Absence of thermalization for systems with long-range interactions coupled to a thermal bath
We investigate the dynamics of a small long-range interacting system, in contact with a large long-range thermal bath. Our analysis reveals the existence of striking anomalies in the energy flux between the bath and the system. In particular, we find that the evolution of the system is not influenced by the kinetic temperature of the bath, as opposed to what happens for short-range collisional systems. As a consequence, the system may get hotter also when its initial temperature is larger than the bath temperature. This observation is explained quantitatively in the framework of the collisionless Vlasov description of toy models with long-range interactions and shown to be valid whenever the Vlasov picture applies, from cosmology to plasma physics.
In the recent past, several theoretical and experimental studies have been devoted to exploring dynamical and thermodynamic properties of long-range interacting systems (LRIS’s) Campa et al. (2009). In such systems, energy is not additive. This fact, together with a possible break of ergodicity, is at the origin of a large gallery of peculiar thermodynamic behaviors: the specific heat of LRIS’s can be negative in the microcanonical ensemble Barré et al. (2001) and temperature jumps may appear at microcanonical first-order phase transitions. These systems also display remarkable non-equilibrium dynamical features. For example, it is well known that under particular conditions isolated LRIS’s may get trapped in long-lasting quasi-stationary states (QSS’s), whose lifetime diverges with system size Yamaguchi et al. (2004); Antoniazzi et al. (2007a). Importantly, when performing the limit ( being the number of particles), the system remains permanently confined in QSS’s Antoniazzi et al. (2007b, c). As a consequence, for large long-range interacting systems, QSS’s are directly accessible through experiment Barré et al. (2004); Bonifacio and Salvo (1994); Bachelard et al. (2010).
Until today, the large majority of studies aimed at elucidating the fundamental properties of LRIS’s have been carried out on isolated systems, i.e. under the assumption that the system properties are not influenced by the external environment. However, recognizing whether a non-equilibrium QSS is stable to an external perturbation is of great importance Nardini et al. (2012), both from a theoretical and an experimental point of view. A related fundamental problem concerns the mechanism through which a LRIS exchanges energy with the surroundings. These questions epitomize the main motivation of the present work.
The non-equilibrium dynamical properties of the LRIS’s in contact with a thermal bath have been studied for the first time only recently Baldovin and Orlandini (2006); Baldovin et al. (2009); Chavanis et al. (2011). As a possible realization of thermal bath, these authors considered a large Hamiltonian system with nearest-neighbor interactions, coupled to a fraction of the spins in the system. They concluded that the coupling with the bath introduces a new time scale in the evolution of the system: the weaker the coupling strength, the longer the system remains trapped in a QSS before relaxing to equilibrium.
At variance with the above studies, we investigate here the dynamics of a LRIS in long-range contact with an additional large system trapped in a QSS. This interaction scheme can be regarded as a more clear-sighted realization of a thermal bath for a LRIS, opening the way to applications in diverse fields such as cosmology and plasma physics. For example, one may think of the collisionless mixing between plasmas, or the operation of magnetic fusion devices for energy production or the merging of globular clusters to a self-gravitating galaxy. Furthermore, it is also tempting to speculate that our simple scheme could be somehow relevant for the self-consistent interaction between dark (the bath) and baryonic (the system) matter in the universe (see, i.e. Ref. Springel et al. (2005)).
Ii The long range thermal bath and the canonical QSS
As a reference case, we have selected the Hamiltonian Mean Field (HMF) model Antoni and Ruffo (1995), widely regarded as a prototype LRI system. The HMF Hamiltonian describes the one-dimensional motion of rotators coupled through a mean field cosine-like interaction,
where is the orientation of the -th rotator and its conjugated momentum. To monitor the evolution of the system, it is customary to introduce the magnetization , an order parameter defined as
The infinite-range coupling between rotators is responsible for the emergence of rather intriguing behaviors, including the existence of QSS’s. In a QSS the system displays non-Gaussian velocity distributions and it takes values of different than those predicted by equilibrium thermodynamics Antoniazzi et al. (2007a, b); da C. Benetti et al. (2012).
