# Fluctuation–Dissipation Relations in Active Matter Systems

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## Abstract

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## 1. Introduction

## 2. Self-Propelled Particles

## 3. Generalized Fluctuation–Dissipation Relation for Self-Propelled Particles

## 4. Numerical Results

#### 4.1. Quartic Potential

#### 4.1.1. One-Dimensional System

#### 4.1.2. Higher-Dimensional Systems

#### 4.2. Double-Well Potential

#### 4.3. Measuring the Non-Equilibrium in Active Systems

## 5. Failure of the Approximated Approaches

#### 5.1. Assuming the Detailed Balance

#### 5.2. Unified Colored Noise Approximation

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the FDR for Active Particles

## Appendix B. The Active Harmonic Oscillator

## Appendix C. Response Function with the UCNA Approach

## References

- Onsager, L. Reciprocal Relations in Irreversible Processes. I. Phys. Rev.
**1931**, 37, 405–426. [Google Scholar] [CrossRef] - Kubo, R. Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems. J. Phys. Soc. Jpn.
**1957**, 12, 570. [Google Scholar] [CrossRef] - Marconi, U.M.B.; Puglisi, A.; Rondoni, L.; Vulpiani, A. Fluctuation–dissipation: Response theory in statistical physics. Phys. Rep.
**2008**, 461, 111–195. [Google Scholar] [CrossRef] [Green Version] - Cugliandolo, L.F. The effective temperature. J. Phys. A Math. Theor.
**2011**, 44, 483001. [Google Scholar] [CrossRef] [Green Version] - Puglisi, A.; Sarracino, A.; Vulpiani, A. Temperature in and out of equilibrium: A review of concepts, tools and attempts. Phys. Rep.
**2017**, 709, 1–60. [Google Scholar] [CrossRef] [Green Version] - Agarwal, G.S. Fluctuation-Disipation Theorems for Systems in Non-Thermal Equilibrium and Applications. Z. Phys.
**1972**, 252, 25. [Google Scholar] [CrossRef] - Falcioni, M.; Isola, S.; Vulpiani, A. Correlation functions and relaxation properties in chaotic dynamics and statistical mechanics. Phys. Lett. A
**1990**, 144, 341. [Google Scholar] [CrossRef] - Gnoli, A.; Puglisi, A.; Sarracino, A.; Vulpiani, A. Nonequilibrium Brownian Motion beyond the Effective Temperature. PLoS ONE
**2014**, 9, e93720. [Google Scholar] [CrossRef] - Speck, T.; Seifert, U. Restoring a fluctuation-dissipation theorem in a nonequilibrium steady state. Europhys. Lett.
**2006**, 74, 391. [Google Scholar] [CrossRef] - Seifert, U.; Speck, T. Fluctuation-dissipation theorem in nonequilibrium steady states. EPL Europhys. Lett.
**2010**, 89, 10007. [Google Scholar] [CrossRef] - Warren, P.B.; Allen, R.J. Malliavin Weight Sampling: A Practical Guide. Entropy
**2014**, 16, 221. [Google Scholar] [CrossRef] [Green Version] - Novikov, E.A. Functionals and the random-force method in turbulence theory. Sov. Phys. JETP
**1965**, 20, 1290. [Google Scholar] - Cugliandolo, L.F.; Kurchan, J.; Parisi, G. Off equilibrium dynamics and aging in unfrustrated systems. J. Phys. I Fr.
**1994**, 4, 1641. [Google Scholar] [CrossRef] - Baiesi, M.; Maes, C.; Wynants, B. Fluctuations and response of nonequilibrium states. Phys. Rev. Lett.
**2009**, 103, 010602. [Google Scholar] [CrossRef] [Green Version] - Maes, C. Response theory: A trajectory-based approach. Front. Phys.
**2020**, 8, 00229. [Google Scholar] [CrossRef] - Lippiello, E.; Corberi, F.; Sarracino, A.; Zannetti, M. Nonlinear response and fluctuation-dissipation relations. Phys. Rev. E
**2008**, 78, 041120. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Marchetti, M.; Joanny, J.; Ramaswamy, S.; Liverpool, T.; Prost, J.; Rao, M.; Simha, R.A. Hydrodynamics of soft active matter. Rev. Mod. Phys.
**2013**, 85, 1143–1189. [Google Scholar] [CrossRef] [Green Version] - Bechinger, C.; Di Leonardo, R.; Löwen, H.; Reichhardt, C.; Volpe, G.; Volpe, G. Active particles in complex and crowded environments. Rev. Mod. Phys.
**2016**, 88, 045006. [Google Scholar] [CrossRef] - Elgeti, J.; Winkler, R.G.; Gompper, G. Physics of microswimmers—Single particle motion and collective behavior: A review. Rep. Prog. Phys.
