The vibrational dynamics of solids is described by phonons constituting basic collective excitations in equilibrium crystals. Here, we consider a non-equilibrium active solid, formed by self-propelled particles, which bring the system into a non-equilibrium steady-state. We identify novel vibrational collective excitations of non-equilibrium (active) origin, which coexist with phonons and dominate over them when the system is far from equilibrium. These vibrational excitations are interpreted in the framework of non-equilibrium physics, in particular, stochastic thermodynamics. We call them “entropons” because they are the modes of spectral entropy production (at a given frequency and wave vector). The existence of entropons could be verified in future experiments on dense self-propelled colloidal Janus particles and granular active matter, as well as in living systems, such as dense cell monolayers.
Active matter1,2 includes a broad range of systems composed of particles that locally convert energy from the environment into directed motion.3,4 The energy exchange with the environment, often induced by chemical reactions or self-imposed gradients, leads to self-propulsion of the particles and internally drives an active system out of equilibrium.
Dense systems of self-propelled particles are rather ubiquitous in nature and often form crystalline structures. Cell monolayers of the human body5–8 and dense colonies of bacteria9–11 are common examples. Moreover, active colloidal Janus particles may cluster and form dense crystallites,12 named “living crystals.”13,14 These active crystals show fascinating phenomena uncommon for equilibrium solids15–18 ranging from intrinsic spatial velocity correlations,19–22 spontaneous velocity alignment,23–25 and non-Gaussian velocity distributions25,26 to “traveling” crystals,27–29 collective rotations,16,30–32 and even flocking.33 Activity also shifts the equilibrium freezing transition,34–37 affects the nature of the two-dimensional melting scenario,35,38–41 and changes the relaxation properties of amorphous solids.42–47 Systematic studies of collective excitations in active solids are still in their infancy,48 and they remain poorly explored even in general non-equilibrium solids (beyond active matter). Unveiling how the nature of the vibrations of a solid changes out of equilibrium represents an open issue relevant in statistical as well as solid-state physics.
External forces or internal mechanisms that dissipate energy drive a system away from equilibrium and spontaneously produce entropy. While self-propulsion is generated by an uptake of energy from the environment, likewise active particles dissipate energy, leading to local entropy production.49–54 Quantifying the non-equilibrium character of active systems via this observable has represented a topic of central interest in recent years.55–58 This subject has been investigated numerically, simulating both active field theories,59–62 active particle dynamics in interacting systems,63–65 as well as colloids in the presence of an active bath.66,67 Particular attention has been devoted to phase-separated configurations where the main contribution to the spatial profile of the entropy production has been observed at the interface between dense and dilute phases.59,68,69 Conversely, analytical results for entropy production have been only derived for simple cases, such as the potential-free particle,70–72 and for near-equilibrium regimes through perturbative methods.73,74 Entropy production in active crystals, however, remains unexplored.
In this paper, we fill this gap by discovering novel vibrational excitations characterizing non-equilibrium active solids and responsible for entropy production. Following the standard nomenclature of solid-state physics,75 we term these new modes “entropons” because each of them represents a spectral contribution to the entropy production rate. Unlike phonons, which describe the vibrational dynamics of equilibrium solids, entropons exist only in non-equilibrium, i.e., they are purely induced by activity. Entropons coexist with phonons but dominate over them for large activity and, therefore, represent the thermodynamically most relevant modes of an active crystal far from equilibrium. The underlying basic picture for active solids is shown in Fig. 1: in equilibrium, the thermal bath coupled to a crystal induces collective vibrational excitations like phonons (yellow color in Fig. 1); in non-equilibrium active solids, the self-propelling force injects energy into the crystal, this energy is dissipated in the environment, and the system produces entropy. In this process, entropons are generated as new collective vibrations, encoding the entropy production (orange color in Fig. 1). Our analysis is based on analytical theory combined with particle-resolved computer simulations and can, in principle, be verified in experiments with dense self-propelled Janus colloids12,76 or vibrated granular particles,29,77–79 as well as in living systems, such as confluent cell monolayers,6,8 which can exhibit crystalline patterns.
A. Collective excitations and spectral entropy production
The decomposition of the collective excitations in phonons and entropons has a deep physical meaning directly linked to emergent collective phenomena. Indeed, active systems at high density show velocity patterns and spatial velocity correlations, independently observed in numerical simulations in Refs. 20 and 23 and in experiments based on cell monolayers in Ref. 8. Since thermally excited phonons do not produce spatial patterns in real space, entropons are clear evidence of novel excitations in active solids.
B. Properties of entropons
A typical shape of σ(ω, q)T/Ta is shown in Fig. 2(a) as a function of ω/ωE for a given q. A sharp peak occurs at a characteristic frequency ω*(q). We identify this peak with an elementary excitation in the crystal and coin the term “entropon” to describe it, following the standard notation of elementary excitations in solids:75 Each entropon is identified with a peak of σ(ω, q).
