# Phase transitions, approach to equilibrium, and structural stability

• Gérard G. Emch
Invited Lectures D. Symmetry Breaking in Statistical Mechanics and Field Theory
Part of the Lecture Notes in Physics book series (LNP, volume 79)

## Keywords

Thermodynamical Limit Thermal Bath Thermodynamical Phasis Equilibrium Statistical Mechanics Hyperbolic Point
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## Notes and References

1. [1]
For a review see for instance: G.G. Emch, Lecture Notes, Tübingen International Summer School on Groups and Many-Body Physics, July 11–16, 1977, to be published.Google Scholar
2. [2]
For a simple model, see for instance: G.G. Emch, Lecture Notes, Schladming Internationale Universitätswochen, Feb. 16–27, 1976, published as Acta Physica Austriaca, Suppl. X V (1976) 79–131.Google Scholar
3. [3]
The KMS condition is called so after R. Kubo, J. Phys. Soc. Japan 12 (1957) 570–586, and P.C. Martin and J. Schwinger, Phys. Rev. 115 (1959) 1342–1373. Its importance in the algebraic approach to statistical mechanics followed here has been recognized by R. Haag, N. Hugenholtz and M. Winnink, Commun. Math. Phys. 5 (1967) 215–236; see also [4] below.Google Scholar
4. [4]
W. Pusz and S. L. Woronowicz, in Rome International Conference on the Mathematical Problems in Theoretical Physics, June 6–15, 1977, to be published; and Marseille International Colloquium on Algebras of Operators and their Applications to Mathematical Physics, June 20–24, 1977, to be published. See also, in both of these proceedings, the reviews by A. Kastler, R. Haag, and by A. Verbeure.Google Scholar
5. [5]
See for instance, C.N. Yang and T.D. Lee, Phys. Rev. 87 (1952) 404–409 and 410–419.Google Scholar
6. [6]
Z. Takeda, Tohoku Math. Journ. 7 (1955) 68–86.Google Scholar
7. [7]
This is the famous GNS construction, named after I. Gelfand and M.A. Naimark, Mat.Sborn.N.S. 12 54 (1943) 197–217, and I.E. Segal, Ann. Math. 48 (1947) 930–948. Its importance is central in the algebraic approach followed here; for more details, see G.G. Emch Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.Google Scholar
8. [8]
G.G. Emch and H.J.F. Knops, Journ.Math.Phys. 11 (1970) 3008–3018.Google Scholar
9. [9]
M. Kac, in Brandeis Summer Institute, 1966 Lecture Notes, M. Chretien, E.P. Gross and S. Deser, Eds., Gordon & Breach, New York, 1968.Google Scholar
10. [10]
R. Brout, Phase Transitions, Benjamin, New York 1965; also E. Lieb Journ. Math. Phys. 7 (1966) 1016–1024.Google Scholar
11. [11]
As it was not central to the questions discussed in this lecture, we did not want to mention in the main text one nevertheless interesting feature of ref.[8]; this paper indeed illustrates, and gets over the fact that the evolution cannot always be uniquely defined, in the thermodynamical limit, as an automorphism of the C*-algebra $$\mathfrak{A}$$ of the quasi-local observables; a proper definition often involves, as it does in this model, careful consideration of the von Neumann algebra πθ($$\mathfrak{A}$$)″ generated by the representation πθ($$\mathfrak{A}$$) constructed from a properly chosen state θ. See also D.W. Robinson, preprint, Bielefeld, 1977. In the present model, this phenomenon is due to the long-range nature of the forces involved, and both its physical background and its analytic expression can be physically understood. In continuous, non-relativistic systems, this sort of difficulty also occur, although for a different reason; this has been already pointed out by D.A. Dubin and G. Sewell, Journ.Math.Phys. 11 (1970) 2990–2998. Models have also been constructed for the singular dynamics of classical, infinite mechanical systems; see for instance O.E. Lanford, Lecture Notes, Battelle Rencontres in Mathematics and Physics. Still another kind of continuous, classical and quantum, mean-free field models have recently been constructed by G. Battle, Ph.D. dissertation, Duke University, 1977; there again the time-evolution has to be defined with much care when the thermodynamical limit is carried out.Google Scholar
12. [12]
Ph.A. Martin, Journ. Stat. Phys. 16 (1977) 149–168.Google Scholar
13. [13]
R.J. Glauber, Journ.Math.Phys. 4 (1963) 294–307.Google Scholar
14. [14]
L. van Hove, Physica 21 (1955) 517–540; 23 (1957) 441–480. See E.B. Davies, Quantum Theory of Open Systems, Academic Press, 1976.Google Scholar
15. [15]
Amongst the other things also proven in [12], it should be worth pointing out here again that the approach to equilibrium being asymptotically exponential, the relaxation time so obtained diverges as | T − Tc|−1, thus giving rise to the expected “critical slowing down” characteristic of the bifurcation point βc J = 1; moreover, the analysis is there completed by a discussion of finite volume effects and fluctuations.Google Scholar
16. [16]
M. Giuffre, Ph.D. dissertation, University of Rochester, 1977.Google Scholar
17. [17]
The model can [16] further be generalized by letting the lattice constant tend to zero, thus providing a continuous model for which the system of ordinary differential equations governing {mk(τ)|k=1,..., K} now becomes a partial-differentio integral equation for the “coarse-grained” observable m(x,τ). Not surprisingly, but still gratifyingly enough, the latter exhibits the same kind of approach to equilibrium in the presence of phase transitions as that shown by the discrete models described in this lecture.Google Scholar