19  Active matter

TipLearning outcomes

1. Understand how local energy consumption in the form of self-propulsion leads to new, nonequilibrium physics, with the help of example minimal models.
2. Understand how self-propulsion modifies the scaling behaviour of the mean squared displacement in active systems.
3. Describe how self-propulsion leads to nonequilibrium phase separation.

19.1 Beyond thermal systems

All the systems we have considered up to now are composed of naturally occurring or synthetic molecules (or atoms) which interact via conservative forces (potentials) and are subject to thermal fluctuations. Essentially, energy is either stored in the system as potential energy or exchanged with the environment as heat, leading to equilibrium dynamics governed by the laws of thermodynamics and statistical mechanics. In these systems, the random motion of particles arises from thermal noise, and the system eventually relaxes to a Boltzmann distribution.

However, many systems in nature and technology are driven out of equilibrium by continuous energy consumption at the microscopic scale. These are known as active matter systems. Here, each constituent (such as a bacterium, a synthetic microswimmer, or a molecular motor) converts energy from its surroundings into a variety of mechanisms:

• motion
• sensing
• processing
• growth and deformation

This is what is typically performed by many living organisms: by consuming energy locally (hence dissipating heat) they perform some function. Here we focus on one of the most fundamental kinds of functions which is motion. A bacterium uses energy to propel itself into space, explore it and eventually interact with the environment, including other bacteria. Active systems can reach steady states, whose observables do not change in time, but unlike equilibrium states these have positive entropy production and break detailed balance (compare with the arrested states chapter).

We could, at this stage, say that these interactions and the behaviour emerging from them are inherently biological and governed solely by mechanisms that transcend the physical correlations (behavioural science). As physicists, we do not negate the existence of such dimensions (especially when communication and sensing are involved) but aim to run a different research programme: we want to gauge the level of biological surprise¹, as nicely put by Andrea Cavagna, a complex system physicist.

To do so we need to understand how systems that dissipate energy locally to produce self-propelled motion can lead to the emergence of non-trivial phase behaviour and how this differs from systems that are either at thermal equilibrium (e.g. colloidal fluids) or are slowly relaxing towards it (e.g. glasses and gels).

The field of research that studies such systems is called physics of active matter and in this chapter we will describe some reference model systems and main results.

19.2 Flocking: the Vicsek model

The Vicsek model (Vicsek et al. 1995) is one of the simplest models of active matter. Its key ingredients are self-propulsion and alignment: \(N\) particles move at constant speed \(v_0\) and, at each time step, take the average direction of their neighbours within a radius \(R\), up to some noise: \[ \mathbf{r}_i(t+1)=\mathbf{r}_i(t)+v_0\hat{\mathbf{n}}_i(t),\qquad \theta_i(t+1)=\arg\Big(\sum_{|\mathbf{r}_j-\mathbf{r}_i|<R}e^{i\theta_j(t)}\Big)+\xi_i, \] where \(\xi_i\) is uniform in \([-\eta/2,\eta/2]\).

• At high noise \(\eta\) (or low density) the particles move in random directions.
• At low noise (or high density) they spontaneously move together: the system flocks.

The transition is measured by the polar order parameter \(\phi=\big|\frac1N\sum_i e^{i\theta_i}\big|\), which goes from \(0\) (disordered) to \(1\) (all particles moving together). Careful simulations show that the transition is discontinuous: close to it, the ordered phase forms dense travelling bands (Grégoire and Chaté 2004).

19.3 Life-inspired motion: run and tumble

The run-and-tumble model describes the motion of active particles, such as bacteria, that alternate between two types of movement: “runs” (straight-line motion) and “tumbles” (random reorientation). This model captures the behavior of microorganisms like E. coli.

Figure 19.1: Swimming Escherichia coli bacteria with fluorescently-labeled flagellar filaments, showing individual run-and-tumble events with flagellar filaments unbundling, cells turning, and flagellar filaments rejoining the bundle. The details of the experimental procedure are given in Turner et al. (2000).

viewof simulation = {
    const n = 100, W = 1000, H = 300, dt = 0.5, v0 = 1.5;

    const tumbleSlider = Inputs.range([0.01, 1.0
    ], {step:0.01, value:0.1, label:"Tumble rate λ"});
    const button = html`<button>Pause</button>`;
    let running = true;
    button.onclick = () => {
        running = !running;
        button.textContent = running ? "Pause" : "Resume";
    };

    let state = Array.from({length: n}, () => ({
        x: Math.random() * W,
        y: Math.random() * H,
        theta: Math.random() * 2 * Math.PI
    }));

    // Each particle gets a trace array
    let traces = Array.from({length: n}, () => []);