Figures 2(b) and 2(c) show σ(ω, q)T/Ta as a function of ω/ωE for different values of q, revealing a good agreement between theory, Eq. (3), and numerical simulations. Close to the equilibrium, in the regime of small persistence time such that τ = 1/Dr ≪ 1/ωE [Fig. 2(b)], the peaks of σ(ω, q)T/Ta occur at the phonon frequency . In this regime, entropons have the same properties of phonons, but their amplitudes are small and proportional to τ (because of the prefactor Ta): The active force behaves as an additional thermal source at effective temperature Ta. In the opposite regime of large persistence time, τ = 1/Dr ≫ 1/ωE [Fig. 2(c)], the peaks of σ(ω, q) are shifted to . As a consequence, entropons are different from phonons since the crystal vibrations are now peaked at frequencies not coinciding with those typical of equilibrium solids. The frequency ω*(q) that maximizes σ(ω, q) is reported in Fig. 2(d) as a function of q for different rescaled persistence time, τωE, while the difference is shown in Fig. 2(e) as a function of τωE for several values of q. Despite ω*(q) linearly increasing with q in the small persistence regime, a clear discrepancy from the linear law emerges in the large persistence regime for small q, where entropons follow a non-linear dispersion relation. As a consequence, the difference grows when τωE is increased much more as q is decreased. Finally, the width of the peaks of σ(ω, q) increases with τωE [compare the curves with the same colors in Figs. 2(b) and 2(c)], implying shorter-lived excitations at high activities.
Finally, we recall that the present analysis has been obtained in the underdamped regime, when the inertial time τI is of the same order or larger than the typical relaxation time of the solid, given by the inverse of the Einstein frequency 1/ωE. Indeed, in the overdamped regime for τI ≪ 1/ωE, the spectrum shows only a single peak around ω ≈ 0, similarly to the case of equilibrium solids. This can be understood by looking at the denominator of Eq. (5) or Eq. (6) that suppresses the dependence from for vanishing τI.
C. Total entropy production rate
In conclusion, we have predicted new elementary vibrational excitations in active solids, termed entropons, because they are modes of the spectral entropy production. Our combined numerical and theoretical study revealed the properties of entropons, which coexist with phonons but dominate over phonons far from equilibrium when the active temperature is larger than the thermal one.
The concept of “entropons” as additional lattice vibrations has a broad generality that goes beyond monodisperse active crystals. It will certainly apply to binary crystals composed of active and passive particles.90,91 Moreover, we expect that in disordered dense systems, such as active glasses43,45–47,92,93 and dense active liquid crystals,94 entropons could play the dominant role of system excitations in determining entropy production. For instance, they could shed light on the activity-induced shift of the glass transition temperature.
Many experimental realizations of active crystals are available. Examples include confluent cell monolayers,6,8,95 dense assemblies of active colloids,76 as well as highly packed active granular systems,29,77,79 for which the solid structure has been achieved by connecting Hexbug particles by springs.48 Therefore, the existence of entropons can, in principle, be verified by analyzing particle trajectories in real space. To identify the contribution of entropons, it is crucial to have experimental access to the measurements of velocities and active forces, i.e., orientational angles. In this way, one can directly calculate the spectral entropy production in experiments by using Eq. (3). Alternatively, entropons can be measured indirectly without knowing the active forces through Eq. (5) by evaluating the dynamical correlations of the particle displacement and subtracting the contribution of thermal phonons, i.e., the response function due to a small perturbation.
Despite derived in the active case, entropons will characterize more general non-equilibrium crystals, as shown in Appendix E for a class of non-active solids driven out of equilibrium by temperature gradients. Therefore, the present paper reveals how concepts in the field of stochastic thermodynamics could have a determinant role in solid physics. For instance, we strongly believe that our theoretical approach can be applied to active solids characterized by polar, nematic, or even implicit alignment interactions between velocity and self-propelled angle. As a consequence, we speculate that entropons could be helpful to interpret and systematically analyze the collective excitations in a solid consisting of active granular particles, as the one experimentally realized in Ref. 48. Future studies on entropons can be manifold, for instance, the scattering of entropons near crystalline defects and the shifted spectra of entropons in a crystal exposed to a temperature gradient. This could lead to possible applications such as shock absorbers and active mass dampers as well as controlled heat radiators obtained by active crystals.
L.C. acknowledges support from the Alexander Von Humboldt foundation. U.M.B.M. and A.P. acknowledge support from MIUR PRIN 2017 Project No. 201798CZLJ. H.L. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through the SPP 2265 under Grant No. LO 418/25-1.
Conflict of Interest
The authors have no conflicts to disclose.
Lorenzo Caprini: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). U. Marini Bettolo Marconi: Formal analysis (supporting); Investigation (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). A. Puglisi: Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Hartmut Löwen: Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.
APPENDIX A: DERIVATION OF THE SPECTRAL ENTROPY PRODUCTION
1. Dynamics in the Fourier space
2. Definition and implicit expression for the spectral entropy production
a. Discretization scheme for the path-integral approach
To employ the path-integral approach, i.e., in writing the expressions for and , we have adopted a continuous time formalism, assuming the Itô convention. Of course, since our system does not involve any multiplicative noise, the resulting entropy production rate is not affected by the choice of the integration scheme: Itô and Stratonovich conventions lead to the same results. The difference between Itô and Stratonovich conventions in the calculation of entropy production rate using path-integral approaches is described in detail in Refs. 49 and 85.
