We report results of computer simulations of a three-dimensional lattice gas of interacting parti... more We report results of computer simulations of a three-dimensional lattice gas of interacting particles subject to a uniform external field E. The dynamics of the system is given by hoppings of particles to nearby empty sites with rates biased for jumps in the direction of E. As for the two-dimensional system we find that here too there exists a critical temperature, Tc(E ) such that for T < T~(E) the systems orders in a very anisotropic phase with striplike typical configurations parallel to the field. Tc(E ) increases with E but substantially less strongly than in two dimensions. There is a break in the slope of the saturation current at Tc(E ). Our data are consistent with the critical exponent fl being mean field.
We consider a low density gas and its description by means of the Boltzmann equation. This topic ... more We consider a low density gas and its description by means of the Boltzmann equation. This topic will be covered at considerable detail, simply because (at least at present) it is one of the most beautiful examples for the emergence of an autonomous large scale dynamics which can be studied with mathematical rigor and at great depth.
Time Correlations and Fluctuations
Springer eBooks, 1991
We developed a simple picture of the macroscopic, large scale description of fluids: on a suffici... more We developed a simple picture of the macroscopic, large scale description of fluids: on a sufficiently coarse space-time scale the locally conserved fields are deterministic and are governed by the hydrodynamic equations. A very natural idea is to refine now the scale of observation such as to discern the fluctuating deviations from the deterministic law. We consider only the case of thermal equilibrium. The extension to local equilibrium states is then a small further step. We will explain the theory for the case of fluctuating hydrodynamics. But in fact the method to be developed is very general and in particular applicable to the various kinetic equations.
Point scatterers are placed on the real line such that the distances between scatterers are indep... more Point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables (stationary renewal process). For a fixed configuration of scatterers a particle performs the following random walk: The particle starts at the point x with velocity v, Iv[ = 1. In between scatterers the particle moves freely. At a scatterer the particle is either transmitted or reflected, both with probability 1/2. For given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations. If this variable is averaged over the distribution of scatterers, it decays not faster than t -3/2.
We study the stationary nonequilibrium states of the van Beijeren/Schulman model of a driven latt... more We study the stationary nonequilibrium states of the van Beijeren/Schulman model of a driven lattice gas in two dimensions. In this model, jumps are much faster in the direction of the driving force than orthogonal to it. Van Kampen's O-expansion provides a suitable description of the model in the high-temperature region and specifies the critical temperature and the spinodal curve. We find the rate dependence of Tc and show that independently of the jump rates the critical exponents of the transition are classical, except for anomalous energy fluctuations. We then study the stationary solution of the deterministic equations (zeroth-order Q-expansion). They can be obtained as trajectories of a dissipative dynamical system with a three-dimensional phase space. Within a certain temperature range below To, these equations have a kink solution whose asymptotic densities we identify with those of phase coexistence. They appear to coincide with the results of the "Maxwell construction." This provides a dynamical justification for the use of this construction in this nonequilibrium model. The relation of the Freidlin-Wentzell theory of small random perturbations of dynamical systems to the steady-state distribution below To is discussed.
The proper form of the generator in the weak coupling limit
Zeitschrift für Physik, Dec 1, 1979
... with PW=(TrbW)®CO p for an arbitrary density ma-trix W of the joint system• Tt is a non-Marko... more ... with PW=(TrbW)®CO p for an arbitrary density ma-trix W of the joint system• Tt is a non-Markovian evolution as can be seen from the memory term in (2). For small 2 (weak coupling) Tt a can be approximated by a Markovian evolution of the form TtZ p = e (-ig~+ ,t2K)t D. ...
Entropy production for quantum dynamical semigroups
Journal of Mathematical Physics, May 1, 1978
In analogy to the phenomenological theory of irreversible thermodynamics we define the entropy pr... more In analogy to the phenomenological theory of irreversible thermodynamics we define the entropy production for an arbitrary quantum dynamical semigroup with a stationary state. We prove that the entropy production is convex and positive and that the entropy production is a measure of dissipativity of the semigroup. The entropy production is used to prove the approach to equilibrium and to classify the stationary states of semigroups arising in the weak coupling limit.
Dynamical processes in macroscopic systems are often approximately described by kinetic and hydro... more Dynamical processes in macroscopic systems are often approximately described by kinetic and hydrodynamic equations. One of the central problems in nonequilibrium statistical mechanics is to underst'and the approximate validity of these equations starting from a microscopic model. %'e discuss a variety of classical as well as quantum-mechanical models for which kinetic equations can be derived rigorously. The probabilistic nature of the problem is emphasized: The approximation of, the microscopic dynamics by either a kinetic or a hydrodynamic equation can be understood as the approximation of a non-Markovian stochastic process by a Markovian process. A. Microscopic dynamics and kinetic equations B. Some classical continuous models A. The Lorentz gas 1. The weak coupling limit 2. Convergence of the weak coupling limit 3. The low-density (Grad) limit 4. Convergence of the low-density limit 5. The mean-field limit 6. Convergence of the mean-field limit B. The Rayleigh gas 1. The Brownian motion limit 2. Convergence of the low-density limit of a tag- ged particle in an ideal fluid 3. Convergence of the low-density limit for a tagged sphere in a hard sphere fluid 4. One-dimensional hard rod systems C. The hydrodynamic limit 1. The diffusion approximation for the Lorentz gas with a periodic configuration of scatterers The physical meaning of the weak coupling and low-density limits E. The problem of the existence of transport coefficients F. Fluctuations III. Interacting Particle Systems A. Nonlinear Markov processes B. The Landau equation C. The Boltzmann equation 1. Convergence of the solution of the BBGKY hier- archy to the solution of the Boltzmann hier achy 2. Convergence to a nonlinear Markov process D. The Vlasov equation E. Fluctuations IV. Lattice Models A. The harmonic lattice B. The anharmonic lattice V. Quantum-Mechanical Models A. Some general remarks B. Quantum-dynamical semigroups C. System coupled to a thermal reservoir 1. The weak coupling limit 2. The singular coupling limit 3. N-level system in a low-density gas 4. The polaron D. Effective Hamiltonians E. The mean-field limit for interacting quantum sys- tems Reference s 569 569
Sample-to-sample fluctuations in the conductivity of a disordered medium
Journal of Statistical Physics, Dec 1, 1992
ABSTRACT
Tracer dynamics in Dyson's model of interacting Brownian particles
Journal of Statistical Physics, Jun 1, 1987
ABSTRACT
Equilibrium states for mean field models
Journal of Mathematical Physics, Feb 1, 1980
We rigorously characterize the KMS and the limiting Gibbs states for mean field models. As an app... more We rigorously characterize the KMS and the limiting Gibbs states for mean field models. As an application we prove the convergence of the Gibbs states for the Dicke Maser model in the infinite volume limit.
Quantum tunneling with dissipation and the Ising model over ?
Journal of Statistical Physics, Nov 1, 1985
ABSTRACT
Long range correlations for stochastic lattice gases in a non-equilibrium steady state
Journal of physics, Dec 21, 1983
ABSTRACT
Large Scale Stochastic Dynamics
Oberwolfach Reports, 2013
... 16 Kay Kirkpatrick (joint with Sourav Chatterjee) Bose-Einstein condensation and probabilisti... more ... 16 Kay Kirkpatrick (joint with Sourav Chatterjee) Bose-Einstein condensation and probabilistic methods for nonlinear Schrödinger equations ..... 18 Jochen Voss (joint with Martin Hairer) Approximations to the Stochastic Burgers Equation ..... 22 ...
Microscopic Time Reversibility and the Boltzmann Equation
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Papers by Herbert Spohn