**2015**, 78, 056601. [Google Scholar] [CrossRef] [PubMed] - Gompper, G.; Winkler, R.G.; Speck, T.; Solon, A.; Nardini, C.; Peruani, F.; Löwen, H.; Golestanian, R.; Kaupp, U.B.; Alvarez, L.; et al. The 2020 motile active matter roadmap. J. Phys. Condens. Matter
**2020**, 32, 193001. [Google Scholar] [CrossRef] - Shaebani, M.R.; Wysocki, A.; Winkler, R.G.; Gompper, G.; Rieger, H. Computational models for active matter. Nat. Rev. Phys.
**2020**, 2, 181–199. [Google Scholar] [CrossRef] [Green Version] - Fodor, É.; Marchetti, M.C. The statistical physics of active matter: From self-catalytic colloids to living cells. Physica A Stat. Mech. Its Appl.
**2018**, 504, 106–120. [Google Scholar] [CrossRef] [Green Version] - Caprini, L.; Marconi, U.M.B.; Vulpiani, A. Linear response and correlation of a self-propelled particle in the presence of external fields. J. Stat. Mech. Theory Exp.
**2018**, 2018, 033203. [Google Scholar] [CrossRef] [Green Version] - Sarracino, A.; Vulpiani, A. On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems. Chaos Interdiscip. J. Nonlinear Sci.
**2019**, 29, 083132. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fodor, É.; Nardini, C.; Cates, M.E.; Tailleur, J.; Visco, P.; van Wijland, F. How far from equilibrium is active matter? Phys. Rev. Lett.
**2016**, 117, 038103. [Google Scholar] [CrossRef] [Green Version] - Szamel, G. Evaluating linear response in active systems with no perturbing field. EPL Europhys. Lett.
**2017**, 117, 50010. [Google Scholar] [CrossRef] [Green Version] - Sarracino, A. Time asymmetry of the Kramers equation with nonlinear friction: Fluctuation-dissipation relation and ratchet effect. Phys. Rev. E
**2013**, 88, 052124. [Google Scholar] [CrossRef] [Green Version] - Berthier, L.; Kurchan, J. Non-equilibrium glass transitions in driven and active matter. Nat. Phys.
**2013**, 9, 310–314. [Google Scholar] [CrossRef] - Levis, D.; Berthier, L. From single-particle to collective effective temperatures in an active fluid of self-propelled particles. EPL Europhys. Lett.
**2015**, 111, 60006. [Google Scholar] [CrossRef] [Green Version] - Nandi, S.K.; Gov, N. Effective temperature of active fluids and sheared soft glassy materials. Eur. Phys. J. E
**2018**, 41, 117. [Google Scholar] [CrossRef] - Cugliandolo, L.F.; Gonnella, G.; Petrelli, I. Effective temperature in active Brownian particles. Fluct. Noise Lett.
**2019**, 18, 1940008. [Google Scholar] [CrossRef] - Preisler, Z.; Dijkstra, M. Configurational entropy and effective temperature in systems of active Brownian particles. Soft Matter
**2016**, 12, 6043–6048. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Petrelli, I.; Cugliandolo, L.F.; Gonnella, G.; Suma, A. Effective temperatures in inhomogeneous passive and active bidimensional Brownian particle systems. Phys. Rev. E
**2020**, 102, 012609. [Google Scholar] [CrossRef] [PubMed] - Villamaina, D.; Baldassarri, A.; Puglisi, A.; Vulpiani, A. Fluctuation dissipation relation: How to compare correlation functions and responses? J. Stat. Mech.
**2009**, 2009, P07024. [Google Scholar] [CrossRef] - Dal Cengio, S.; Levis, D.; Pagonabarraga, I. Linear response theory and Green-Kubo relations for active matter. Phys. Rev. Lett.
**2019**, 123, 238003. [Google Scholar] [CrossRef] [Green Version] - Dal Cengio, S.; Levis, D.; Pagonabarraga, I. Fluctuation-Dissipation Relations in the absence of Detailed Balance: Formalism and applications to Active Matter. arXiv
**2020**, arXiv:2007.07322. [Google Scholar] - Burkholdera, E.W.; Brady, J.F. Fluctuation-dissipation in active matter. J. Chem. Phys.
**2019**, 150, 184901. [Google Scholar] [CrossRef] - Maes, C. Fluctuating motion in an active environment. Phys. Rev. Lett.
**2020**, 125, 208001. [Google Scholar] [CrossRef] - Berthier, L.; Flenner, E.; Szamel, G. How active forces influence nonequilibrium glass transitions. New J. Phys.