3. Explicit expression for the spectral entropy production
To obtain an analytical expression for σ(ω, q), it is necessary to calculate the cross-correlation involved in Eq. (A11). In order to do that, we shall make two simplifying assumptions in the dynamics, Eq. (1) of the main text:
- Each particle performs small oscillations around a node of a triangular lattice so that the total inter-particle potential can be approximated as the sum of quadratic terms. Introducing the displacement ui of the particle i with respect to its equilibrium position, , namely, , the pair potential, in the harmonic approximation, readswhere ωE is the Einstein frequency of the solid, given by(A14)ωE is determined by the first and second derivative of the interacting potential, U, calculated at , i.e., the lattice constant of the solid, e.g., the average distance between neighboring particles in the solid.(A15)
- We approximate the active Brownian particle (ABP) active force, , as an Ornstein–Uhlenbeck process for each particle evolving aswhere ζi is a vector of white noises such that ⟨ζi(t)ζi(0)⟩ = δ(t).(A16)
4. Derivation of the positional dynamical correlation
APPENDIX B: DERIVATION OF THE TOTAL ENTROPY PRODUCTION OF THE SYSTEM
The prediction for the spectral entropy production σ(ω, q) allows us to calculate the entropy production of the system, , explicitly as a function of the parameters of the model.
APPENDIX C: NUMERICAL DETAILS
1. Numerical details of the simulations
2. Measure of the spectral entropy production and error estimate
To measure the spectral entropy production σ(ω, q), we have considered Eq. (3) (first line) of the main text. To do so, we have calculated the Fourier transform of data both in frequency ω and wave vector q, applying the definition on displacement, velocity, and active force by using Eqs. (A1a)–(A1c). Then, we have calculated the average dynamical correlation in the steady-state performing the average over the realization of the noise.
In the main text, Figs. 5(b) and 5(c) plot the spectral entropy production σ(ω, q) as a function of ω/ωE for three values of the rescaled wave vector qd0 and two values of the rescaled persistence time τωE. Data are reported without error bars for presentation reasons. Figure 2 contains the same data reported in Fig. 5 of the main text but split into several panels to confirm the agreement within the statistical error between data (points with error bars) and theoretical predictions (solid lines), given by Eq. (3) of the main text. In particular, panels (a)–(c) are realized with τωE = 7 [Fig. 5(c) of the main text], while panels (d)–(f) are realized with τωE = 7 × 10−2 [Fig. 5(b) of the main text]. Finally, panels (a) and (d), (b) and (e), and (c) and (f) plot qd0 = 0.1, 0.2, and 0.5, respectively.
APPENDIX D: VISUALIZATION OF THE SYSTEM: SNAPSHOTS AND MEAN-SQUARE DISPLACEMENT
In this section, we provide a visualization of the system in real space. In Figs. 3(a)–3(d), we report several snapshot configurations in the plane of motion of the system for different values of the reduced persistent time τωE = 7 × 101, 7, 7 × 10−1, 7 × 10−2. All of them display the typical hexagonal order characterizing two-dimensional crystals, as explicitly illustrated in Fig. 3, while the color gradient confirms the absence of spatial order in the direction of the active force, as expected.
In Fig. 6, we also report the displacement maps Δxi(t) = xi(t) − xi(0) for different values of the reduced persistent time τωE and for different duration times ΔtωE, where Δt = t − t0. For the smaller value of τωE, the displacement maps show weak structures that become more evident when ΔtωE increases. Spatial structures in the displacement maps are more evident and larger when τωE increases.
APPENDIX E: THE CONCEPT OF ENTROPONS BEYOND ACTIVE SOLIDS
In this section, we show the generality of the concept of entropons that, in general, characterize a broad class of far from equilibrium solids, governed by a breaking of the fluctuation–dissipation theorem. This emphasizes the generality of entropons beyond active systems supporting our claim of generality expressed in the last sentence of the main paper.
The expression for the dynamical correlations, contained in Eq. (E11), provides the information on the vibrational excitations of the non-equilibrium solids described by Eq. (E2). This observable is formed by the sum of two terms: (i) an equilibrium term, proportional to the response matrix, that provides the contribution of phonons along x and y directions and (ii) a second term that is proportional to the entropy production of the system and can be identified as the contribution of entropons, being proportional to the spectral entropy production. Consistently with our picture, this term disappears in the equilibrium limit for λ = 0 and/or Tx = Ty, i.e., when the entropy production rate vanishes.
The results of this section show that the picture of entropons goes beyond the case of active solids and characterizes more, in general, non-equilibrium solids reaching a steady state. In addition, in the case of solids driven out of equilibrium by the presence of different thermal baths, new vibrational excitations, encoding the spectral entropy production, coexist with thermal phonons.