Rigorous mathematical results Braun and Hepp (1977) indicate that in the limit the discrete HMF dynamics reduces to the continuum Vlasov equation
where is the microscopic one-particle distribution function, , and . The specific energy is a conserved quantity. The Vlasov equation defines the natural framework to address the puzzle of QSS’s emergence Yamaguchi et al. (2004); Antoniazzi et al. (2007b). Specifically, QSS’s are connected to the stable stationary solutions of the Vlasov equation. This observation suggests a statistical mechanics approach, inspired by the seminal work of Lynden-Bell Lynden-Bell (1967), to characterize analytically the QSS properties. Lynden-Bell’s approach is based on the definition of a locally-averaged (“coarse-grained”) distribution, yielding an entropy functional defined from first-principle statistical-mechanics prescriptions. By constrained maximization of such an entropy, one obtains closed analytical expressions for the single-particle distribution in the QSS regime Antoniazzi et al. (2007b, c). As a natural consequence, the QSS’s can be equally interpreted as equilibrium configurations of the corresponding continuous description Staniscia et al. (2010). Hence, the QSS thermal bath that we consider here corresponds to a magnetized equilibrium solution of the underlying Vlasov equation (3).
Let be the normalized single-particle distribution that characterizes the QSS bath. Such a function is obtained as the stationary solution of the Vlasov equation (3) corresponding to a water-bag initial distribution, for and zero elsewhere. Note that the initial magnetization of the bath and its energy density can be expressed in terms of and , as and . This in turn implies that the initial water-bag profile is uniquely determined by and , in agreement with the Lynden-Bell theory 111The Lynden-Bell theory provides a quantitatively correct description of macroscopic observables, such as the average QSS magnetization. Alternative approaches accounting explicitly for non-ergodicity yield more accurate predictions da C. Benetti et al. (2012)..
At this point, in our discussion, another HMF system with water-bag profile is injected and let evolve consistently with the bath. This system, in the following, is described in terms of its associated single-particle distribution . Clearly the bath should be significantly larger than the system to which it is coupled. This can be accomplished through the following normalization condition
where sets the relative size of the two mutually interacting and HMF systems. We are interested in tracking the time evolution of the distribution under the constraint (4). From the physical point of view, we are reproducing the microcanonical dynamics of one isolated HMF system (), composed of two sub-systems supposed as distinguishable: the larger subsystem (the bath ) has already relaxed to its QSS equilibrium. The system is initially confined in an out-of-equilibrium configuration of the water-bag type. To monitor the evolution of both subsystems, we follow the kinetic temperatures , with and the corresponding magnetizations . Here, and . We emphasize that are average kinetic energies per particle and not true thermodynamic temperatures. In fact, our results highlight the crucial fact that the appropriate definition of the true thermodynamic temperature associated with a QSS is not known.
A typical time evolution of these observables, obtained by numerical integration of the Vlasov equation (3), is illustrated in Fig. 1 222See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevE.87.042110 for a movie depicting the time evolution of the bath and system phase portraits together with plots of the magnetization and the temperature.. Before injecting the system (i.e.. at ), the bath is first prepared in a water-bag initial condition and then allowed to evolve towards a QSS. After the bath has relaxed well into its QSS (this is ), the interaction is switched on, meaning that a new HMF combined system is evolved, comprising bath and system. After a short transient, the system reaches a quasi-equilibrium state where the mean value of the kinetic temperature is different from the temperature of the bath. In other words, the bath and the systems do not thermalize. Similarly, the two magnetizations converge to different values. Importantly, we note that the specific values of temperature and magnetization attained by the system spotlight a non-trivial interaction with the bath. and are indeed substantially different from the values that the system would reach when evolved microcanonically from the same initial condition. We obtain equivalent results upon simulating the discrete -body dynamics (1). In this case, after a transient that gets progressively longer as the system size is increased, and tend to zero. Thus, granularity causes thermalization, which is instead prevented in the continuum (Vlasov) limit. We term canonical QSS’s the quasi-equilibrium configurations that the system explores when in long-range contact with a QSS thermal bath in the zero energy-flux regime.
