Equilibrium and nonequilibrium properties of systems with long-range interactions

Equilibrium and nonequilibrium properties of systems with long-range interactions

Stefano Ruffo Dipartimento di Energetica “S. Stecco” and CSDC, Università di Firenze via s. Marta 3, I-50139 FIRENZE, Italy INFN, Sezione di Firenze
Received: date / Revised version: date

We briefly review some equilibrium and nonequilibrium properties of systems with long-range interactions. Such systems, which are characterized by a potential that weakly decays at large distances, have striking properties at equilibrium, like negative specific heat in the microcanonical ensemble, temperature jumps at first order phase transitions, broken ergodicity. Here, we mainly restrict our analysis to mean-field models, where particles globally interact with the same strength. We show that relaxation to equilibrium proceeds through quasi-stationary states whose duration increases with system size. We propose a theoretical explanation, based on Lynden-Bell’s entropy, of this intriguing relaxation process. This allows to address problems related to nonequilibrium using an extension of standard equilibrium statistical mechanics. We discuss in some detail the example of the dynamics of the free electron laser, where the existence and features of quasi-stationary states is likely to be tested experimentally in the future. We conclude with some perspectives to study open problems and to find applications of these ideas to dipolar media.

05.20.-yClassical statistical mechanics and 05.70.FhPhase transitions:general studies and 05.45.-aNonlinear dynamics and chaos

1 Introduction

For systems with long-range interaction the pair potential decays at large distances with a weak power law , with , the space dimension Leshouches (). Examples are: gravity, whose statistical mechanics is made more complex by the unremovable singularity of the potential at the origin; Coulomb interactions, with the phenomenon of charge screening which allows the treatment of globally neutral systems; dipolar media, that display the well known feature of shape dependence; vortices interacting with logarithmic potential in , for which Onsager first discussed microcanonical features, like the presence of negative temperatures, at the first Statphys meeting in Florence Onsager ().

Long-range interactions can be made extensive, but are intrinsically non-additive. Let us consider the simplest model of ferromagnetic systems, the Curie-Weiss mean-field Hamiltonian


in which the spins are globally coupled with strength . The energy scales with system size, , and an intensive energy density can be defined in the thermodynamic limit, but, due to the presence of links, the sum of the energies of two subsystems and is never giving the total energy . Besides that, for mean-field models of this kind the set of accessible macrostates in the space of intensive parameters and , can be non convex (see Fig. 1). These two mathematical properties, that can be present only at finite for short-range interactions (), have important physical consequences. The specific heat Lynden (), and other quantities related to the curvature of the entropy, like susceptibility Touchette (), can become negative in certain energy ranges; jumps in temperature can be realized at first-order microcanonical transitions Barre-01 (); broken ergodicity can be present both for finite and in the thermodynamic limit Mukamel-05 () (see Sect. 5). All this is inscribed inside the general framework of ensemble inequivalence Ensemble (); Ispolatov (). We will briefly illustrate these features for a mean-field XY model with two-spin and four-spin interactions in Sect. 2.