**2017**, 19, 125006. [Google Scholar] [CrossRef] - Mandal, D.; Klymko, K.; DeWeese, M.R. Entropy production and fluctuation theorems for active matter. Phys. Rev. Lett.
**2017**, 119, 258001. [Google Scholar] [CrossRef] [Green Version] - Caprini, L.; Marconi, U.M.B. Active particles under confinement and effective force generation among surfaces. Soft Matter
**2018**, 14, 9044–9054. [Google Scholar] [CrossRef] [Green Version] - Wittmann, R.; Brader, J.M.; Sharma, A.; Marconi, U.M.B. Effective equilibrium states in mixtures of active particles driven by colored noise. Phys. Rev. E
**2018**, 97, 012601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bonilla, L.L. Active ornstein-uhlenbeck particles. Phys. Rev. E
**2019**, 100, 022601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dabelow, L.; Bo, S.; Eichhorn, R. Irreversibility in active matter systems: Fluctuation theorem and mutual information. Phys. Rev. X
**2019**, 9, 021009. [Google Scholar] [CrossRef] [Green Version] - Martin, D.; O’Byrne, J.; Cates, M.E.; Fodor, É.; Nardini, C.; Tailleur, J.; van Wijland, F. Statistical Mechanics of Active Ornstein Uhlenbeck Particles. arXiv
**2020**, arXiv:2008.12972. [Google Scholar] - Woillez, E.; Kafri, Y.; Gov, N.S. Active Trap Model. Phys. Rev. Lett.
**2020**, 124, 118002. [Google Scholar] [CrossRef] [Green Version] - Wu, X.L.; Libchaber, A. Particle diffusion in a quasi-two-dimensional bacterial bath. Phys. Rev. Lett.
**2000**, 84, 3017. [Google Scholar] [CrossRef] [Green Version] - Maggi, C.; Paoluzzi, M.; Pellicciotta, N.; Lepore, A.; Angelani, L.; Di Leonardo, R. Generalized energy equipartition in harmonic oscillators driven by active baths. Phys. Rev. Lett.
**2014**, 113, 238303. [Google Scholar] [CrossRef] [Green Version] - Maggi, C.; Paoluzzi, M.; Angelani, L.; Di Leonardo, R. Memory-less response and violation of the fluctuation-dissipation theorem in colloids suspended in an active bath. Sci. Rep.
**2017**, 7, 17588. [Google Scholar] [CrossRef] - Chaki, S.; Chakrabarti, R. Effects of active fluctuations on energetics of a colloidal particle: Superdiffusion, dissipation and entropy production. Physica A Stat. Mech. Its Appl.
**2019**, 530, 121574. [Google Scholar] [CrossRef] [Green Version] - Caprini, L.; Hernández-García, E.; López, C.; Marconi, U.M.B. A comparative study between two models of active cluster crystals. Sci. Rep.
**2019**, 9, 16687. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Das, S.; Gompper, G.; Winkler, R.G. Confined active Brownian particles: Theoretical description of propulsion-induced accumulation. New J. Phys.
**2018**, 20, 015001. [Google Scholar] [CrossRef] [Green Version] - Caprini, L.; Marconi, U.M.B. Active chiral particles under confinement: Surface currents and bulk accumulation phenomena. Soft Matter
**2019**, 15, 2627–2637. [Google Scholar] [CrossRef] [Green Version] - Farage, T.F.; Krinninger, P.; Brader, J.M. Effective interactions in active Brownian suspensions. Phys. Rev. E
**2015**, 91, 042310. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Maggi, C.; Paoluzzi, M.; Crisanti, A.; Zaccarelli, E.; Gnan, N. Universality class of the motility-induced critical point in large scale off-lattice simulations of active particles. arXiv
**2020**, arXiv:2007.12660. [Google Scholar] - Caprini, L.; Marconi, U.M.B.; Maggi, C.; Paoluzzi, M.; Puglisi, A. Hidden velocity ordering in dense suspensions of self-propelled disks. Phys. Rev. Res.
**2020**, 2, 023321. [Google Scholar] [CrossRef] - Caprini, L.; Marconi, U.M.B. Time-dependent properties of interacting active matter: Dynamical behavior of one-dimensional systems of self-propelled particles. Phys. Rev. Res.
**2020**, 2, 033518. [Google Scholar] [CrossRef] - Puglisi, A.; Marini Bettolo Marconi, U. Clausius relation for active particles: What can we learn from fluctuations. Entropy
**2017**, 19, 356. [Google Scholar] [CrossRef] [Green Version] - Caprini, L.; Marconi, U.M.B.; Puglisi, A.; Vulpiani, A. The entropy production of Ornstein—Uhlenbeck active particles: A path integral method for correlations. J. Stat. Mech. Theory Exp.