In the continuum limit, when the system is trapped in a canonical QSS, we find that the average energy flux between the bath and the system indeed vanishes, making the two subsystems by all means decoupled and thus preventing thermalization (see Appendix A for a more detailed analysis). It is remarkable that a zero-flux steady state is reached for in the non-collisional continuum limit, at variance with what is normally found in most collisional systems.
Iii The energy flux between the system and the bath
Even more surprising is the behavior of the system during the “violent relaxation” stage towards the canonical QSS, which is characterized by a net energy flux from the (cold) bath to the (hot) system. To better illustrate this observation, we plot and versus time in Fig. 2. Note that is larger than at , the time of injection. As time progresses, the difference increases even further, resulting in an anomalous energy transfer from the bath to the system. In short, and counter-intuitively, the hot system gets hotter when placed in contact with a large long-range QSS reservoir. This observation, although fighting intuition, does not violate any laws of physics, as the second law of thermodynamics is only expected to hold at thermal equilibrium.
Once the system has settled down in its canonical QSS at zero average energy flux, and are found to be different from zero. In order to pinpoint the relation between and , we performed a series of simulations for the same bath conditions as specified in the caption of Fig. 1, and varying the initial energy of the system . Different energies lead to distinct canonical QSS’s, as it happens to isolated systems trapped in microcanonical QSS’s. At first glance, it is tempting to speculate that canonical QSS’s might originate from a net balance of two opposing thermodynamic forces, presumably related to and . However, we find that the dynamical evolution of is not influenced by the temperature of the bath , at least for , but only responds to its magnetization . Therefore, provided is kept fixed, can be set to an arbitrary value, without significantly altering the system dynamics. This is illustrated by the data collapse reported in Fig. 3.
Iv Beyond the HMF model: a theoretical interpretation based on the Vlasov equation
This striking observation is unintuitive as compared to the case of short-range systems. Even more interestingly, it is by no means restricted to the HMF. In order to illustrate this fact, we note that in the Vlasov limit the distribution functions obey to
where is a generic mean-field potential (the prime defining ordinary differentiation with respect to ), defined as
being the two-body potential. Since the system/bath relative size , we can treat it as a perturbative parameter, with and . Expanding eqs. (5) and keeping only terms that cause changes in the physical observables, we are led to the two following coupled equations
The equation for the bath implies that this is frozen in its initial configuration, a stable equilibrium of the Vlasov equation, at all times 333A similar scenario is expected for baths at thermal equilibrium, which is also a stable state of the Vlasov equation.. The equation for is the Liouville equation for a distribution of uncoupled particles moving in an external potential, being constant. These conclusions are utterly general and should apply to any physical system whose density is governed by the Vlasov equation. For the HMF model, thanks to its rotational invariance, one has with no loss of generality
which is simply the Liouville equation for a set of uncoupled pendula. Hence, the leading-order evolution of depends only on and not on . As it is shown in Appendix B, sets the width of the resonance of the pendulum along , which scales as . This implies that the temperature should be proportional to , as can be also appreciated by dividing eq. (8) in the stationary state by .
Consistently with the above scaling arguments, we plot in Fig. 3 as a function of the rescaled width of the initial water-bag , for different values of the bath magnetization and temperature. The data refer to direct integration of the (constrained) Vlasov equations (5) and of eqs. (8). In all cases, the data collapse nicely on a single master curve, which confirms the validity of our reasoning. An analytical calculation of for yields , in excellent agreement with the result of direct integration of eq. (8) (see Appendix B and also Ref. de Buyl et al. (2011)). The inset further shows that -body simulations agree with all results obtained in the continuum limit.