All the above is for equilibrium properties (i.e. maximal entropy states), but systems with long-range interactions also show a very slow approach to equilibrium. For systems with short-range interactions it has been definitely assessed that, appropriately selecting the initial state, the subsequent relaxation to equilibrium takes place on a finite time in the thermodynamic limit Lichtenberg (). On the contrary, since the seminal paper of Lynden-Bell Lynden (), it has been proposed that systems with long-range interactions can display a two-step relaxation. In a first stage, the system relaxes rapidly (“violently”, according to Lynden-Bell) to a quasi-stationary state whose lifetime increases with system size: two types of dependencies have been proposed, either power-law Yama-04 () or logarithmic Mukamel-05 (), depending on some detailed property of the initial state. In a second stage, the system begins a slow approach to equilibrium that may be either direct or proceed through successive relaxations to different quasi-stationary states. The fact that the lifetime of quasi-stationary states diverges with system size allows to obtain a separation of the two time scales. This scenario, originally proposed for gravity, has been extended to the two-dimensional Euler equation, the original Onsager’s system, by Chavanis Chava-96 (). A theoretical proposal has been advanced by Lynden-Bell in order to interpret the initial relaxation to a quasi-stationary state: that the system tries to maximize an entropy which takes into account additional “constraints” appearing in the thermodynamic limit, the most relevant of those being the normalization of the one-body distribution function Lynden (). Although the initial numerical tests of this theory gave some hope of success Hohl (), its predictions were subsequently disproved for several systems, even for the simple one-dimensional self-gravitating system Sakagami () and the theory fell into some discredit. We will show in Sect. 3 results for the so-called Hamiltonian Mean Field (HMF) model Antoni () (a mean-field XY model) for which Lynden-Bell theory has a straightforward application and leads to predictions that are in reasonable agreement with numerical simulations Chava-06 (). A recent domain of application of Lynden-Bell’s ideas is to the free electron laser FEL (), where this approach allows to predict the features of the intensity saturation of the laser, simply knowing the initial conditions, without explicitly solving the equations of motion, as was tipycally done before: in a sense, this constitutes the statistical mechanics of the free electron laser. This application is likely to produce in the future experimental tests of the features of quasi-stationary states, as we discuss in Sect. 6.

Furthermore, the macrostates that maximize Lynden-Bell’s entropy for a given energy depend on the initial magnetization of the HMF model. The system undergoes a nonequilibrium phase transition that can be of the second or the first order, and hence a nonequilibrium tricritical point is present (see Sect. 4). It is interesting that the concepts of equilibrium phase transition can be extended to states that are not in equilibrium. It means that one can assume mixing limited to the phase-space visited by the trajectories on a time scale that is small with respect to the relaxation time to equilibrium.

Sect. 7 is devoted to some final remarks and perspectives of applications to wave-particle systems and to dipolar media.

Figure 1: The set of accessible macrostates in the space can have a non-convex shape for systems with long-range interactions, such that if and can be realized macroscopically, this is not necessarily true for all the states joining these two along the straight dashed line.

2 Phase diagram of a mean-field XY model

Let us consider the following Hamiltonian


which can be thought as representing a system of spins with all-to-all two-spin, , and four-spin, , interactions. A kinetic energy term is added, considering as canonically conjugate to . Because of this addition, Hamiltonian (2) can also represent a system of unit mass particles moving on a circle without collisions, interacting only through a mean-field type potential. This model can be solved in both the canonical and the microcanonical ensemble Debuyl (). The resulting phase diagram is shown in Fig. 2. For both ensembles a tricritical point is present, but its location is different in the two ensembles. The behavior of the order parameter in the two ensembles is also shown, in order to highlight the striking difference in the predictions. The so-called caloric curve (kinetic temperature vs. energy density ) is reported in Fig. 3. The microcanonical ensemble (full line) predicts a region of negative specific heat, where kinetic temperature decreases as the energy density is increased. Moreover, a temperature jump is present at the transition energy.

Figure 2: Phase diagram of Hamiltonian (2). The canonical second order transition line (full horizontal line starting at ) becomes first order (dotted line) at the canonical tricritical point. The microcanonical second order transition line coincides with the canonical one up to but it extends further towards the microcanonical tricritical point, located at . At this latter point, the transition line bifurcates in two first order microcanonical lines (dashed), corresponding to a temperature jump. There are no microcanonical macrostates for parameter values within the shaded region.
Figure 3: Caloric curve for . The full line is the theoretical prediction in the microcanonical ensemble. The dashed line represents the first order phase transition in the canonical ensemble. The points are obtained from a molecular dynamics simulation of Hamiltonian (2) with .