    // Reset traces when tumble rate changes
    let lastLambda = tumbleSlider.value;
    tumbleSlider.addEventListener("input", () => {
        traces = Array.from({length: n}, () => []);
        lastLambda = tumbleSlider.value;
    });

    function step(state, λ) {
        return state.map((p, i) => {
            if (Math.random() < λ * dt) p.theta = Math.random() * 2 * Math.PI;

            // Predict next position
            let nx = p.x + v0 * Math.cos(p.theta) * dt;
            let ny = p.y + v0 * Math.sin(p.theta) * dt;

            // Bounce at left/right walls
            if (nx < 0) {
                p.x = 0;
                p.theta = Math.PI - Math.random() * Math.PI; // random angle pointing right
            } else if (nx > W) {
                p.x = W;
                p.theta = Math.PI + Math.random() * Math.PI; // random angle pointing left
            } else {
                p.x = nx;
            }

            // Bounce at top/bottom walls
            if (ny < 0) {
                p.y = 0;
                p.theta = (Math.random() * Math.PI); // random angle pointing down
            } else if (ny > H) {
                p.y = H;
                p.theta = Math.PI + (Math.random() * Math.PI); // random angle pointing up
            } else {
                p.y = ny;
            }

            // Add current position to trace
            traces[i].push([p.x, p.y]);
            // Limit trace length for performance
            if (traces[i].length > 200) traces[i].shift();
            return p;
        });
    }

    const container = html`<div></div>`;
    container.append(tumbleSlider, button);
    const svg = d3.select(container).append("svg")
        .attr("width", W)
        .attr("height", H)
        .style("background", "#f8f8f8");

    while (true) {
        const λ = tumbleSlider.value;
        if (running) state = step(state, λ);

        svg.selectAll("*").remove();

        // Draw traces
        traces.forEach(trace => {
            if (trace.length > 1) {
                svg.append("path")
                    .attr("d", d3.line()(trace))
                    .attr("stroke", "#90caf9")
                    .attr("stroke-width", 1)
                    .attr("fill", "none");
            }
        });

        // Draw particles
        svg.selectAll("circle")
            .data(state)
            .enter().append("circle")
            .attr("cx", d => d.x)
            .attr("cy", d => d.y)
            .attr("r", 3)
            .attr("fill", "#1976d2");

        svg.selectAll("line")
            .data(state)
            .enter().append("line")
            .attr("x1", d => d.x)
            .attr("y1", d => d.y)
            .attr("x2", d => d.x + 10 * Math.cos(d.theta))
            .attr("y2", d => d.y + 10 * Math.sin(d.theta))
            .attr("stroke", "#1976d2")
            .attr("stroke-width", 1.2);

        yield container;
        await Promises.delay(16);
    }
}

Figure 19.2: Non-interacting run and tumble particles.

The model is designed to encode a minimal set of ingredients that make the characteristic trajectories of the bacteria quite different from an ordinary random walk. We can formulate this kind of model via a simple algorithm.

1. Run Phase: During a run, the particle moves in a straight line with constant velocity \(v_0\): \[ \mathbf{r}(t + \Delta t) = \mathbf{r}(t) + v_0 \hat{\mathbf{n}}(t) \Delta t \]

   where:

   • \(\mathbf{r}(t)\) is the position of the particle at time \(t\),
   • \(v_0\) is the constant speed,
   • \(\hat{\mathbf{n}}(t)\) is the unit vector indicating the direction of motion.

2. Tumble Phase: During a tumble, the particle randomly reorients. The new direction \(\hat{\mathbf{n}}(t)\) is chosen from a uniform distribution over the unit sphere (in 3D) or circle (in 2D).

We can imagine a schematic algorithm: - Initialize the particle’s position \(\mathbf{r}(0)\) and direction \(\hat{\mathbf{n}}(0)\). - For each time step \(\Delta t\): - With probability \(\lambda \Delta t\), perform a tumble (randomize \(\hat{\mathbf{n}}\)). - Update the position using the run equation. - Repeat for the desired simulation duration.

Here, \(\lambda\) is the tumble rate, which determines the frequency of reorientation events.

The mean squared displacement (MSD) of run-and-tumble particles exhibits two regimes, connected by a crossover:

1. Ballistic regime (short times):
   At very short times, before the first tumble occurs, particles move in straight lines at constant speed. The MSD grows quadratically with time: \[ \langle \Delta r^2(t) \rangle \sim v_0^2 t^2 \]

2. Diffusive regime (long times):
   At times much longer than the average run time (\(t \gg 1/\lambda\)), the direction of motion has been randomized many times, and the motion becomes diffusive: \[ \langle \Delta r^2(t) \rangle \sim 4 D_{\text{eff}} t \] where \(D_{\text{eff}} = \frac{v_0^2}{2\lambda}\) in 2D.