Summarizing, we have proposed an implementation of long-range QSS bath. We showed that a small system in true long-range contact with a large, long-range reservoir reaches a zero-flux steady state, that we term canonical quasi-stationary state. These are stationary states of the system-bath coupled Vlasov equations, but quasi-stationary solutions of the associated -body problem. Remarkably, in the explored range of parameters, we find that hotter-than-bath systems become hotter in canonical QSS’s. In the context of the HMF model, based on simple scaling arguments, we have unveiled how the system anomalously increases its kinetic temperature as the fraction of its particles trapped in the resonance set by the bath magnetization gain energy. The kinetic energy gain is proportional to the value of and independent of the bath temperature at the leading order in . We stress here, that this observation does not violate any fundamental laws of physics. Indeed, the average kinetic energy of the system does not coincide with its thermodynamic temperature. In this respect, our work raises the following central, yet unanswered, question: what is the correct thermodynamic measure of temperature for a system frozen in a QSS? Notice that in the present work, the energy of the thermal bath was chosen to lie in the part of the (microcanonical) phase diagram corresponding to a magnetized QSS. As regards the system, we considered initial energies leading to both magnetized and non magnetized (microcanonical) QSS’s.
In conclusion, and based on the theoretical analysis that we have carried out, we argue that the results illustrated in this paper are general and extend beyond the HMF case-study, whenever the collisionless Vlasov picture is a good description of the dynamics.
Acknowledgements.Dicussions during the workshop “Equilibrium and out-of-equilibrium properties of systems with long-range interactions” held at the Centre Blaise Pascal, ENS-Lyon in August 2012 are acknowledged. F.P., D.F. and G.D.N. would like to thank D. Hane for insightful discussions.
Appendix A The energy flux
The energy flux from the bath to the system is defined as , where is the total energy of . In order to derive an explicit expression for , we start by calculating , the rate of energy loss of the -the particle. Denoting by its energy, we have
where and , being the global magnetization
Here is the total number of particles, i.e. the sum of those belonging to the bath, , and those in the system, . Summing over all particles belonging to the bath in eq. (9), one eventually obtains
where is the magnetization of the bath and the time derivative of reads
In the continuum limit the sums are replaced by integrals
with , .
According to the adopted sign convention, is positive if the bath cedes energy to the system. In Fig. 5 the instantaneous energy flux (upper panel) is plotted versus time for a typical realization of the Vlasov dynamics. After an initial transient, oscillates around zero, implying that the bath and the system have established a zero-average-flux dynamical equilibrium. This condition corresponds to the emergence of the canonical QSS. Furthermore, the net energy flux is positive, a fact that can be appreciated by looking at the evolution of the cumulated flux (see lower panel of Fig. 5). This implies a net transfer of energy from the bath to the system.
We stress that the system gets hotter as its total energy increases after putting it in contact with the bath. The total energy of the system increases when it is put in contact with the bath, as it is clearly proved by looking at the energy flux (the time derivative of the total energy) versus time in Fig. 5. The cumulated flux is positive, which, according to our conventions, attests to a flow of total energy from the bath to the system. In order to make this point even more clear, we show in Fig. 4 the total energy of the system versus time from the moment of the injection. The total energy of system and bath stays constant, while there is a clear flux of total energy from the bath to the tiny system, which is left permanently hotter as a result.
Appendix B On the analytic estimate of the asymptotic temperature .
The phase space of the pendulum is foliated in trajectories with constant energy
hence, . We want to discuss an analytic estimate of the quantity for and for an initial homogeneous system, . This calculation has the merit of enabling one to gain insight into the nature of the canonical QSS and further clarify the scaling adopted in Fig. 4. This analysis can be extended to cover the case , and also , a generalization to which we shall return in a separate contribution.
We note that for . To evaluate , we first consider the average kinetic temperature of the particles which are assigned a given energy . In formulae
where indicates a time average over one period
being the complete elliptic integral of the first kind. Expression (16) takes the equivalent form
where is the angle of inversion of the selected (closed) trajectory. By performing the integral one eventually gets
where is the incomplete elliptic integral of the second kind. The final temperature of the system can now be evaluated as
where is the density of states of the system, which is univocally fixed by the initial condition. The integral in eq. (20) extends from to , i.e. the energies that identify the separatrix of the pendulum. In fact, the system is trapped inside the separatrix , given the specific condition selected here (, hence no particle lies outside the resonance at ). Recalling eq. (15), the distribution can be calculated easily, as
Numerical integration gives , in excellent agreement with the data reported in Fig. 3. In the general case (), . The scaling suggested by eq. (22) implies , which in turn explains the origin of the reduced variables used in Fig. 3.
