Thermal Quorum-Sensing Active Particles

• Thermal Quorum-Sensing Active Particles (tQSAPs) are motile agents that adjust their self-propulsion based on local density detected through thermal diffusion or chemical signals.
• The models employ Langevin dynamics with density-dependent speeds and various sensing kernels to implement sharp or graded quorum thresholds.
• tQSAPs exhibit rich nonequilibrium behaviors such as phase separation, annular swarming, and absorbing transitions, promising advances in adaptive material design and biological probing.

Thermal Quorum-Sensing Active Particles (tQSAPs) are a class of synthetic or biological motile systems in which individual agents modulate their self-propulsion in response to local density, mediated by thermal diffusion and/or chemical signaling. The defining property is that each particle senses the surrounding environment—either through direct neighbor count, nonlocal thermal/chemical fields, or fluctuating order-parameter kernels—and adapts its motility according to a threshold or graded rule. This density-dependent feedback yields rich nonequilibrium phase behavior, including clustering, annular swarming, absorbing transitions, and re-entrant order–disorder regimes inaccessible in constant-motility active matter.

1. Microscopic Dynamics and Governing Equations

Each tQSAP is modeled as an overdamped, self-propelled agent subject to thermal noise and a quorum-sensing regulated motility function. Common forms include:

• Langevin Dynamics: For particle $i$ at position $\mathbf{r}_i$, orientation $\mathbf{u}_i$,

$$\dot{\mathbf r}_i = v[\rho(\mathbf r_i)]\,\mathbf u_i + \sqrt{2D_t}\,\boldsymbol\xi_i(t) \quad;\quad \dot{\mathbf u}_i = -D_r\,\mathbf u_i + \sqrt{2D_r/d}\,\boldsymbol\eta_i(t)$$

where $v[\rho]$ is the local density-dependent speed, $D_t$ the translational diffusivity, and $D_r$ the rotational diffusivity (Dinelli et al., 2024, Lefranc et al., 19 Feb 2025, Bureković et al., 23 Jan 2026).

• Quorum-Sensing Rule: Motility is suppressed or activated when a sensed quantity—neighbor number or coarse-grained density—crosses a threshold:

$v(S) = v_0\,H(S - S_c)$

where $S$ is generated as a convolution,

$$S(x, t) = \int_{\mathbb R^2} K(x - y)\,\rho(y, t)\,d y$$

with $K$ a sensing kernel (top-hat, Gaussian, etc.) and $H$ the Heaviside step function (Smith et al., 22 Apr 2025).

• Thermal Quorum Sensing in Janus Colloids: Quorum-sensing feedback is coupled to thermally mediated forces where collective heating modulates both motility and particle interactions. Stochastic equations link self-propulsion, thermodiffusion (Soret effect), and heat-mediated drift (Golestanian, 2011).

2. Kernel Functions, Agent-Based Rules, and Activation Schemes

The precise mechanism for quorum sensing depends on the model and experimental context:

• Sensing Kernel: Either a finite-radius top-hat ($K(x) = \frac{1}{\pi R^2}\,\chi_{|x| < R}$) for local neighbor count, or a Gaussian kernel for graded, distance-dependent sensing. The range parameter $R$ or $\sigma$ sets the spatial scale for thresholding (Smith et al., 22 Apr 2025).
• Agent-Based Steps: Each agent updates its position and orientation according to whether their sensed local signal $S_i$ exceeds the threshold $P^*$. Activation is instantaneous in the sharp-threshold models, or can be softened to a continuous sigmoid function. Updates combine propulsive drift (active only if $A_i = 1$) and Brownian diffusion (Smith et al., 22 Apr 2025, Velasco et al., 2018).
• Bimodal Motility: In simpler cases, a binary speed is assigned depending on quorum status ("in quorum" vs "out of quorum" states). The ratio $X = V_i/V_o$ and persistence lengths $\ell_{i,o}$ are key control parameters for clustering and dispersal transitions (Thapa et al., 2023).

3. Phase Behavior, Clustering, and Pattern Formation

tQSAP systems generically admit several nonequilibrium regimes as control parameters (Péclet number, packing fraction, persistence time, quorum threshold, kernel range) are varied:

• Active Glass, Phase Separation, and Active Liquid: As persistence time of the active force increases, the system passes from glassy arrest to phase separation (hexatic/solid-liquid coexisting domains) and finally to an active liquid (Singh et al., 2019).
• Motility-Induced Phase Separation (MIPS): Density-dependent suppression of motility promotes cluster nucleation and arrest, with coexistence curves measured by bimodal local density histograms, bond-orientational order, and structure factors. Phase separation in tQSAPs occurs at much lower packing fraction than in conventional active Brownian particle systems (Jose et al., 2020).
• Annular Swarming: For intermediate thresholds and large drift-to-diffusion ratio ($v_0/D$), clusters organize into hollow rings, with a sharp outer rim determined by an integral equation for the steady-state density (Smith et al., 22 Apr 2025).

