 
Summary: Nuclear Physics B 587 (2000) 403418
www.elsevier.nl/locate/npe
On thermalization in classical scalar field theory
Gert Aarts , Gian Franco Bonini, Christof Wetterich
Institut für theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
Received 29 March 2000; revised 21 May 2000; accepted 11 July 2000
Abstract
Thermalization of classical fields is investigated in a 4 scalar field theory in 1 + 1 dimensions,
discretized on a lattice. We numerically integrate the classical equations of motion using initial con
ditions sampled from various nonequilibrium probability distributions. Timedependent expectation
values of observables constructed from the canonical momentum are compared with thermal ones. It
is found that a closed system, evolving from one initial condition, thermalizes to high precision in the
thermodynamic limit, in a timeaveraged sense. For ensembles consisting of many members with the
same energy, we find that expectation values become stationary  and equal to the thermal values
 in the limit of infinitely many members. Initial ensembles with a nonzero (noncanonical) spread
in the energy density or other conserved quantities evolve to noncanonical stationary ensembles. In
the case of a narrow spread, asymptotic values of primary observables are only mildly affected. In
contrast, fluctuations and connected correlation functions will differ substantially from the canonical
values. This raises doubts on the use of a straightforward expansion in terms of 1PIvertex functions
to study thermalization. © 2000 Elsevier Science B.V. All rights reserved.