- Campa et al. (2009) A. Campa, T. Dauxois, and S. Ruffo, Phys. Rep. 480, 57 (2009).
- Barré et al. (2001) J. Barré, D. Mukamel, and S. Ruffo, Phys. Rev. Lett. 87, 030601 (2001).
- Yamaguchi et al. (2004) Y. Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois, and S. Ruffo, Physica A 337, 36 (2004).
- Antoniazzi et al. (2007a) A. Antoniazzi, D. Fanelli, J. Barré, P.-H. Chavanis, T. Dauxois, and S. Ruffo, Phys. Rev. E 75, 011112 (2007a).
- Antoniazzi et al. (2007b) A. Antoniazzi, F. Califano, D. Fanelli, and S. Ruffo, Phys. Rev. Lett. 98, 150602 (2007b).
- Antoniazzi et al. (2007c) A. Antoniazzi, D. Fanelli, S. Ruffo, and Y. Y. Yamaguchi, Phys. Rev. Lett. 99, 040601 (2007c).
- Barré et al. (2004) J. Barré, T. Dauxois, G. D. Ninno, D. Fanelli, and S. Ruffo, Phys. Rev. E 69, 045501(R) (2004).
- Bonifacio and Salvo (1994) R. Bonifacio and L. D. Salvo, Nucl. Instrum. Methods Phys. Res. A 341, 360 (1994).
- Bachelard et al. (2010) R. Bachelard, T. Manos, P. de Buyl, F. Staniscia, F. S. Cataliotti, G. D. Ninno, D. Fanelli, and N. Piovella, J. Stat. Mech. 2010, P06009 (2010).
- Nardini et al. (2012) C. Nardini, S. Gupta, S. Ruffo, T. Dauxois, and F. Bouchet, J. Stat. Mech. 2012, L01002 (2012).
- Baldovin and Orlandini (2006) F. Baldovin and E. Orlandini, Phys. Rev. Lett. 96, 240602 (2006).
- Baldovin et al. (2009) F. Baldovin, P.-H. Chavanis, and E. Orlandini, Phys. Rev. E 79, 011102 (2009).
- Chavanis et al. (2011) P.-H. Chavanis, F. Baldovin, and E. Orlandini, Phys. Rev. E 83, 040101(R) (2011).
- Springel et al. (2005) V. Springel, S. D. M. White, A. Jenkins, C. S. Frenk, N. Yoshida, L. Gao, J. Navarro, R. Thacker, D. Croton, J. Helly, J. A. Peacock, S. Cole, P. Thomas, H. Couchman, A. Evrard, J. Colberg, and F. Pearce, Nature 435, 629 (2005).
- Antoni and Ruffo (1995) M. Antoni and S. Ruffo, Phys. Rev. E 52, 2361 (1995).
- da C. Benetti et al. (2012) F. P. da C. Benetti, T. Teles, R. Pakter, and Y. Levin, Phys. Rev. Lett. 108, 140601 (2012).
- Braun and Hepp (1977) W. Braun and K. Hepp, Commun. Math. Phys. 56, 101 (1977).
- Lynden-Bell (1967) D. Lynden-Bell, Mon. Not. R. Astron. Soc. 136, 101 (1967).
- Staniscia et al. (2010) F. Staniscia, A. Turchi, D. Fanelli, P. H. Chavanis, and G. De Ninno, Phys. Rev. Lett. 105, 010601 (2010).
- (20) The Lynden-Bell theory provides a quantitatively correct description of macroscopic observables, such as the average QSS magnetization. Alternative approaches accounting explicitly for non-ergodicity yield more accurate predictions da C. Benetti et al. (2012).
- (21) See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevE.87.042110 for a movie depicting the time evolution of the bath and system phase portraits together with plots of the magnetization and the temperature.
- (22) A similar scenario is expected for baths at thermal equilibrium, which is also a stable state of the Vlasov equation.
- de Buyl et al. (2011) P. de Buyl, D. Mukamel, and S. Ruffo, Phys. Rev. E 84, 061151 (2011).