities $P_-$ or $P_+$ respectively). For hard disks, shrinking diameter never requires the demon particle to exert work on the surrounding gas, but expansions are only possible if no other obstacles (i.e. particles) are nearby (probability $P_<$). b. Condensed two-state Markov chain model transition diagram for the one-bit measurement that demon particles collect to determine which diameter state to adopt. In the steady state, the measurement sequence $M=[m_t,m_{t+t_m},\dots]$ has a mean entropy per measurement, $H(M)$. c. Work done to propel a single demon particle through a passive gas per measurement cycle (duration $t_m$). Power dissipated in demon particle motion is estimated from simulation by finding the density-dependent drift velocity $\bm{u}$ and mobility coefficient $\mu_v$. The black curve is the average free energy dissipated in deleting the demon particle's memory of prior measurements, obtained from a Markov chain model of the measurement process (see SI section S4 for additional details), where $\sigma$ is the intrinsic efficiency of rectification by collisional biasing and $\eta$ is a fitted constant of order one. Numerical and theoretical estimations agree on the location of the work maxima at a density of $\rho=\rho^* = 1/A$, where $A$ is the excluded volume change as the demon particle changes between its two diameter states.