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Out-of-equilibrium quasi-stationary states (QSS)
Studying the dynamics of Hamiltonian systems with a large number of degrees of freedom and its connection to equilibrium statistical mechanics has been a long standing problem. The relaxation to statistical equilibrium has been under scrutiny ever since the pioneering work of Fermi and the FPU problem [40]. Moreover, since the advent of powerful computers and for specific systems within a class of initial conditions, integrating numerically Hamiltonian dynamics has proven to be competitive in regards to Monte-Carlo schemes for the study of statistical properties (see for instance [41, 42] and references therein). The assumption made is that since the system admits only a few conserved quantities for generic initial conditions, once the dimensions of phase space are large enough, microscopic Hamiltonian chaos should be at play and be sufficiently strong to provide the foundation for the statistical approach within the micro-canonical ensemble. In order to attain thermodynamic equilibrium one typically has to wait a long enough time t for the system to relax to its final maximum entropy state. However recent studies have shown that there is an increase of regularity with the system size in the microscopic dynamics when considering systems with long-range interactions [43, 44, 45, 46]. Indeed, the statistical and dynamical properties of these systems are still under debate. For instance we can measure negative microcanonical specific heat [47]. Moreover, phase transitions for one dimensional systems are also observed [9]. Given a vectorx (t) representing the state of the system at time t, whose size depend on the number of degrees of freedom, the evolution of a generic dynamical system may be represented by an application Φ(t,x ), called flux, that transforms the initial conditionx 0 at t = 0 into the evolved statex t: x t = Φ(t,x 0) . (1.30).
From the microscopic configurationx we can then compute macroscopic quanti-ties Θ(x ) (which can be every typical quantity like temperature or energy). A stationary equilibrium state requires that fluctuations of Θ would be negligible and, apart from integrable or strongly non-ergodic cases, this is generally true in the limit N → ∞, where small contributions cancels out thanks to the increased statistics.
Lynden-Bell approach to metastable QSS
The distribution function f of a system obeying the Vlasov equation (3.12) has a collisionless time evolution. The system undergoes an initial violent relaxation, governed by Landau damping [8], which develops through a peculiar process called filamentation due to the collisionless mixing: the fine-grained distribution function f (q, p, t) will never reach a stationary state, but instead it will be stretched and stirred into filaments mixed at smaller and smaller scales [73]. From an observer point of view, this never-ending evolution is not physically relevant, since we have a limited resolution on our observation of the fine-grain of the phase space detail. An interesting approach to this problem was initially proposed by Lynden-Bell, in the framework of galactic dynamics [8]. At the
time there were experimental observations that the radiation signals emitted by elliptic galaxies were almost regular, implying that the galaxy was in an equilibrium state. However, these observations clashed with the analytical estimates for the typical relaxation times of the galaxies due to the two-body collisional effects. It was hence proposed that the manifest regularity could be due to a sort of out-of-equilibrium stationary state [74, 75].
With reference to cosmological applications, Lynden-Bell proposed a maximum-entropy approach to determine the stationary solutions of the Vlasov equation, pioneering the theory that it is nowadays referred to as to the violent relaxation theory. The very basic idea behind his work is to introduce a cutoff to the scale of observation. He first considered the coarse grained distribution f , obtained by averaging the microscopic f (q, p, t) over a finite grid. Each element of this grid will contain a large enough portion of phase space that encompass multiple filaments. The evolution will continue at smaller scales, but the coarse-grained distribution f (q, p, t) will reach an equilibrium state very fast. In order to analytically describe this equilibrium, the key passage is to associate to f a mixing entropy S[f ], via a rigorous counting of the microscopic configu-rations that are compatible with a given macroscopic state. While following the discussion that can be found in reference [8], we shall reformulate the original derivation in order to accommodate for the extended initial condition that we will introduce in the next section. To apply the statistical mechanics machinery, we divide the phase space into a very large number of micro-cells each of volume ω˜. The micro-cells define an hyper-fine support that can be invoked to obtain an adequate representation of the fine grained function f , provided the mass of the phase element that occupies each cell is given. Consider a discretization of the original function f into n levels of phase density fJ , J = 1, …, n. Then the phase element mass is fJ ω˜ or 0. Under the effect of the equation (3.30), the distribution function f (q′, p′, t) will be mixed and filamented. Still Vlasov dynamics conserve all the Casimirs Cm[f ] = R f m dqdp, and, as a consequence, it will conserve independently the total mass of each level. Lynden-Bell suggested to group these micro-cells into coarse grained macro-cells, very small, but sufficiently large to contain several micro-cells. Let us call ν the number of micro-cells inside the macro-cell, the latter having therefore volume νω. Define niJ the number of elements with phase density fJ that populate cell i, located in (qi, pi). P Clearly i niJ = NJ , where NJ stands for number of micro-cells occupied by level J.