3 Quasi-stationary states

Hamiltonian (2) reduces, for and adding a constant to shift the energy of the ground state to zero, to that of the HMF model Antoni ()


whose equilibrium properties are standard: the system undergoes a mean-field second order phase transition in both the microcanonical and the canonical ensemble at the energy , corresponding to the temperature (see Vatteville () for a comprehensive recent review of the model). In order to study nonequilibrium properties, it has been customary to prepare an initial state of the “water-bag” type where all the particles are uniformly distributed in a rectangular domain of width and height centered around the origin in the single-particle phase space . Once the size of the domain is given, energy and magnetization are uniquely determined: , . During microcanonical time evolution, energy and total momentum remain constant, but magnetization varies, and one expects that it reaches the equilibrium value compatible with the given energy. This is not what happens, as shown in Fig. 4. One observes an initial “violent” relaxation to a plateau value (closer and closer to as the number of particles is increased), corresponding to the quasi-stationary state, followed by a second relaxation to equilibrium, which takes place on longer and longer times as increases. The lifetime of the quasi-stationary state has been fitted with a power-law in Ref. Yama-04 ().

Figure 4: Magnetization vs. time for the HMF model with energy , vanishing total momentum and “water-bag” initial condition with (). The value of increases from left to right: .

How does one explain such a behaviour? It is crucial to realize that the mean-field dynamics (3) is well represented in the large limit by the following Vlasov equation BraunHepp ()


where is the single-particle distribution function and the potential is given by

Besides energy, Vlasov equation also conserves the norm of the distribution function , which for the “water-bag” initial condition we have considered implies that the distribution function remains two-level at all times. To take this into account, Lynden-Bell has proposed Lynden () that the system adapts itself on macrostates that maximize the following “fermionic” entropy


where is the coarse-grained distribution function. The Pauli principle that is implicit in this approach refers to the fact that a “fluid element” in the -space cannot occupy a cell which is already occupied by another fluid element, just because the distribution must remain two-level in the course of time. It turns out that the maximization of at fixed energy, momentum and norm can be explicitly performed for the HMF model, giving the following solution


where , and are Lagrange multipliers corresponding to the conservation of energy, momentum and norm. As it is clear from equation (6), the quasi-stationary distribution has to be determined self-consistently, since it depends on the distribution itself through the magnetization . We have chosen the subscript QSS to mean quasi-stationary-state, because we propose Chava-06 () that the states that maximize Lynden-Bell entropy are indeed the quasi-stationary states observed, e.g., in the numerical simulation reported in Fig. 4. The magnetization in the QSS, , and the values of the Lagrange multipliers are obtained by solving numerically a set of implicit equations Chava-06 (), once the energy and the initial magnetization are given (or alternatively ). Since we look for solutions whose total momentum vanishes, . In Fig. 5 we show the comparison of the predictions of the theory concerning momentum distributions with the numerical simulations performed integrating the equations of motion given by Hamiltonian (3) with . The agreement is quite good, if one takes into account that the predictions have no adjustable parameter and are strictly determined by the choice of the initial condition. However, expecially the plot in lin-lin scale (panel (d) in Fig. 5) reveals that our theory is unable to reproduce the double bump obtained in numerical simulations.

Figure 5: (color online) Comparison of the theoretical predictions for the quasi-stationary momentum distribution (dashed line) with the numerical simulations (points) performed using Hamiltonian (3) for and (a), (b), (c) . The case is also shown in lin-lin scale in panel (d). For all these cases (homogeneous quasi-stationary states).

The presence of this bumpy feature is confirmed by simulating the Vlasov equation (4) (see Fig. 6) Califano (). The double bump in momentum distribution is the result of the presence of two sliding resonances (the two vortices in the lower right panel), for which an explanation that takes into account the specific dynamical properties of the model is necessary Chava-06 (); Califano ().

