By Elliott H. Lieb
''why are nontrivial versions of actual platforms so usually soluble in a single measurement yet no longer in 3 dimensions''
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Extra resources for Mathematical Physics in One Dimension. Exactly Soluble Models of Interacting Particles
Sample text
RN) XexpZ-ß{VN+Vw)lhdRi. (2) The most general conveniently applicable method for evaluating multiple integrals of this type is that at7 tributable to Giirsey. This is restricted to potentials that may be expressed as sums of pair potentials be«T. Nagamiya, Proc. -Math. Soc. Japan 22,705 (1940); 1034 (1940). • J . Frenkel, Kinetic Theory of Liquids (Oxford University Press, New York, 1940), p. 126. • G. Rushbrookc and H. Ursel, Proc. Cambridge Phil. Soc. 44, 2637 (1948). F. Gürscy, Proc.
DXl . . V(Xn — Xo)]/kT] dxn. Setting Xl — XO = Τι, Xo — Xl = To, . . , Xn — Xn-l = Tn and observing that r2, . . , r„) = 1, d(xi, X2, . . Γ2 . . ,Γη \-[U(n) + U(r2) + . . + U{rn) >0 + pin + r 2 + . . Γ2 . . ,Γη >0 / e x p \-[U(n) + prJ/fcT} exp 1 - [tf(r,) + pr 2]/fcT} . . exp \-[U(rn) = (/ o °° exp I ~[U{r) + pr]/kT} dn drt . . drn + prn]/A:T} rfri dr2 . . drn n dr) . On the other hand Φ(ρ) = j i * Ζ exp [-pZ/fcT] dxn = with exp [—(F* + pZ)/fcT] dx» I = xn — XO, and where F* represents the configurational part of the free energy.
Vp[- fR-ßV(K)yiR ONE DIMENSION 1106 S A L S B U R G , Z W A N Z I G , The second line follows from the definition of ß(c), (l) Eq. (7). From the normalization condition on p , A N D K I R K W O O D B y using Eq. ( B l ) we find \g™(x-u)du x-k~\ we see that limp<»(r)-l/la. (A10) After changing the summation in E q . *. I ( A )1 where r > r ' and n=*k—j. If we adopted the hard core model, E q . (21), then n<(r—r')/a «<(r— r')/a where α is the the t t diameter of the hard core. =r— r' and infinite we find that Ρ ( 2 > 1 « A(R-na) (*)«Σ ; 1 χ / .