1,720,997 research outputs found
Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation
Full text linkWe suggest a new derivation of a wave kinetic equation for the spectrum of the weakly nonlinear Schrödinger equation with stochastic forcing. The kinetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size of the nonlinearity and of the forcing to zero, while the time is correspondingly rescaled; then, the size of the box is sent to infinity (with a suitable rescaling of the solution). We report here the results of the first limiting procedure, analysed with full rigour in Kuksin and Maiocchi (0000), and show how the second limit leads to a kinetic equation for the spectrum, if some further hypotheses (commonly employed in the weak turbulence theory) are accepted. Finally we show how to derive from these equations the Kolmogorov-Zakharov spectra
Exponentially long stability times for a nonlinear lattice in the thermodynamic limit
No full textIn this paper, we construct an adiabatic invariant for a large 1-d lattice of particles, which is the so called Klein Gordon lattice. The time evolution of such a quantity is bounded by a stretched exponential as the perturbation parameters tend to zero. At variance with the results available in the literature, our result holds uniformly in the thermodynamic limit. The proof consists of two steps: first, one uses techniques of Hamiltonian perturbation theory to construct a formal adiabatic invariant; second, one uses probabilistic methods to show that, with large probability, the adiabatic invariant is approximately constant. As a corollary, we can give a bound from below to the relaxation time for the considered system, through estimates on the autocorrelation of the adiabatic invariant
Replacement of the Lorentz law for the shape of the spectral lines in the infrared region
Full text linkWe propose a new phenomenological law for the shape of the spectral lines in the infrared, which accounts for the exponential decay of the extinction coefficient in the high-frequency region, observed in many spectra. We apply this law to the measured infrared spectra of LiF, NaCl, and MgF2, finding good agreement over a wide range of frequencies
The limit of small Rossby numbers for randomly forced quasi-geostrophic equation on β-plane
Full text linkWe consider the 2d quasigeostrophic equation on the β-plane for the stream function ψ, with dissipation and a random force: (+ K)ψt - pJ(ψψ) - βψx = lang;random force〉 ? k2ψ +ψ. (∗) Here ψ = ψ(t, x, y), x εR/2πLZ, y ε R/2πZ. For typical values of the horizontal period L we prove that the law of the action-vector of a solution for (∗) (formed by the halves of the squared norms of its complex Fourier coefficients) converges, as β → ∞, to the law of an action-vector for solution of an auxiliary effective equation, and the stationary distribution of the action-vector for solutions of (∗) converges to that of the effective equation. Moreover, this convergence is uniform in κ ∈ (0, 1]. The effective equation is an infinite system of stochastic equations which splits into invariant subsystems of complex dimension 3; each of these subsystems is an integrable hamiltonian system, coupled with a Langevin thermostat. Under the iterated limits limL=ρ→∞limβ→∞ and limκ→0limβ→∞ we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies
PERTURBATION THEORY AT THE THERMODYNAMIC LIMIT
Full text linkLa presente tesi propone un'estensione della teoria delle perturbazioni Hamiltoniana al limite termodinamico (ossia, per sistemi con un numero infinito di gradi di libertà e una temperatura, o energia specifica, finita), nello spirito della teoria ergodica (cioè, in presenza di una misura invariante). Per un sistema concreto, il modello Φ4 discreto, si ottiene una versione debole del teorema di Nehoroscev. Il risultato è che, nel limite termodinamico, esiste almeno un'osservabile, indipendente dall'energia, la cui funzione di
autocorrelazione temporale rimanga significativamente discosta da zero
fino a tempi esponenzialmente lunghi nei parametri perturbativi. La tesi è completata dalla discussione di argomenti correlati, ossia le proprietà analitiche generali delle funzioni di autocorrelazione temporale e un'applicazione euristica della teoria perturbativa al problema del limite di densità nei plasmi magnetizzati.The present thesis provides an extension of Hamiltonian perturbation
theory to the thermodynamic limit (i.e., for systems with an infinite
number of degrees of freedom and a finite temperature or specific energy), in the spirit of ergodic theory (i.e., in the presence of an invariant measure). For a concrete model, which is the discrete Φ4 model, a weaker version of classical Nekhoroshev theorem is obtained. The result is that, at the thermodynamic limit, there exists at least one observable, independent of energy, such that its
time–autocorrelation function does not relax to zero up to times
exponentially long in the perturbation parameters. In the thesis, further related subjects are discussed, namely, analytical properties of generic time-autocorrelation functions and a heuristic application of perturbation theory to the problem of the density limit in magnetized plasmas
Relaxation times for Hamiltonian systems
Full text linkUsually, the relaxation times of a gas are estimated in the frame of the Boltzmann equation. In this paper, instead, we deal with the relaxation problem in the frame of the dynamical theory of Hamilto-nian systems, in which the definition itself of a relaxation time is an open question. We introduce a lower bound for the relaxation time, and give a general theorem for estimating it. Then we give an appli-cation to a concrete model of an interacting gas, in which the lower bound turns out to be of the order of magnitude of the relaxation times observed in dilute gases.
An Averaging Theorem for FPU in the Thermodynamic Limit
Full text linkConsider an FPU chain composed of N≫1 particles, and endow the phase space with the Gibbs measure corresponding to a small temperature β^(-1). Given a fixed K , we construct K packets of normal modes whose energies are adiabatic invariants (i.e., are approximately constant for times of order β^(1−a), a>0 ) for initial data in a set of large measure. Furthermore, the time autocorrelation function of the energy of each packet does not decay significantly for times of order β. The restrictions on the shape of the packets are very mild. All estimates are uniform in the number N of particles and thus hold in the thermodynamic limit N→∞ , β>0
Statistical thermodynamics for metaequilibrium or metastable states
Full text linkWe show how statistical thermodynamics can be formulated in situations of metaequilibrium or metastability (as in the cases of supercooled liquids or of glasses respectively). By analogy with phenomenological thermodynamics, the primary quantities considered are the heat Q absorbed and the work W performed by the system of interest. These are defined through the energy exchanges which occur when the system is put in contact with a thermostat and with a barostat, the whole system being dealt with as a global Hamiltonian dynamical system. The coefficients of the fundamental form (Formula presented.) turn out to have such expressions that the closure of the form is manifest: this gives the first principle. A further step is performed by making use of time reversibility. This provides new expressions for the coefficients, such that the second principle in the form of Clausius is also manifest. Such coefficients are expressed in terms of time-autocorrelations of suitable dynamical variables, in a way analogous to that of fluctuation dissipation theory for equilibrium states. All these results are independent of the ergodicity properties of the global dynamical system
A series expansion for the time autocorrelation of dynamical variables
No full textWe present here a general iterative formula which gives a (formal) series expansion for the time autocorrelation of smooth dynamical variables, for all Hamiltonian systems endowed with an invariant measure. We add some criteria, theoretical in nature, which enable one to decide whether the decay of the correlations is exponentially fast or not. One of these criteria is implemented numerically for the case of the Fermi-Pasta-Ulam system, and we find indications which might suggest a sub-exponential decay of the time autocorrelation of a relevant dynamical variable
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