Pressure is the mechanical force per unit area that a confined system exerts on its container. In thermal equilibrium, it depends only on bulk properties—such as density and temperature—through an equation of state. Here we show that in a wide class of active systems the pressure depends on the precise interactions between the active particles and the confining walls. In general, therefore, active fluids have no equation of state. Their mechanical pressure exhibits anomalous properties that defy the familiar thermodynamic reasoning that holds in equilibrium. The pressure remains a function of state, however, in some specific and well-studied active models that tacitly restrict the character of the particle–wall and/or particle–particle interactions.
This is a preview of subscription content
Subscribe to Journal
Get full journal access for 1 year
only $8.25 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Allen, M. P. & Tidlesley, D. J. Computer Simulation of Liquids (Oxford Univ. Press, 1987).
Cugliandolo, L. F. The effective temperature. J. Phys. A 44, 483001 (2011).
Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143 (2013).
Berg, H. E. coli in Motion (Springer, 2001).
Cates, M. E. Diffusive transport without detailed balance in motile bacteria: Does microbiology need statistical physics? Rep. Prog. Phys. 75, 042601 (2012).
Palacci, J., Cottin-Bizonne, C., Ybert, C. & Bocquet, L. Sedimentation and effective temperature of active colloidal suspensions. Phys. Rev. Lett. 105, 088304 (2010).
Fily, Y. & Marchetti, M. C. Athermal phase separation of self-propelled particles with no alignment. Phys. Rev. Lett. 108, 235702 (2012).
Buttinoni, I. et al. Dynamical clustering and phase separation in suspensions of self-propelled colloidal particles. Phys. Rev. Lett. 110, 238301 (2013).
Kudrolli, A., Lumay, G., Volfson, D. & Tsimring, L. S. Swarming and swirling in self-propelled polar granular rods. Phys. Rev. Lett. 100, 058001 (2008).
Narayan, V., Ramaswamy, S. & Menon, N. Long-lived giant number fluctuations in a swarming granular nematic. Science 317, 105–108 (2007).
Deseigne, J., Dauchot, O. & Chaté, H. Collective motion of vibrated polar disks. Phys. Rev. Lett. 105, 098001 (2010).
Ballerini, M. et al. Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc. Natl Acad. Sci. USA 105, 1232–1237 (2008).
Mallory, S. A., Saric, A., Valeriani, C. & Cacciuto, A. Anomalous thermomechanical properties of a self-propelled colloidal fluid. Phys. Rev. E 89, 052303 (2014).
Fily, Y., Baskaran, A. & Hagan, M. F. Dynamics of self-propelled particles under strong confinement. Soft Matter 10, 5609–5617 (2014).
Fily, Y., Baskaran, A. & Hagan, M. F. Dynamics and density distribution of strongly confined noninteracting nonaligning self-propelled particles in a nonconvex boundary. Phys. Rev. E 91, 012125 (2015).
Ni, R., Stuart, M. A. C. & Bolhuis, P. G. Tunable long range forces mediated by self-propelled colloidal hard spheres. Phys. Rev. Lett. 114, 018302 (2015).
Yang, X., Manning, M. L. & Marchetti, M. C. Aggregation and segregation of confined active particles. Soft Matter 10, 6477–6484 (2014).
Takatori, S. C., Yan, W. & Brady, J. F. Swim pressure: Stress generation in active matter. Phys. Rev. Lett. 113, 028103 (2014).
Takatori, S. C. & Brady, J. F. Towards a thermodynamics of active matter. Phys. Rev. E 91, 032117 (2015).
Ginot, F. et al. Nonequilibrium equation of state in suspensions of active colloids. Phys. Rev. X 5, 011004 (2015).
Solon, A. P. et al. Pressure and phase equilibria in interacting active Brownian spheres. Phys. Rev. Lett. 114, 198301 (2015).
Palacci, J., Sacanna, S., Steinberg, A. P., Pine, D. J. & Chaikin, P. M. Living crystals of light-activated colloidal surfers. Science 339, 936–939 (2013).