Quasi-stationary states have been demonstrated to be robust to both the application of and external bath Baldovin () and to the addition of a small nearest neighbor interaction Giansanti (). The Vlasov equation is exact in the limit, and people have tried to address the question of finite corrections. In this respects the most interesting progress is registered in papers by Bouchet and Dauxois BouchetDauxois () and Chavanis Chava-Vlasov (). For the sake of commpleteness, it should be mentioned that a different approach, based on Tsallis statistics, has been proposed to describe quasi-stationary states Rapisarda (). Recently, these authors have concentrated their attention on the behaviour of correlations in the single particle dynamics, that can give rise to non Gaussian distributions for sums of variables (generalized central limit theorems).

Figure 6: (color online) Time evolution of the single-particle distribution function according to the Vlasov equation (4) for a rectangular “water-bag” initial state with and .

4 Nonequilibrium tricritical point

The maximization of Lynden-Bell entropy reserves another surprise. Since this variational problem introduces another control parameter besides energy, the magnetization of the initial state, one obtains a phase transition at an energy that depends on . The transition energy coincides with that given by Boltzmann entropy (to which Lynden-Bell’s entropy reduces in the diluted limit ) only for . Not only, below the transition becomes first order. The full phase diagram is plotted in Fig. 7. A nonequilibrium tricritical point is present in the phase diagram. The specific heat is negative along all the transition line and a temperature jump appears when the transition change to first order.

Figure 7: (color online) Theoretical phase diagram on the control parameter plane : second order phase transition line (dashed); first order phase transition line (full); tricritical point (full dot). Inset: magnification of the first order phase transition region and limits of the metastability region (dash-dotted).

Numerical simulations confirm this extremely interesting theoretical finding Antoniazzi (). For instance, we plot in Figs. 8,9 the order parameter as a function of energy density at fixed , finding signature of a second order (Fig. 8) or first order (Fig. 9) phase transition. As previously explained, these are “short-time” phase transitions, at variance with the “long-time” phase transitions of equilibrium statistical mechanics. Hence, in order to reveal them numerically, one has to average over a short initial time and take many initial instances.

Figure 8: (color online) as a function of for where the phase transition is second order. The full line is the theoretical prediction and the points are simulations of the HMF model for which the number of initial realizations is (), (), () and the averaging time .
Figure 9: (color online) as a function of for where the phase transition is first order. The results are plotted as in Fig. 8.

5 Broken ergodicity

After this excursion in nonequilibrium, let us come back to equilibrium properties. The XY model (2), for some specific choice of antiferromagnetic and ferromagnetic couplings has energy density values where regions with different magnetization are disconnected if energy is fixed (as shown in Fig. 1) broken (). An example is given in Fig. 10, where we show two different situations. In panel a) the “standard” ergodicity breaking associated with a phase transition is revealead by successive magnetization switches between local entropy maxima: the number of particles is small () such that the entropy barrier (see inset) is not insurmontable. In panel b) we display the new type of microcanonical broken ergodicity first discussed in Refs. Mukamel-05 (); Borgonovi (): now magnetization cannot switch because there is a gap with no macrostates.

Figure 10: Time evolution of the magnetization of model (2) for: a) , and ; b) , and , obtained integrating the equations of motion derived from Hamiltonian (2) with . In panel a) magnetization switches between the two most probable values (see the inset for the dependence of entropy on ). In panel b), two different initial conditions are plotted simultaneously, corresponding to two different values of the initial magnetization, and . The inset shows the entropy, which now vanishes in the interval . Here and , which are shown to bound from above (dotted line) and from below (dashed line) the magnetization of the two initial conditions.

