# Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation

## Abstract

We here discuss the emergence of quasistationary states (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian mean-field model, numerical simulations are performed based on both the original N-body setting and the continuum Vlasov model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously demonstrates that the Vlasov-wave system provides the correct framework to address the study of QSS. Further, analytical calculations based on Lynden-Bell's theory of violent relaxation are shown to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical solutions are shown to be affected by small scale fluctuations, a finding that points to the need for novel schemes able to account for particle correlations.

- Authors:

- Dipartimento di Energetica and CSDC, Universita di Firenze, and INFN, via S. Marta, 3, 50139 Florence (Italy)
- Dipartimento di Fisica 'E.Fermi' and CNISM, Universita di Pisa, Largo Pontecorvo, 3 56127 Pisa (Italy)
- (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 20951225

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review Letters; Journal Volume: 98; Journal Issue: 15; Other Information: DOI: 10.1103/PhysRevLett.98.150602; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOLTZMANN-VLASOV EQUATION; COMPARATIVE EVALUATIONS; CONVERGENCE; CORRELATIONS; FLUCTUATIONS; HAMILTONIANS; INTERACTION RANGE; MATHEMATICAL SPACE; MEAN-FIELD THEORY; NUMERICAL SOLUTION; RELAXATION; SIMULATION

### Citation Formats

```
Antoniazzi, Andrea, Ruffo, Stefano, Califano, Francesco, Fanelli, Duccio, and Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL.
```*Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVLETT.98.150602.

```
Antoniazzi, Andrea, Ruffo, Stefano, Califano, Francesco, Fanelli, Duccio, & Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL.
```*Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation*. United States. doi:10.1103/PHYSREVLETT.98.150602.

```
Antoniazzi, Andrea, Ruffo, Stefano, Califano, Francesco, Fanelli, Duccio, and Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL. Fri .
"Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation". United States.
doi:10.1103/PHYSREVLETT.98.150602.
```

```
@article{osti_20951225,
```

title = {Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation},

author = {Antoniazzi, Andrea and Ruffo, Stefano and Califano, Francesco and Fanelli, Duccio and Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL},

abstractNote = {We here discuss the emergence of quasistationary states (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian mean-field model, numerical simulations are performed based on both the original N-body setting and the continuum Vlasov model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously demonstrates that the Vlasov-wave system provides the correct framework to address the study of QSS. Further, analytical calculations based on Lynden-Bell's theory of violent relaxation are shown to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical solutions are shown to be affected by small scale fluctuations, a finding that points to the need for novel schemes able to account for particle correlations.},

doi = {10.1103/PHYSREVLETT.98.150602},

journal = {Physical Review Letters},

number = 15,

volume = 98,

place = {United States},

year = {Fri Apr 13 00:00:00 EDT 2007},

month = {Fri Apr 13 00:00:00 EDT 2007}

}