Stochastic Geometry in Active Matter

• Stochastic geometry of active matter is the study of self-driven units whose random dynamics lead to unique spatial structures and macroscopic observables.
• It establishes a rigorous thermodynamic framework that quantifies fluctuating work, heat, and pressure even in systems lacking traditional free energy.
• The framework explains how active phase separation and negative interfacial tension emerge from persistent energy dissipation and capillary fluctuations.

Active matter comprises assemblies of self-driven units that consume energy to generate motion, leading to distinctive nonequilibrium behaviors not found in passive materials. The stochastic geometry of active matter interrogates how spatial structures, fluctuations, and macroscopic observables emerge from the interplay of stochastic dynamics, energy dissipation, and microscopic activity. Recent developments have established rigorous stochastic thermodynamic frameworks for active systems, enabling precise definitions of fluctuating work, heat, pressure, and interfacial tension—even when traditional free energy concepts fail. These advances reveal unique implications in non-equilibrium steady states, particularly concerning phase separation, interfacial fluctuations, and the emergent spatial organization of driven particles.

1. Stochastic Thermodynamics of Active Matter

The stochastic thermodynamic formalism for active matter is built upon trajectory-level definitions of fluctuating work and heat. For a given microstate trajectory, the infinitesimal work increment is

$$\delta \hat{w} = \left(\frac{\partial H}{\partial X}\right)\cdot dX + f\cdot dr$$

where $H$ is the effective Hamiltonian (including both ideal and interaction terms), $X$ denotes external parameters (e.g., volume), and $f$ is an effective, generally non-conservative (active) force. The first law at the trajectory level reads

$\delta \hat{q} = \delta \hat{w} - dH$

Heat $\delta \hat{q}$ and work $\delta \hat{w}$ thus become fluctuating, stochastic quantities that enable a thermodynamic description irrespective of equilibrium or the existence of a free energy.

For active matter, dissipation is described by effective non-conservative forces that drive persistent non-equilibrium motion. For active Brownian particles (ABPs) with propulsion speed $v_0$ and mobility $\mu_0$, the active force on particle $k$ is $f_k = -f_0 e_k = -(v_0/\mu_0) e_k$, where $e_k$ is the unit orientation. The total work splits into "excess" work (due to system deformation) and "housekeeping" work (maintaining non-equilibrium via persistent propulsion): $$\delta \hat{w} = \delta \hat{w}_{ex} + \delta \hat{w}_{hk},\quad \delta \hat{w}_{hk} = \sum_k f_k\cdot dr_k$$ This decomposition captures both deformation-induced and steady-state energetic contributions.

2. Non-Equilibrium Steady States and Conjugate Macroscopic Observables

Unlike equilibrium systems, where free energy gradients define macroscopic observables, active matter in non-equilibrium steady states (NESSs) lacks a free energy. Nevertheless, via virtual work, one can construct intensive and extensive conjugate variables. For example, a virtual volume deformation leads to a microscopic stress tensor for a system with non-conservative forces: $$\hat{\sigma} = -\frac{NB}{V} I + \frac{1}{V} \sum_k (f_k + \nabla_k U) r_k^T$$ where $B$ is the ideal gas factor, $I$ the identity, and $U$ the interaction potential. The conjugate quantity to system volume $V$—the pressure—is thereby defined via: $\delta w_{ex} = -p\,dV$ This definition applies even without reference to equilibrium free energy, generalizing the concept of pressure to NESSs and ensuring a consistent thermodynamic interpretation within the stochastic geometry framework.

3. Pressure, Interfacial Tension, and Virtual Work

For ABP systems, pressure and interfacial tension are derived from the excess work performed during virtual deformations:

• Pressure: Combining the ideal, conservative, and active force contributions, the pressure is given by:

$p = p^{(\mathrm{id})} + p^{(U)} + p^{(f)}$

where $p^{(\mathrm{id})}$ arises from ideal gas behavior, $p^{(U)}$ incorporates pair-conservative forces ($p^{(U)} = \frac{1}{2A}\sum_{k<l} w(|r_k - r_l|)$ with $w(r) = -r u'(r)$), and $p^{(f)}$ is the active contribution ($$p^{(f)} = \frac{1}{2A}(v_0/\mu_0)\sum_k e_k \cdot r_k$$).