The full expression in 2D, \[ \langle \Delta r^2(t) \rangle = \frac{2v_0^2}{\lambda^2}\left(\lambda t-1+e^{-\lambda t}\right), \] interpolates smoothly between the two regimes.

19.4 Active Brownian particle and motility-induced phase separation

Active Brownian particles (ABPs) are a minimal model for self-propelled colloids, such as Janus particles, which move due to chemical reactions at their surfaces. For example, a colloid half-coated with platinum can catalyze the decomposition of hydrogen peroxide in solution, generating local gradients that propel the particle forward.

The ABP model captures the essential physics of these systems. Each particle moves with a constant speed in a direction that undergoes rotational diffusion. This leads to persistent motion at short times and diffusive behavior at long times, similar to the run-and-tumble model but with continuous reorientation.

The dynamics of an ABP can be described by the following equations:

1. Translational Motion: The position \(\mathbf{r}(t)\) of the particle evolves as: \[ \frac{d\mathbf{r}(t)}{dt} = v_0 \hat{\mathbf{n}}(t) + \sqrt{2D_t} \boldsymbol{\xi}(t), \] where:
   • \(v_0\) is the self-propulsion speed,
   • \(\hat{\mathbf{n}}(t)\) is the unit vector indicating the particle’s orientation,
   • \(D_t\) is the translational diffusion coefficient (for a sphere, given by the Stokes-Einstein relation),
   • \(\boldsymbol{\xi}(t)\) is a Gaussian white noise term with zero mean and correlations \(\langle\xi_i(t)\xi_j(t')\rangle=\delta_{ij}\delta(t-t')\).
2. Rotational Motion: The orientation \(\hat{\mathbf{n}}(t)\) undergoes rotational diffusion, described by: \[ \frac{d\hat{\mathbf{n}}(t)}{dt} = \sqrt{2D_r}\, \boldsymbol{\eta}(t)\times\hat{\mathbf{n}}(t), \] where:
   • \(D_r\) is the rotational diffusion coefficient,
   • \(\boldsymbol{\eta}(t)\) is a Gaussian white noise term as above (the cross product keeps \(\hat{\mathbf{n}}\) a unit vector).
   In 2D, the orientation \(\hat{\mathbf{n}}(t)\) can be expressed in terms of an angle \(\theta(t)\): \[ \hat{\mathbf{n}}(t) = (\cos\theta(t), \sin\theta(t)), \] and the rotational dynamics reduce to: \[ \frac{d\theta(t)}{dt} = \sqrt{2D_r} \eta_\theta(t), \] where \(\eta_\theta(t)\) is a scalar Gaussian white noise term.

The relative importance of self-propulsion and thermal motion is measured by the Péclet number \(\mathrm{Pe}=v_0/\sqrt{D_tD_r}\): for \(\mathrm{Pe}\ll1\) the particles behave as passive Brownian colloids, while for \(\mathrm{Pe}\gg1\) they are strongly active.

We can analyse this a little more in detail by considering the characteristic features of the displacements of (dilute or noninteracting) active Brownian particles.

19.4.1 Mean squared displacement of Active Brownian Particles

Indeed, it can be shown (see below) that the mean squared displacement for active Brownian particles displays three regimes: \[ \operatorname{MSD}(\tau) = \left[4 D_t + 2 v_0^2 \tau_R \right] \tau + 2 v_0^2 \tau_R^2 \left( e^{-\tau / \tau_R} - 1 \right) \]

where: - \(D_{\mathrm{eff}} = D_t + \frac{v_0^2}{2 D_r}\) is the effective long-time diffusion coefficient (with \(\tau_R = 1/D_r\)),

This clearly shows that there are three main regimes:

• a very short time diffusive regime \(\operatorname{MSD}(\tau) \simeq 4D_t\tau\)
• an intermediate regime, \(4D_t/v_0^2 \ll \tau \ll \tau_R\), before any reorientation has taken place: \[ \operatorname{MSD}(\tau) = 4 D_t \tau + v_0^2 \tau^2 \] leading to super-diffusive (ballistic) motion
• a final regime where the reorientation has taken place and where a new diffusive regime is reached where the MSD is proportional to time, i.e. \(\operatorname{MSD}(\tau)=\left[4 D_t+2 v_0^2 \tau_{\mathrm{R}}\right] \tau\) but with a new diffusion constant.

Note

We derive the MSD expression for 2d Active Brownian particles.

