Knauf, A.: On a Ferromagnetic Spin Chain. Part II: Thermodynamic Limit. J. Math. Phys. 35, 228--236 (1994)  Home/Search   Document Not in Database   Summary   Related Articles   Check

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Knauf, A.: On a Ferromagnetic Spin Chain. Part II: Thermodynamic Limit. J. Math. Phys. 35, 228--236 (1994)

....more is known about ferromagnetic spin systems than about non ferromagnetic ones (GKS inequalities, Lee Yang theorem, etc. So there is some hope that one may prove new results on Riemann s zeta function by applying ideas from statistical mechanics. The interested reader should consult [6] and [7] for more information. See also the related work [3] of Bost and Connes on Hecke C algebras and the Riemann zeta function. Among the thermodynamical quantities describing a spin chain of length k there are the density Fk of the free energy, the expectation value Uk of 4 the energy density and ....

.... (h G l (oe) Gammafi for h G l (oe) exp(H G l (oe) Moreover, h G l (oe) h C l 1 (oe; 1) h C l (oe) h C l (1 Gamma oe) 6) h G l 1 (oe; 0) 2h C l (oe) h C l (1 Gamma oe) 7) and h G l 1 (oe; 1) h C l (oe) 2h C l (1 Gamma oe) 8) In [7] we proved the existence of the thermodynamic limit F (fi) lim k 1 F k (fi) By E 4 concavity E 4 of E 4 fi E 5 Delta E 4 F E 5 (fi) E 4 the E 4 thermodynamic E 4 limit E 4 U E 5 (fi) E 4 of E 4 U E 5 k E 9 (fi) E 4 exists E 4 for E 4 almost E 15 all E 15 ....

Knauf, A.: On a Ferromagnetic Spin Chain. Part II: Thermodynamic Limit. SFB 288, Preprint No. 19

.... has a thermodynamic limit in the sense 0 (j k 1 (0; t) Gamma j k (t) Delta 2 s C Delta 2 Gammad (t 2 G k n f0g) 5 For fi 0 the thermodynamic limit F (fi) lim k 1 F k (fi) with F k (fi) Gamma 1 fi Delta k ln (Z k (fi) 28) of the free energy per spin exists [25]. 4. The interaction is ferromagnetic, that is, j k (t) 0 (t 2 G k n f0g) 29) 5. The effective interaction A k (l; r) X t 0 2G r Gammal Gamma1 j k (0; 0; 1; t l 1 ; t r Gamma1 ; 1; 0; 0) between spins at positions l and r decays quadratically with their ....

Knauf, A.: On a Ferromagnetic Spin Chain. Part II: Thermodynamic Limit. J. Math. Phys. 35, 228--236 (1994)

.... H G k : G k R, H G k : ln(h G k ) the values h G k (oe) h C k 1 (oe; 1) 3) being the new elements appearing in each row of Fig 1b) They have the symmetries H G k (1 Gamma oe 1 ; 1 Gamma oe k ) H G k (oe 1 ; oe k ) H G k (oe k ; oe 1 ) In [10] the existence of the thermodynamic limit F (fi) lim k 1 F C k (fi) of the free energy F C k (fi) Gamma(fi Delta k) Gamma1 ln(Z C k (fi) was shown (due to the lack of strict translation invariance of the interaction standard estimates cannot be applied here) In [9] thermodynamic ....

....and t are non zero, then (19) simply states that j G k l (s; t) 0, i.e. the ferromagnetic property. If s or t is zero, then all connected multi polymers contributing to the right hand side of (19) contribute to l.h.s. too (after a translation by k for the t terms) 2. Remark. Proposition 3 of [10] says that for non zero t 2 G n j C n 1 (t; 0) j C n (t) j C n 1 (0; t) This implies in particular that the canonical interaction coefficients neither increase nor decrease under concatenation. 5 Estimates for the Free Energy In this section we will show that the limit free energy is ....

Knauf, A.: On a Ferromagnetic Spin Chain. Part II: Thermodynamic Limit. Journal of Mathematical Physics 35, 228--236 (1994)

.... interaction has a thermodynamic limit in the sense 0 (j k 1 (0; t) Gamma j k (t) Delta 2 s C Delta 2 Gammad (t 2 G k n f0g) For fi 0 the thermodynamic limit F (fi) lim k 1 F k (fi) with F k (fi) Gamma 1 fi Delta k ln (Z k (fi) 7) of the free energy per spin exists [7]. ffl The interaction is ferromagnetic, that is, j k (t) 0 (t 2 G k n f0g) 8) This is in accordance with earlier speculations (see Ruelle [14] Note, however, that the system is not of Ising type, since multi body interactions are present. ffl The effective interaction A k (l; r) ....

Knauf, A.: On a Ferromagnetic Spin Chain. Part II: Thermodynamic Limit. J. Math. Phys. 35, 228--236 (1994)

.... Gamma l) 2 . It turned out to be useful to introduce the grand canonical energy functions H G k : G k R, H G k (oe) H C k 1 (oe; 1) partly because of their symmetries H G k (1 Gamma oe 1 ; 1 Gamma oe k ) H G k (oe 1 ; oe k ) H G k (oe k ; oe 1 ) In [12] the existence of the thermodynamic limit F (fi) lim k 1 F k (fi) of the free energy F k (fi) Gamma(fi Delta k) Gamma1 ln(Z k (fi) was shown (due to the lack of strict translation invariance of the interaction standard estimates cannot be applied here) In [11] thermodynamic ....

Knauf, A.: On a Ferromagnetic Spin Chain. Part II: Thermodynamic Limit. Journal of Mathematical Physics 35, 228--236 (1994)

....paper the system was shown to exhibit a phase transition at s = 1 with type I states (resp. type III) at low (resp. high) temperature. In the second approach mentioned the quotient i(s Gamma1) i(s) is interpreted as a partition function at the inverse temperature s, see Cvitanovi c [8] Knauf [17, 18, 19, 20], Contucci and Knauf [7] and Guerra and Knauf [12] It was shown that the partition function described a spin chain with asymptotically translation invariant long range ferromagnetic interaction (the number theoretical spin chain) The point s = 2 corresponds to a phase transition where ....

Knauf, A.: On a Ferromagnetic Spin Chain. Part II: Thermodynamic Limit. Journal of Mathematical Physics 35, 228--236 (1994)

....(k 1) st row. Notice that these sequences of integers coincide with the denominators of the modified Farey sequence. For n k 1 the multiplicity k (n) jfoe 2 G k j h k (oe) ngj of n equals (n) This implies (4) since Z k (s) X n2N k (n)n Gammas : As has been worked out in [10, 11, 12], in the articles [5, 6] with Contucci, and the one with Guerra [9] this Number Theoretical Spin Chain has the properties of typical systems considered in statistical mechanics. It has exactly one phase transition, at s = 2. But from the point of view of number theory the most important point ....

Knauf, A.: On a Ferromagnetic Spin Chain. Part II: Thermodynamic Limit. J. Math. Phys. 35, 228--236 (1994)

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