Regime	Characterization Criteria	Order Parameters
Active glass	Low $\tau_p$; immobilized, slow particles	MSD, incoherent scattering
Phase separation	Intermediate $\tau_p$; dense & dilute coexistence	Cell order $\Phi$, bond $\Psi_6$
Active liquid	Large $\tau_p$; homogeneous, rapid dynamics	Structure factor, vitrification

4. Continuum Theories: Hydrodynamics and Cahn–Hilliard Models

Coarse-graining the microscopic agent rules yields field theories for the density $\rho(\mathbf r, t)$:

• Advection–Diffusion PDEs: General form,

$$\partial_t \rho = D \nabla^2 \rho - \nabla \cdot (v(S)\,\rho)$$

clustering is triggered when global or local density crosses $P^*/K(0)$ (Smith et al., 22 Apr 2025).

• Active Cahn–Hilliard Theory $O(\nabla^4)$: Advanced particle-to-field derivations capture higher-order gradient effects essential for accurate binodal and interfacial tension calculation. Five density-dependent coefficient functions $$\mu(\rho), K(\rho), \lambda(\rho), \zeta(\rho), \nu(\rho)$$ set the macroscopic transport and pattern-formation properties. Multiple-scale analysis is required to correctly eliminate fast-evolving orientational moments (Bureković et al., 23 Jan 2026).
• Fluctuating Hydrodynamics: Universal scaling laws for static structure factor $S(k)$, intermediate scattering function $F(k, t)$, and effective diffusivity arise from considering both active and thermal contributions to noise and drift. The criterion $1 + \rho_0 v'(\rho_0)/v(\rho_0) < 0$ defines the onset of linear instability and spinodal decomposition (Dinelli et al., 2024).

5. Absorbing Phase Transitions, Mechanical Equilibria, and Pressure Drops

Motility regulation via quorum sensing can yield additional nonequilibrium phenomena:

• Absorbing Transition: For two-state systems (active/inactive), a critical global density $\rho_c$ marks the point above which all particles become arrested, as established by mean-field and hydrodynamic equation analysis (Lefranc et al., 19 Feb 2025).
• Flat Interface with Pressure Drop: Unlike equilibrium phase coexistence, tQSAP models predict flat interfaces between active and arrested phases with finite bulk pressure differences, set by the non-divergence flux induced by quorum-sensing feedback (Lefranc et al., 19 Feb 2025).
• Hysteresis and Coarsening: Dynamical sweeps and modulations in threshold or activity can produce hysteretic loops and $R(t)\sim t^{1/3}$ coarsening kinetics, distinct from classical Ostwald ripening (Lefranc et al., 19 Feb 2025).

6. Thermally Mediated Quorum Sensing: Janus Colloids and Soret Coupling

In laser-activated or thermophoretic tQSAP systems:

• Heat-Mediated Interaction: Asymmetric heating yields local temperature fields that modulate colloidal drift via the Soret effect. The mobility response is described by $S_T = D_T / D$, and the system is mapped onto nonlinear Poisson–Boltzmann equations for density (Golestanian, 2011).
• Instability Thresholds: Depending on the sign of the Soret coefficient, systems undergo depletion (repulsion, $S_T > 0$) or collapse (attraction, $S_T < 0$), with critical laser intensity $I_c$ marking the instability (supernova-like explosion).
• Programmable Self-Organization: Thermal feedback provides a macroscopic "switch"—quorum thresholds control transitions between benign, hollowed density profiles and catastrophic collapse (Golestanian, 2011).

7. Structural Statistics, Universalities, and Design Implications

• Cluster Size Distributions and Radial Correlations: tQSAP models generate master curves connecting average cluster size, short-range density correlations, and activation ratios. The transition between “social” ($X < 1$) and “anti-social” ($X > 1$) regimes is governed by persistence and speed scales (Thapa et al., 2023).
• Phase Diagram Control: Parameters such as sensing radius, propulsion decay rate, base activity, and thermal noise can be tuned to select between swarming, asters, crystalline clusters, or active jets (Velasco et al., 2018, Jose et al., 2020).
• Universality: At the hydrodynamic scale, systems with quorum-sensing or chemotactic interactions exhibit indistinguishable pattern formation, implying that observation of clustering is insufficient to diagnose the underlying microscale physics (Dinelli et al., 2024).
• Material Design: By engineering sensing and feedback functions, tQSAPs offer pathways to adaptive materials, smart glasses, and biological probes whose collective organization is programmable by density and thermal or chemical signals (Singh et al., 2019, Golestanian, 2011).

In summary, thermal quorum-sensing active particles integrate density-regulated motility with thermal fluctuations to produce emergent collective phenomena—clustering, phase separation, annular swarms, and absorbing transitions—captured quantitatively by a hierarchy of agent-based, hydrodynamic, and active Cahn–Hilliard models. These systems afford robust control of active matter organization via kernel design and feedback function selection, with deep connections to biofilm formation, colloid thermophoresis, and programmable soft-matter engineering (Smith et al., 22 Apr 2025, Bureković et al., 23 Jan 2026, Lefranc et al., 19 Feb 2025, Dinelli et al., 2024, Jose et al., 2020, Singh et al., 2019, Golestanian, 2011).