Bricard, A., Caussin, J. B., Desreumaux, N., Dauchot, O. & Bartolo, D. Emergence of macroscopic directed motion in populations of motile colloids. Nature 503, 95–98 (2013).
Stenhammar, J. et al. Continuum theory of phase separation kinetics for active Brownian particles. Phys. Rev. Lett. 111, 145702 (2013).
Redner, G. S., Hagan, M. F. & Baskaran, A. Structure and dynamics of a phase-separating active colloidal fluid. Phys. Rev. Lett. 110, 055701 (2013).
Schnitzer, M. J. Theory of continuum random walks and application to chemotaxis. Phys. Rev. E 48, 2553 (1993).
Tailleur, J. & Cates, M. E. Statistical mechanics of interacting run-and-tumble bacteria. Phys. Rev. Lett. 100, 218103 (2008).
Cates, M. E., Marenduzzo, D., Pagonabarraga, I. & Tailleur, J. Arrested phase separation in reproducing bacteria creates a generic route to pattern formation. Proc. Natl Acad. Sci. USA 107, 11715–11729 (2010).
Theveneau, E. et al. Collective chemotaxis requires contact-dependent cell polarity. Dev. Cell 19, 39–53 (2010).
Sepulveda, N. et al. Collective cell motion in an epithelial sheet can be quantitatively described by a stochastic interacting particle model. PLoS Comp. Biol. 9, e1002944 (2013).
Elgeti, J. & Gompper, G. Self-propelled rods near surfaces. Europhys. Lett. 85, 38002 (2009).
Tailleur, J. & Cates, M. E. Sedimentation, trapping, and rectification of dilute bacteria. Europhys. Lett. 86, 60002 (2009).
Solon, A. P., Cates, M. E. & Tailleur, J. Active Brownian particles and run-and-tumble particles: A comparative study. Preprint at http://arXiv.org/abs/1504.07391 (2015).
Liu, C. et al. Sequential establishment of stripe patterns in an expanding cell population. Science 334, 238–241 (2011).
Bialké, J., Lowen, H. & Speck, T. Microscopic theory for the phase separation of self-propelled repulsive disks. Europhys. Lett. 103, 30008 (2013).
Wysocki, A., Winkler, R. G. & Gompper, G. Cooperative motion of active Brownian spheres in three-dimensional dense suspensions. Europhys. Lett. 105, 48004 (2014).
Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, 2000).
Cates, M. E. & Tailleur, J. When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-induced phase separation. Europhys. Lett. 101, 20010 (2013).
Berke, A. P., Turner, L., Berg, H. C. & Lauga, E. Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101, 038102 (2008).
We thank K. Keren, M. Kolodrubetz, C. Marchetti, A. Polkovnikov, J. Stenhammar, R. Wittkowski and X. Yang for discussions. This work was funded in part by EPSRC EP/J007404. M.E.C. holds a Royal Society Research Professorship. Y.K. was supported by the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation. M.K. is supported by NSF grant No. DMR-12-06323. A.B. and Y.F. acknowledge support from NSF grant DMR-1149266 and the Brandeis Center for Bioinspired Soft Materials, an NSF MRSEC, DMR-1420382. Their computational resources were provided by the NSF through XSEDE computing resources and the Brandeis HPCC. Y.K., A.P.S. and J.T. thank the Galileo Galilei Institute for Theoretical Physics for hospitality. A.B., M.E.C., Y.F., A.P.S., M.K. and J.T. thank the KITP at the University of California, Santa Barbara, where they were supported through National Science Foundation Grant NSF PHY11-25925.
The authors declare no competing financial interests.
About this article
Cite this article
Solon, A., Fily, Y., Baskaran, A. et al. Pressure is not a state function for generic active fluids. Nature Phys 11, 673–678 (2015). https://doi.org/10.1038/nphys3377
Scientific Reports (2021)
Nature Nanotechnology (2021)
Scientific Reports (2021)
Communications Physics (2021)
Nature Communications (2020)