By Barry M McCoy

Statistical Mechanics is the learn of structures the place the variety of interacting debris turns into countless. within the final fifty years large advances were made that have required the discovery of totally new fields of arithmetic corresponding to quantum teams and affine Lie algebras. they've got engendered awesome discoveries touching on non-linear differential equations and algebraic geometry, and feature produced profound insights in either condensed subject physics and quantum box conception. regrettably, none of those advances are taught in graduate classes in statistical mechanics. This ebook is an try to right this challenge. It starts with theorems at the life (and lack) of order for crystals and magnets and with the idea of serious phenomena, and maintains via offering the tools and result of fifty years of analytic and machine computations of section transitions. It concludes with an intensive presentation of 4 of crucial of precisely solved difficulties: the Ising, eight vertex, tough hexagon and chiral Potts types.

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D. N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev. 87 (1952) 410–419. W. Atkins, Physical Chemistry (Oxford University Press 1982) edn. 2. 3 Stability, existence and uniqueness In chapter 1 we saw that the macroscopic thermodynamic Helmholtz free energy A(V, T ) is computed classically from the partition function QN (V, T ) as QN (V, T ) = e−A(V,T )/kB T , where QN (V, T ) = 1 N! 2) and H(r, p) is the Hamiltonian of the system.

The lattice gas of Lee and Yang [5] models the short range repulsion by forbidding two or more particles to be at the same site, and models the attractive force by having a negative interaction energy − between particles which are nearest neighbors on the lattice. 15) U (R) = 0 otherwise Lattice gases are also used to gain insight into the phase transitions of hard core systems such as hard spheres and discs. 16) If the lattice is square (so that γ = 4) the model is called hard squares; if the lattice is triangular (so that γ = 6) the model is called hard hexagons.

26) The numbers N+ , N− , N++ , N−− , N+− and N are not independent. Clearly N− = N − N+ . 27) To ﬁnd other relations consider drawing a line from each + spin to its γ nearest neighbors. There are γN+ such lines. Between every ++ pair there are two lines, and between every +− pair there is one line. Thus γN+ = 2N++ + N+− and similarly γN− = 2N−− + N+− . 26) as EI = −4EN++ + 2(Eγ − H)N+ − ( γE − H)N. 31) N++ =0 and the free energy FI and magnetization MI are given by βFI = − MI = 1 ln ZI (H, T, N ) N 1 (N+ − N− ).