• Interfacial Tension: For inhomogeneous phase-separated states, the conjugate work associated with interface extension yields the interfacial tension:

$$\hat{\gamma} = \int dx [-\sigma_\perp(x) + \sigma_\parallel(x)]$$

This is closely related to the Kirkwood–Buff formula but remains valid in NESSs, thus generalizing equilibrium mechanical definitions to active contexts.

4. Phase Separation, Interfacial Fluctuations, and Negative Tension

A key result is that, under motility-induced phase separation, the interfacial tension $\gamma$ calculated via virtual work can be negative, juxtaposed with stable phase separation—a scenario forbidden in equilibrium thermodynamics. This is reconciled by considering capillary wave theory for the interface:

• Interface profile: $h(y) = \sum_q h_q\,e^{iqy}$
• Interfacial width: For strong fluctuations, $w^2 = w_0^2 + (L_y/12)$ in 2D, demonstrating that interface roughness grows with system size.
• Fluctuations and stability: The negative $\gamma$ does not signal instability; instead, strong interfacial fluctuations, set by the rate of dissipative housekeeping work from active propulsion, determine a finite interface "stiffness" and stabilize the phase-separated configuration.

This scenario suggests that energetic costs for interface deformation in active NESSs are dictated by the dissipation rate (housekeeping work per unit time per particle), not by thermal equilibrium energy.

5. Fluctuation-Dissipation Relation and Housekeeping Work

In equilibrium, the fluctuation–dissipation theorem (FDT) links system stiffness to energy fluctuations and determines interfacial tension: $$\gamma_{eq} = \frac{w_{ex}}{\Delta\ell} = (\mathrm{stiffness}) \times (\mathrm{thermal~energy})$$ In active matter, a corresponding relation ties interface energetics to the dissipation rate of active work: $$\gamma \approx \frac{w_{ex}}{\Delta\ell} \approx \frac{\dot{w}_{hk}\,\tau}{N} \approx -f_0 p$$ where $\dot{w}_{hk}$ is the housekeeping work rate, $\tau$ a characteristic interface fluctuation timescale, and $p$ an effective particle displacement. Thus, the interface's mechanical response and spatial fluctuations are fundamentally set by non-equilibrium dissipation.

6. Theoretical Insights, Contradictions, and Measurement Strategies

• Negative interfacial tension is a haLLMark of active NESSs and should not be interpreted as mechanical instability, but rather as an emergent property resulting from the interplay of sustained dissipation and capillary (geometric) fluctuations.
• Stochastic geometry in active matter thus departs from equilibrium expectations: stability can coincide with negative tension due to strong fluctuation-induced interface stiffness and active "extensile" forcing leading to fluctuations and subsequent interface re-stabilization.
• Measuring active observables: Pressure and tension can be consistently extracted from the work required to perform controlled deformations (virtual or real), even in the absence of equilibrium thermodynamic potentials.
• Experimental implications: The framework predicts that large capillary interfacial fluctuations, interface widths growing with system size, and the presence of phase-separated regions with strongly fluctuating morphology are robust signatures in active systems.

7. Broader Implications and Open Directions

This stochastic thermodynamic framework bridges microscopic non-conservative dynamics with macroscopic observables in active matter, codifying energy input, dissipation, and fluctuations as fundamental to the geometry of emergent structures. It establishes that:

• Conjugate variables (pressure, tension) retain physical meaning via work and deformation arguments in active NESSs.
• Phase separation and associated interface properties reflect a modified balance between driving (housekeeping work) and fluctuations (interface roughening), not simply minimized free energy.
• Fluctuation regimes in active systems are determined by the scale of dissipation, implying strong scaling relations between system size, interface width, and fluctuation amplitudes absent in equilibrium.

Unresolved challenges include extending these results to non-scalar active matter with alignment, nematic, or chiral order, and formulating general criteria for the emergence and characterization of stochastic geometric structures in non-equilibrium steady states lacking equilibrium analogs.

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This synthesis establishes a rigorous foundation for the stochastic geometry of active matter by integrating trajectory-scale stochastic thermodynamics, effective non-conservative force modeling, and macroscopic virtual work definitions. It underlines how energy dissipation, non-equilibrium steady states, and strong capillary fluctuations jointly determine the emergent spatial organization of active materials (Speck, 2016).