We first integrate the velocity

\[ \mathbf{r}(\tau) - \mathbf{r}(0) = v_0 \int_0^\tau \hat{\mathbf{n}}(s) ds + \sqrt{2 D_t} \int_0^\tau \boldsymbol{\eta}(s) ds. \]

Since \(\boldsymbol{\eta}(t)\) is independent from \(\hat{\mathbf{n}}(t)\),

\[ \langle \Delta r^2(\tau) \rangle = v_0^2 \left\langle \left| \int_0^\tau \hat{\mathbf{n}}(s) ds \right|^2 \right\rangle + 4 D_t \tau, \]

using \(\langle |\sqrt{2D_t}\int_0^\tau \boldsymbol{\eta}(s) ds|^2 \rangle = 2 d D_t \tau\) with dimension \(d=2\).

We first need to evaluate the orientation correlation integral.

Write the first term as a double integral:

\[ \left\langle \left| \int_0^\tau \hat{\mathbf{n}}(s) ds \right|^2 \right\rangle = \int_0^\tau ds \int_0^\tau ds' \langle \hat{\mathbf{n}}(s) \cdot \hat{\mathbf{n}}(s') \rangle. \]

For 2D rotational diffusion, the orientation correlation is exponential:

\[ \langle \hat{\mathbf{n}}(s) \cdot \hat{\mathbf{n}}(s') \rangle = e^{-|s - s'|/\tau_R}. \]

We calculate the following double integral

\[ I(\tau) = \int_0^\tau ds \int_0^\tau ds' e^{-|s - s'|/\tau_R}. \]

Use symmetry: the integrand depends only on \(|s - s'|\). Express as

\[ I(\tau) = 2 \int_0^\tau ds \int_0^s ds' e^{-(s - s')/\tau_R} = 2 \int_0^\tau ds \int_0^s du\, e^{-u/\tau_R}, \]

where we set \(u = s - s'\).

Integrate over \(u\):

\[ \int_0^s e^{-u/\tau_R} du = \tau_R \left(1 - e^{-s/\tau_R}\right). \]

So

\[ I(\tau) = 2 \tau_R \int_0^\tau \left(1 - e^{-s/\tau_R}\right) ds = 2 \tau_R \left[ \tau - \int_0^\tau e^{-s/\tau_R} ds \right]. \]

By integrating the exponential we get

\[ \int_0^\tau e^{-s/\tau_R} ds = \tau_R \left(1 - e^{-\tau/\tau_R}\right). \]

Hence

\[ I(\tau) = 2 \tau_R \left[ \tau - \tau_R \left(1 - e^{-\tau/\tau_R} \right) \right] = 2 \tau_R \tau - 2 \tau_R^2 \left(1 - e^{-\tau/\tau_R} \right). \]

Putting it all together,

\[ \langle \Delta r^2(\tau) \rangle = v_0^2 I(\tau) + 4 D_t \tau = v_0^2 \left[ 2 \tau_R \tau - 2 \tau_R^2 (1 - e^{-\tau/\tau_R}) \right] + 4 D_t \tau, \]

or equivalently

\[ \boxed{ \langle \Delta r^2(\tau) \rangle = \left(4 D_t + 2 v_0^2 \tau_R \right) \tau + 2 v_0^2 \tau_R^2 \left( e^{-\tau/\tau_R} - 1 \right). } \]

This crossover from ballistic to diffusive behavior is a hallmark of persistent random walks such as ABPs. With \(D_t=0\) it is identical to the run-and-tumble result for \(\lambda=1/\tau_R\): more generally, run-and-tumble and active Brownian particles are equivalent on large scales when \(\lambda=(d-1)D_r\) (Cates and Tailleur 2013). Note that \(\tau_R=1/D_r\) in 2D, while in 3D \(\langle\hat{\mathbf n}(t)\cdot\hat{\mathbf n}(0)\rangle=e^{-2D_rt}\).

19.4.2 Interacting ABPs and motility induced phase separation

When the active Brownian particles are actually interacting with each other, they can give rise to an exceptional nonequilibrium phenomenon of self organisation called motility-induced phase separation (MIPS).

Figure 19.3: Motility induced phase separation in a 3D system of active Brownian particles. As the persistence of the motion increases (i.e. the rotational diffusion decreases), the system phase separates into a dense and a dilute phase, akin to a liquid-gas phase separation in equilibrium systems. Adapted from Turci and Wilding (2021)

The idea is to consider purely repulsive particles (i.e hard spheres) obeying the ABP dynamics. This means that, in equilibrium (i.e. in the absence of self-propulsion) no liquid-gas phase separation is possible (see Section 15.3.1).