6 Application to the free electron laser

Free-Electron Lasers (FELs) are coherent and tunable radiation sources, which differ from conventional lasers because they use a relativistic electron beam as their lasing medium, hence the term “free-electron”. The physical mechanism responsible for light emission and amplification is the interaction between the relativistic electron beam, a magnetostatic periodic field generated in the undulator and an optical wave co-propagating with the electrons. Due to the presence of the magnetic field, electrons are forced to follow oscillating (tipycally sinusoidal) trajectories and emit synchrotron radiation. This spontaneous emission is then amplified along the undulator until the laser effect is reached. Among the different schemes, single-pass high-gain FELs are currently attracting a growing interest. Basic features of the system dynamics are successfully captured by a simple Hamiltonian model introduced by Colson and Bonifacio Bonifacio ()


where is the rescaled longitudinal coordinate, which plays the role of time. Here, is the so-called Pierce parameter, the mean energy of the electrons at the undulator’s entrance, the wave number of the undulator, the plasma frequency, the speed of light, the total electron number density, and respectively the charge and mass of one electron. Furthermore, , where is the rms peak undulator field. Here is the resonant energy, and being respectively the period of the undulator and the wavelenght of the radiation field. Introducing the wavenumber of the FEL radiation, the two canonically conjugated variables are (,), defined as and . corresponds to the phase of the electrons with respect to the ponderomotive wave. The complex amplitude represents the scaled field, transversal to . Finally, the detuning parameter is given by , and measures the average relative deviation from the resonance condition. This succinct, but detailed, description of the model should transmit the idea that, by adjusting model parameters, one can really simulate realistic experimental situations.

It can be easily checked that model (7) can be derived from the Hamiltonian


where is the number of electrons and the intensity  and the phase  of the wave are related to . In addition to the energy, the total momentum is also a conserved quantity.

It should also be mentioned that, remarkably, this simplified formulation applies to other physical systems, provided an identification of the variables involved is performed. As an example, consider the electron beam-plasma instability. When a weak electron beam is injected into a thermal plasma, electrostatic modes at the plasma frequency (Langmuir modes) are destabilized. The interaction of the Langmuir waves and the electrons constituting the beam can be studied in the framework of a self-consistent Hamiltonian picture ElskensBook (), formally equivalent to the one in Bonifacio (). In a recent paper andrea () we have established a bridge between these two areas of investigation and exploited the connection to derive a reduced Hamiltonian model to characterize the saturated dynamics of the laser.

There are many similarities between model (8) and the HMF model (3), and indeed the canonical free energy of one model can be exactly mapped onto the other largedev (). While in the HMF model particles interact directly, in the FEL dynamics electrons interact only through the field , whose dynamics depend in turn on those of the electrons. Therefore, already on this basis one could expect similarities in the behaviours of the two models. Indeed, similarly to the HMF model, Hamiltonian (8) presents a standard second order phase transition of the mean field type. As for the HMF, FEL dynamics is well represented in the limit by the following Vlasov equation


Hence, one can similarly expect the existence of quasi-stationary states FEL (). A direct simulation of Eqs. (7) is shown in Fig. 11 for electrons. Initially, the electrons are uniformly distributed in the interval and their momentum is also uniformly spread in in a “water-bag” distribution: this initial state has a physical meaning, because it is very close to the state in which, experimentally, the electrons are injected into the undulator. The laser intensity is initially set to a small value, in order to switch on the instability, whose initial growth is exponential. On a short time, the intensity reaches a saturation level and performs wide oscillations around it, but these oscillations dump with time, and a well defined asymptotic intensity is reached. This process is the equivalent of Lynden-Bell’s “violent relaxation”. The -dependence of the saturation level is shown in the inset, where three different values of are considered. For the smaller value, (curve 3), the saturated intensity is larger than for higher values. When increases the system remains trapped in the quasi-stationary state for a longer and longer time. The second relaxation to the larger intensity value corresponds to the relaxation to Boltzmann-Gibbs equilibrium. This latter relaxation has no experimental relevance for free electron lasers, because it would take place for enormous undulator lengths. A similar behavior would be observed for the “bunching parameter” , which is the equivalent of the magnetization of the HMF model.

Figure 11: Typical evolution of the radiation intensity of a free electron laser using Eqs. (7); the detuning is set to , the energy per electron and electrons are simulated. The inset presents averaged simulations on longer times for different values of : (curve 1), 400 (curve 2) and 100 (curve 3).

