Download Random Time Evolution of Infinite Particle Systems Frank Spitzer

Proceedings of the International Congress of Mathematicians
Vancouver, 1974
Random Time Evolution of Infinite Particle Systems
Frank Spitzer
This is an outline of progress in thisfieldduring the last four-year period, which
was largely the result of major developments in equilibrium statistical mechanics
during the preceding four-year period. A detailed version of this outline will appear
in [27].
1. In 1968 ([2], [3]), Dobru§in introduced the notions of an infinite Gibbs
state (IGS) and of a Markov random field (MRF), and showed that a MRF is a
natural generalization of a stationary Markov process. Somewhat later [(1], [23],
[22], [6]), it was realized that every MRF is an IGS with nearest neighbor potential,
and vice versa.
2. In 1969, Lanford and Ruelle [15] independently defined IGS and proved a
variational characterization analogous to the classical one for finite Gibbs states.
3. The year 1970 saw the first time evolutions which have a given MRF or IGS
as equilibrium state. These were on one hand birth and death, or spin flip, evolutions [4], already studied in one dimension by Glauber [5]. On the other hand, time
evolutions which preserve the number of particles were proposed in [24].
4. Rigorous existence proofs of such time evolutions as Markovian Feller semigroups Tt91 ^ 0, acting on the continuous function space C{0)9 where Q = {0,1 }z"
is the configuration space, were given during 1971 and 1972 by Dobrusin [4],
Harris [7], Holley [9], Liggett [16]. Such a semigroup is ergodic if there exists a
probability measure v o n ö such that p Tt => v for all//. Then v is the unique equilibrium state.
5. Dobruâin [4] showed that a birth and death evolution with strictly positive
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rates is ergodic if the interaction (dependence of rates on neighboring sites) is weak.
The case of zero birth rate when all neighboring sites are vacant was studied by
Harris [8], who proved ergodicity when the death rates are sufficiently small, nonergodicity in the contrary case. Holley [10], [11], [12], [13] obtained deep and detailed results when the birth rates are such that the equilibrium states are the
equilibrium states for the Ising model. Thus he shows that Tt is nonergodic exactly
when the corresponding Ising model exhibits phase transition. The reason one
obtains a complete theory in this case was clarified [25], [21] by showing that Tt
acts time reversibly in an equilibrium state exactly when the equilibrium state is a
MRF (equilibrum state for the Ising model). A surprising irreversible case (the
voter model) was recently (1974) completely analyzed by Holley and Liggett [14].
6. The jump processes with constant speed and exclusion have been completely
analyzed [17], [26], [18] in the case when the transition function P{x9y) is symmetric.
When P is recurrent or a random walk transition function, then the only equilibrium states are the exchangeable measures (convex combinations of Bernoulli
product measures). When P is unsymmetric, recent work of Liggett [19], [20]
suggests interesting conjectures.
Bibliography
1. M. B. Averintsev (1970), On a method of describing discrete parameter randomfields,Problemy
Peredaci Informacii 6,100-109. (Russian)
2. R. L. Dobrusin (1968), Description of a randomfieldby means of conditional probabilities and
conditions for its regularity, Teor. Verojatnost. i Primenen. 13,201-229 = Theor. Probability Appi.
13, 197-224. MR 37 #6989.
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(1968), Gibbsian randomfieldsfor lattice systems withpairwise interactions, Funkcional.
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(1971), Markov processes with a large number of locally interacting components,
Problemy Peredaöi Informacii 7,70-87. (Russian)
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6. R. G. Grimmett (1973), A theorem about randomfields,Bull. London Math. Soc. 5,81-84.
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(1974), Contact interactions on a lattice (to appear).
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(1971), Free energy in a Markovian model of a lattice spin system, Comm. Math. Phys.
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(1974), Some remarks on the FKG inequality, Comm. Math. Phys. 36, 227-231.
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(1974), Recent results on the stochastic Ising model, Rocky Mountain J. Math. 4,
479-496.
14. R. Holley and T. Liggett (1974), Ergodic theorems for weakly interacting infinite systems
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15. O. E. Lanford, and D. Ruelle (1969), Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys. 13,194-215. MR 41 # 1343.
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Soc. 165,471-481. MR 46 # 8328.
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(1973), A characterization of the invariant measures for an infinite particle system with
interactions, Trans. Amer. Math. Soc. 179,433-453.
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(1974), A characterization of the invariant measures for an infinite particle system with
interactions. II, Trans, Amer. Math. Soc. 198,207-213.
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(1974), Convergence to total occupancy in an infinite particle system with interactions
(to appear).
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(1974), Ergodic theorems for the asymmetric simple exclusion process, Trans. Amer.
Math. Soc. (to appear).
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appear).
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142-154. MR 43 #2773.
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