# Time evolution of infinite anharmonic systems

@article{Lanford1977TimeEO, title={Time evolution of infinite anharmonic systems}, author={Oscar E. Lanford and Joel L. Lebowitz and Elliott H. Lieb}, journal={Journal of Statistical Physics}, year={1977}, volume={16}, pages={453-461} }

We prove the existence of a time evolution for infinite anharmonic crystals for a large class of initial configurations. When there are strong forces tying particles to their equilibrium positions then the class of permissible initial conditions can be specified explicitly; otherwise it can only be shown to have full measure with respect to the appropriate Gibbs state. Uniqueness of the time evolution is also proven under suitable assumptions on the solutions of the equations of motion.

## 62 Citations

### Dynamics of infinite classical anharmonic systems with constraints

- Mathematics
- 1987

For the infinite systems of classical anharmonic oscillators with constraints, one formulates existence and uniqueness theorems of the solution of the motion equations and of the chain of Bogolyubov…

### Hierarchical equations of evolution of an anharmonic system

- Mathematics
- 1980

We investigate the evolution of the states of a system of infinitely many anharmonic oscillators via a hierarchy of equations similar to the BBGKY one. We prove an existence theorem for the solutions…

### On the stationary measures of anharmonic systems in the presence of a small thermal noise

- Mathematics
- 1986

We consider certain small stochastic perturbations of ad-dimensional infinite system of coupled anharmonic oscillators. The evolution law is reversible in the Yaglom sense, thus Gibbs states with the…

### On the Propagation of a Perturbation in an Anharmonic System

- Physics
- 2007

We give a not trivial upper bound on the velocity of disturbances in an infinitely extended anharmonic system at thermal equilibrium. The proof is achieved by combining a control on the non…

### On the dynamics of infinite anharmonic systems

- Physics
- 1981

The nonequilibrium dynamics of a large class of classical systems of anharmonic oscillators is studied. An existence theorem for the solution of the hierarchical equations describing the evolution of…

### Time evolution of Gibbs states for an anharmonic lattice

- Mathematics
- 1979

In this paper we study the time evolution of a regular class of states of an infinite classical system of anharmonic oscillators. The conditional probabilities are investigated and an explicit form…

### Some remarks on nonequilibrium dynamics of infinite particle systems

- Mathematics
- 1984

Classical mechanics of infinitely many particles in dimensions one and two is considered, particles interacting by a superstable pair potential of finite range. The group of motion generated by…

### Convergence to stationary states for infinite harmonic systems

- Mathematics
- 1983

We study the evolution of the states for one-dimensional infinite harmonic systems, interacting through a translation invariant force of rapid decrease. We prove that for a large class of initial…

### Stationary states of random Hamiltonian systems

- Mathematics
- 1994

SummaryWe investigate the ergodic properties of Hamiltonian systems subjected to local random, energy conserving perturbations. We prove for some cases, e.g. anharmonic crystals with random nearest…

### Estimating the Lieb-Robinson Velocity for Classical Anharmonic Lattice Systems

- Physics
- 2009

We estimate the Lieb-Robsinon velocity, also known as the group velocity, for a system of harmonic oscillators and a variety of anharmonic perturbations with mainly short-range interactions. Such…

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