**2019**, 2019, 053203. [Google Scholar] [CrossRef] [Green Version] - Dabelow, L.; Eichhorn, R. Irreversibility in active matter: General framework for active Ornstein-Uhlenbeck particles. arXiv
**2020**, arXiv:2011.02976. [Google Scholar] - Marconi, U.M.B.; Puglisi, A.; Maggi, C. Heat, temperature and Clausius inequality in a model for active Brownian particles. Sci. Rep.
**2017**, 7, 46496. [Google Scholar] [CrossRef] [PubMed] - Caprini, L.; Marconi, U.M.B.; Puglisi, A.; Vulpiani, A. Comment on “Entropy Production and Fluctuation Theorems for Active Matter”. Phys. Rev. Lett.
**2018**, 121, 139801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Martin, D. AOUP in the presence of Brownian noise: A perturbative approach. arXiv
**2020**, arXiv:2009.13476. [Google Scholar] - Szamel, G. Self-propelled particle in an external potential: Existence of an effective temperature. Phys. Rev. E
**2014**, 90, 012111. [Google Scholar] [CrossRef] [Green Version] - Caprini, L.; Marini Bettolo Marconi, U.; Puglisi, A.; Vulpiani, A. Active escape dynamics: The effect of persistence on barrier crossing. J. Chem. Phys.
**2019**, 150, 024902. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Woillez, E.; Kafri, Y.; Lecomte, V. Nonlocal stationary probability distributions and escape rates for an active Ornstein—Uhlenbeck particle. J. Stat. Mech. Theory Exp.
**2020**, 2020, 063204. [Google Scholar] [CrossRef] - Caprini, L.; Marconi, U.M.B.; Puglisi, A. Activity induced delocalization and freezing in self-propelled systems. Sci. Rep.
**2019**, 9, 1386. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stenhammar, J.; Marenduzzo, D.; Allen, R.J.; Cates, M.E. Phase behaviour of active Brownian particles: The role of dimensionality. Soft Matter
**2014**, 10, 1489–1499. [Google Scholar] [CrossRef] [Green Version] - Fily, Y. Self-propelled particle in a nonconvex external potential: Persistent limit in one dimension. J. Chem. Phys.
**2019**, 150, 174906. [Google Scholar] [CrossRef] [Green Version] - Wio, H.S.; Colet, P.; San Miguel, M.; Pesquera, L.; Rodriguez, M. Path-integral formulation for stochastic processes driven by colored noise. Phys. Rev. A
**1989**, 40, 7312. [Google Scholar] [CrossRef] - Bray, A.; McKane, A.; Newman, T. Path integrals and non-Markov processes. II. Escape rates and stationary distributions in the weak-noise limit. Phys. Rev. A
**1990**, 41, 657. [Google Scholar] [CrossRef] - Sharma, A.; Wittmann, R.; Brader, J.M. Escape rate of active particles in the effective equilibrium approach. Phys. Rev. E
**2017**, 95, 012115. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Maggi, C.; Marconi, U.M.B.; Gnan, N.; Di Leonardo, R. Multidimensional stationary probability distribution for interacting active particles. Sci. Rep.
**2015**, 5, 10742. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wittmann, R.; Maggi, C.; Sharma, A.; Scacchi, A.; Brader, J.M.; Marconi, U.M.B. Effective equilibrium states in the colored-noise model for active matter I. Pairwise forces in the Fox and unified colored noise approximations. J. Stat. Mech. Theory Exp.
**2017**, 2017, 113207. [Google Scholar] [CrossRef] [Green Version] - Marconi, U.M.B.; Maggi, C. Towards a statistical mechanical theory of active fluids. Soft Matter
**2015**, 11, 8768–8781. [Google Scholar] [CrossRef] [Green Version] - Marconi, U.M.B.; Gnan, N.; Paoluzzi, M.; Maggi, C.; Di Leonardo, R. Velocity distribution in active particles systems. Sci. Rep.
**2016**, 6, 23297. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Response function, ${R}_{x}\left(t\right)$, for different values of $\tau $ calculated numerically via Equation (3) in the case of a quartic potential, $U\left(x\right)=k{x}^{4}/4$. Times are measured in units of the typical time ${t}^{*}$ (see main text). The dashed black lines are obtained by using the FDR, Equation (5). (