Lynden-Bell microcanonical solution for the HMF model
As we already seen in section 3.1, thanks to the Braun-Hepp theorem, mean-field models can be approximated in the large size limit by the Vlasov equation. It may be argued that Vlasov equation provide the correct framework to address the problem of QSS emergence. The procedure exposed in the previous section allows us to search for an equilibrium solution of a Vlasov system, by maximizing Lynden-Bell entropy. The problem is not easily solvable in a generic context, and one must resort to strict assumptions on the form of the initial condition in order to obtain an equilibrium solution of the distribution function. Lynden-Bell theory was successfully applied in the past to obtain a solution that explains the QSS regime of the HMF model [20]. Such a solution was recovered by using a peculiar initial distribution function known as “water-bag”, which is flat over a bounded domain of phase space (usually called a “level”) and zero outside of it. In the following we will discuss the multi-level water-bag class of initial conditions, which naturally extends beyond the single water-bag case study, so far explicitly considered in the literature. It is our intention to test the predictive ability of the Lynden-Bell theory within such generalized framework. The theory will be developed with reference to the general setting, including n levels. The benchmark with direct simulations will be instead limited to the two-levels case, i.e. n = 2.
The HMF out-of-equilibrium dynamics (QSS) regime in the single level case
First let us briefly review the main results already present in literature and discuss the general features of HMF’s QSS states. As we already mentioned the HMF model describes N identical particles identified by the set of coordinates θ, p, where θ ∈ [0, 2π[ is the angular coordinate over the ring and p is the conjugated momentum.
Table of contents :
Italian abstract
French abstract
Introduction
1 Systems with long-range interactions
1.1 Examples of Long-Range Systems
1.2 Extensivity and Additivity
1.2.1 Extensivity and Kac rescaling
1.2.2 Lack of Additivity
1.3 Thermodynamics
1.3.1 The microcanonical description
1.3.2 The canonical description
1.3.3 Ensemble inequivalence and convexity of the entropy functional
1.4 Out-of-equilibrium quasi-stationary states (QSS)
2 Equilibrium solution of the HMF model
2.1 The HMF model
2.2 Canonical equilibrium solution
2.3 Microcanonical equilibrium solution
3 Out-of-equilibrium thermodynamics
3.1 The Vlasov limit
3.2 Lynden-Bell approach to metastable QSS
3.3 Lynden-Bell microcanonical solution for the HMF model
3.3.1 The HMF out-of-equilibrium dynamics (QSS) regime in the single level case
3.3.2 N-levels extended solution
3.3.3 The case n = 2: theory predictions and numerical simulations.
4 Out-of-equilibrium canonical description of the HMF model
4.1 Out-of-equilibrium thermal bath: questioning temperature definition
4.2 A thermal machine working with non-Maxwellian fluid
4.2.1 Vlasov fluid in an external field
4.2.2 Constructing the thermodynamic cycle
4.2.3 Negative kinetic heat capacity and the violation of the second principle of thermodyamics
4.2.4 Reconciling theory and experience: alternatives to the violation of second law
5 Quasi-stationary states at the short-range threshold
5.1 The α-HMF model
5.2 Equilibrium phase transitions in the short-range regime
5.3 QSS lifetime
6 Self organization in a model of long-range rotators
6.1 An extended model of rotators
6.2 Equilibrium dynamics
6.3 Equilibrium distribution in the thermodynamic limit
6.4 α-HMF limit
6.5 Equilibrium phase transition
Conclusions
Bibliography
