Towards a thermodynamics of active matter

Self-propulsion allows living systems to display self-organization and unusual phase behavior. Unlike passive systems in thermal equilibrium, active matter systems are not constrained by conventional thermodynamic laws. A question arises, however, as to what extent, if any, can concepts from classical thermodynamics be applied to nonequilibrium systems like active matter. Here we use the new swim pressure perspective to develop a simple theory for predicting phase separation in active matter. Using purely mechanical arguments we generate a phase diagram with a spinodal and critical point, and define a nonequilibrium chemical potential to interpret the “binodal.” We provide a generalization of thermodynamic concepts like the free energy and temperature for nonequilibrium active systems. Our theory agrees with existing simulation data both qualitatively and quantitatively and may provide a framework for understanding and predicting the behavior of nonequilibrium active systems.

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Figure 1

Phase diagram in the ${\text{Pe}}_R\text{−}\varphi$ plane in (a) 3D and (b) 2D. The color bar shows the active pressure scaled with the swim energy $k_s{T}_s=\zeta U_0^2{\tau }_R/6$, and the blue and red curves are the binodal and spinodal, respectively. The critical point is shown with a red star. The open and filled symbols are simulation data with a homogeneous and phase-separated state, respectively.

Figure 2

Nonequilibrium chemical potential as a function of ${\Pi }^{\text{act}}$ for ${\text{Pe}}_R=a/\left(U_0{\tau }_R\right)=\zeta U_{0}a/\left(6k_s{T}_s\right)=0.02$, where $k_s{T}_s=\zeta U_0^2{\tau }_R/6$ is the swimmers' energy scale. The symbols are BD simulations [7] and the curve is the model, Eq. (2).

Figure 3

Gibbs FE as a function of $\varphi$ for fixed values of ${\text{Pe}}_R$ and ${\Pi }^{\text{act}}\varphi /\left(n{k}_s{T}_s\right)=0.18$, where $k_s{T}_s=\zeta U_0^2{\tau }_R/6$. The black arrow points towards decreasing ${\text{Pe}}_R$ at fixed ${\Pi }^{\text{act}}$. The filled color circles denote the stable states.