**b**–

**d**) ${R}_{x}\left(t\right)$ (solid yellow lines) compared to the FDR (dashed black lines), Equation (5). Dotted violet lines represent ${C}_{x}=\langle x\left(t\right){U}^{\prime}\left(0\right)\rangle /\left({D}_{a}\gamma \right)+\langle {U}^{\prime}\left(t\right)x\left(0\right)\rangle /\left({D}_{a}\gamma \right)$ while green solid lines represent ${C}_{v}={\tau}^{2}\langle v\left(t\right){U}^{\u2033}\left(0\right)v\left(0\right)\rangle /\left({D}_{a}\gamma \right)+{\tau}^{2}\langle v\left(t\right){U}^{\u2033}\left(t\right)v\left(0\right)\rangle /\left({D}_{a}\gamma \right)$. (

**e**–

**g**) ${D}_{x}=\langle x\left(t\right){U}^{\prime}\left(0\right)\rangle /\left({D}_{a}\gamma \right)-\langle {U}^{\prime}\left(t\right)x\left(0\right)\rangle /\left({D}_{a}\gamma \right)$ (solid red lines) and ${D}_{v}={\tau}^{2}\langle v\left(t\right){U}^{\u2033}\left(t\right)v\left(0\right)\rangle /\left({D}_{a}\gamma \right)-{\tau}^{2}\langle v\left(t\right){U}^{\u2033}\left(0\right)v\left(0\right)\rangle /\left({D}_{a}\gamma \right)$ (solid blue lines). Panels (b,e) are obtained with $\tau ={10}^{-1}$, panels (c,f) with $\tau =1$ and, finally, panels (d,g) with $\tau =10$. The other parameters are ${D}_{a}=1$, $k=3$, and $\gamma =1$.

