By Charles-Edouard Pfister
Summary. major effects are offered during this paper: Isakov’s Theorem
about the impossibility of an analytic continuation of the strain at
a first order part transition aspect and Friedli’s paintings concerning the restoration
of the analytic continuation of the strain within the van der Waals restrict at
such a transition element. An exposition of Pirogov-Sinai thought for models
with periodic ground-states is given in part three, which are read
independently.
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Extra info for On the nature of isotherms at first order phase transitions for classical lattice models
Sample text
18); the size of Un is proportional to θ . 17)). By taking ε smaller, one can choose θ larger. θ is chosen so that θ > θ and ρ(1 − θ ) > ε. 1). 13) Stability of contours is proved inductively as follows. Let β0 be large enough and assume that β ≥ β0 and that for q = 1, 2 the weights ωn−1 (Γq ) are τ (β)-stable, as well as d ωn−1 (Γq ) ≤ e−τ dz (β)|Γq | . 14) and (m = q) (Intm Γq ) − βgqn−1 V (Γq )| ≤ δ C0 |Γq | ln Θn−1 q q n−1 ln Θn−1 V (Γq )| ≤ δ C0 |Γq | . m (Intm Γ ) − βgm Let Γq be a contour with V (Γq ) = n.
If β is sufficiently large, then for all n ≥ 1 d ∗ µ (0; β) = 0 , dν n and 2δ d2 ∗ µn (ν; β) ≤ 2 dν ∆ − 2δ 2δ ∆ − 2δ 2 + 2δ +1 . 27) The first formula is a consequence of the reality of the constrained pressures on the real axis, which implies that ν → µ∗n (ν; β) is an even function. The next proposition gives an estimate of |µ∗ (ν; β) − µ∗n (ν; β)| and improved estimates for the derivative of the weight of a contour (see [FrPf1]). 2. 1, there exist β0 ∈ R+ and a constant D so that the following holds for all β ≥ β0 .
However, the truncated pressures cannot be analytic, so they are inappropriate in the present context. The construction of the phase diagram for the complex variable z is done as follows. It is an iterative procedure. 8)). 12) and lim bqn = 0 , n q = 1, 2 . The constrained pressures pn−1 of order n − 1, q = 1, 2, are analytic on q Un−1 := {z ∈ C : Rez ∈ Un−1 (Imz; β)} . On the nature of isotherms at first order phase transitions 33 So, for each interval Un−1 (ν; β) the constrained pressures pn−1 are wellq defined and the point µ∗n (ν; β) is the (unique) solution of the equation Re pn−1 (µ∗n (ν; β) + iν) − pn−1 (µ∗n (ν; β) + iν) = 0 .