The distribution in the quasi-stationary state can be obtained by maximizing Lynden-Bell entropy (5), keeping energy and momentum fixed. The result of this maximization procedure is very similar to that of the HMF




and , and are the usual Lagrange multipliers, whose value is determined by the initial condition.

The dependence of both the intensity and the bunching parameter in the quasi-stationary state on the detuning parameter for an initially homogeneous state () with zero momentum dispersion is shown in Fig. (12).

Figure 12: Laser intensity and bunching parameter in the quasi-stationary state as a function of detuning : theoretical prediction (solid and dashed lines, respectively) and simulations (symbols). The dotted vertical line, , represents the transition from the low to the high-gain regime.

If FEL’s Hamiltonian is equivalent to HMF, one might wonder if situations exist which produce more complex phase diagrams and dynamical evolutions. Indeed, in the regime where the beam current and the emittance is small, FEL’s dynamics is better described by a model where many planar waves interact with the beam


where the complex amplitudes represent the scaled field, transversal to , and the coupling parameters depend on the experimental setup. We have analysed in some detail Johal () only the case with two odd harmonics , performing the same analysis of the quasi-statonary states as for Eqs. (7). The initial condition is again a homogeneous almost monocromatic beam, but now there are two “order parameters”, and , with the corresponding bunchings and . In region (shaded) of Fig. 13 the quasi-stationary state is dominated by the first wave: hence and . On the contrary, in region , and . The full line corresponds to a nonequilibrium phase transition of the first order.

Figure 13: (color online) Phase diagram in the (,) parameter space for the two-wave model (12), corresponding to a FEL with an initial cold homogeneous beam. Lynden-Bell’s theory predicts a transition from a region (shaded) dominated by the first wave to one, , dominated by its odd harmonic. Above the dashed line the short time growth rate of is predicted to be larger than that of by a linear theory

However, a linear analysis show that in the region between the dashed line and the transition line to (inside the region) the growth rate of is larger than that of . Therefore, at short time the growth of is expected to be faster and the initial evolution of the FEL is dominated by the first odd harmonic. However, this is not the maximal Lynden-Bell’s entropy state, and on a later time, the system is finally going to be dominated by the first wave. This is indeed what happens, as shown in Fig. 14.

Figure 14: (color online) Upper panel: Evolution of the intensities and showing an initial growth of the first odd harmonic followed by the final relaxation to the maximal Lynden-Bell’s entropy state dominated by the first wave. Lower panel: time evolution of Lynden-Bell’s entropy

7 Conclusions and perspectives

We have discussed both equilibrium and nonequilibrium properties of systems with long range interactions with reference to mean-field models. Most of these features should extend to cases where forces weakly decay Leshouches (); largedev (), but this problem remains to be seriously investigated. In this perspective, the general classification of phase transitions obtained in Ref. BarreBouchet () could be extremely useful.

The domain of application of these ideas is vast and the perspective to perform key experiments in the near future is realistic. We have discussed in some detail in Sect. 6 a realistic model of the free electron laser. However, we can foresee applications to all systems where particles interact with waves in a self-consistent way. As an example, let us mention collective atomic recoil lasers, where ultracold atoms interact with the optical modes of the cavity, determining collective effects and first and second order phase transitions CARL ().

Another domain where experimental applications are envisageable is that of layered spin structures, where dipolar interactions dominate over Heisenberg exchange Sievers (). In this case, a mean-field term of the Hamiltonian is shown to depend on the shape of the sample, determining the presence or absence of phase transitions Campa ().

Most of my knowledge of the physics of systems with long-range interactions has been shaped by J. Barré, F. Bouchet, P.H. Chavanis, T. Dauxois, D. Fanelli D.H.E. Gross, D. Mukamel, Y.Y. Yamaguchi. I also thank all the coauthors of my papers on this subject, with whom I have entertained endless discussions. This work is funded by the PRIN05 grant Dynamics and thermodynamics of systems with long-range interactions.


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