**Figure 2.**(

**a**) Response function, ${R}_{x}\left(t\right)$, for two different values of $\tau ={10}^{-3},10$ and dimensions $d=1,2,3$, calculated numerically via Equation (3), in the cases of quartic potentials $U\left(\mathbf{x}\right)={k\left|\mathbf{x}\right|}^{4}/4$. The dashed black lines are obtained by using the FDR, Equation (5). (

**b**,

**c**) ${D}_{x}=\langle x\left(t\right){U}^{\prime}\left(0\right)\rangle /\left({D}_{a}\gamma \right)-\langle {U}^{\prime}\left(t\right)x\left(0\right)\rangle /\left({D}_{a}\gamma \right)$ (panel (

**b**)) and ${D}_{v}={\tau}^{2}\langle v\left(t\right){U}^{\u2033}\left(t\right)v\left(0\right)\rangle /\left({D}_{a}\gamma \right)-{\tau}^{2}\langle v\left(t\right){U}^{\u2033}\left(0\right)v\left(0\right)\rangle /\left({D}_{a}\gamma \right)$ (panel (

**c**)) for $d=1,2,3$, at $\tau =1$. The other parameters are ${D}_{a}=1$, $k=3$, $\gamma =1$.

**Figure 3.**(

**a**) Response function, ${R}_{x}\left(t\right)$, for different values of $\tau $ calculated numerically via Equation (3) in the case of a double-well potential, $U\left(x\right)=k({x}^{4}/4-{x}^{2}/2)$. The dashed black lines are obtained by using the FDR, Equation (5). (

**b**,

**c**) ${\mathcal{D}}_{x}\left(t\right)$ for different values of $\tau $ as reported in the legend. (

**d**,

**e**) ${\mathcal{D}}_{v}\left(t\right)$ for different values of $\tau $ as reported in the legend. The other parameters are ${D}_{a}=1$, $k=3$, and $\gamma =1$.

**Figure 4.**$\mu $, ${\mu}_{x}$ and ${\mu}_{v}$ defined in Equations (8)–(10), as a function of $\tau /{t}^{*}$ for the quartic potential, $U\left(x\right)=k{x}^{4}/4$, and the doublewell potential, $U\left(x\right)=k({x}^{4}/4-{x}^{2}/2)$ for panels (a,b), respectively. The other parameters are ${D}_{a}=1$, $k=3$, $\gamma =1$.

**Figure 5.**Response function, ${R}_{x}\left(t\right)$ as a function of $t/{t}^{*}$ for different values of $\tau $ as indicated in the legend, in the case of a quartic potential, $U\left(x\right)=k{x}^{4}$, and a double-well potential, $U\left(x\right)=k({x}^{4}/4-{x}^{2}/2)$, shown in panels (

**a**,

**b**), respectively. Solid lines are calculated numerically via Equation (3). Dashed lines report the expressions for ${R}^{D}$ (Equation (13)), while dotted lines those for ${R}^{U}$ (Equation (17)). The parameters of the simulations are ${D}_{a}=1$, $k=3$, $\gamma =1$.

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Caprini, L.; Puglisi, A.; Sarracino, A.
Fluctuation–Dissipation Relations in Active Matter Systems. *Symmetry* **2021**, *13*, 81.
https://doi.org/10.3390/sym13010081

**AMA Style**

Caprini L, Puglisi A, Sarracino A.
Fluctuation–Dissipation Relations in Active Matter Systems. *Symmetry*. 2021; 13(1):81.
https://doi.org/10.3390/sym13010081

**Chicago/Turabian Style**

Caprini, Lorenzo, Andrea Puglisi, and Alessandro Sarracino.
2021. "Fluctuation–Dissipation Relations in Active Matter Systems" *Symmetry* 13, no. 1: 81.
https://doi.org/10.3390/sym13010081