However, as we reduce the rotational diffusion, the persistent motion becomes more important, the system becomes more out of equilibrium and new physics comes to play.

In particular, the collision between particles are no longer leading to quick decorrelation: on the contrary, head-to-head collisions between the particles mean that there is a finite residence time for a pair of particles to stay in each other neighborhoods. This effect is amplified by multiple collisions, leading to many-body caging effects that promote density heterogeneities in the fluid.

The mechanism is a positive feedback (Cates and Tailleur 2015). First, active particles accumulate where they move slowly: non-interacting active particles with a slowly varying speed \(v(\mathbf r)\) reach a steady state with \[ \rho(\mathbf r)\propto\frac{1}{v(\mathbf r)}, \] a non-uniform density without any potential, which no Boltzmann distribution can produce. Second, collisions make the speed decrease with density, \(v(\rho)\). Together, a denser region slows particles down, which makes it denser still. The homogeneous state becomes unstable (a spinodal instability) when \[ \frac{d\ln v}{d\ln\rho}<-1 . \]

The result is striking: for sufficiently low rotational diffusions, the fluid of active Brownian particles spontaneously phase separates into a dilute and a dense phase, akin to the gas and liquid phase of equilibrium systems. As in equilibrium liquid-gas coexistence, the coexistence region ends at a critical point, where density fluctuations are strongly enhanced.

This has been examined in detail in computer simulations, both in two dimensional systems and in three dimensional systems (where the physics is even richer and presents parallels with the situation of colloid-polymer mixtures due to the very short range nature of the effective interactions between active particles).

19.5 Summary

The table collects the main results of this chapter. Follow the links to go back to the relevant part.

Topic	Key result	Remember
Active matter	Local energy consumption → self-propulsion	No Boltzmann distribution
Vicsek model	Self-propulsion + alignment, noise \(\eta\); polar order \(\phi=\lvert\frac1N\sum_ie^{i\theta_i}\rvert\)	Flocking at low noise or high density
Run-and-tumble	Straight runs at \(v_0\), random tumbles at rate \(\lambda\)	\(D_{\rm eff}=v_0^2/2\lambda\) (2D); equivalent to ABPs for \(\lambda=(d-1)D_r\)
Active Brownian particles	\(\dot{\mathbf r}=v_0\hat{\mathbf n}+\sqrt{2D_t}\,\boldsymbol\xi\), \(\dot\theta=\sqrt{2D_r}\,\eta_\theta\); \(\mathrm{Pe}=v_0/\sqrt{D_tD_r}\)	Continuous reorientation, \(\tau_R=1/D_r\)
MSD	\(4D_t\tau\) → \(v_0^2\tau^2\) → \(4D_{\rm eff}\tau\), with \(D_{\rm eff}=D_t+v_0^2/2D_r\)	Diffusive, then ballistic, then enhanced diffusion
MIPS	Purely repulsive ABPs phase separate at low \(D_r\)	Dense \(\to\) slow \(\to\) denser; unstable if \(d\ln v/d\ln\rho<-1\); critical point

References

Cates, Michael E., and Julien Tailleur. 2013. “When Are Active Brownian Particles and Run-and-Tumble Particles Equivalent? Consequences for Motility-Induced Phase Separation.” EPL 101: 20010. https://doi.org/10.1209/0295-5075/101/20010.

Cates, Michael E., and Julien Tailleur. 2015. “Motility-Induced Phase Separation.” Annual Review of Condensed Matter Physics 6: 219–44. https://doi.org/10.1146/annurev-conmatphys-031214-014710.

Grégoire, Guillaume, and Hugues Chaté. 2004. “Onset of Collective and Cohesive Motion.” Physical Review Letters 92: 025702. https://doi.org/10.1103/PhysRevLett.92.025702.

Turci, Francesco, and Nigel B. Wilding. 2021. “Phase Separation and Multibody Effects in Three-Dimensional Active Brownian Particles.” Phys. Rev. Lett. 126 (January): 038002. https://doi.org/10.1103/PhysRevLett.126.038002.

Turner, Linda, William S Ryu, and Howard C Berg. 2000. “Real-Time Imaging of Fluorescent Flagellar Filaments.” Journal of Bacteriology 182 (10): 2793–801.

Vicsek, Tamás, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet. 1995. “Novel Type of Phase Transition in a System of Self-Driven Particles.” Physical Review Letters 75 (6): 1226–29. https://doi.org/10.1103/PhysRevLett.75.1226.

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1. Andrea Cavagna, a complex system physicist, describes “biological surprise” as the extent to which observed behaviour cannot be explained by physical interactions alone.