By Dr. David A. Lavis, Professor George M. Bell (auth.)
ISBN-10: 3642084117
ISBN-13: 9783642084119
ISBN-10: 3662038439
ISBN-13: 9783662038437
This two-volume paintings presents a complete research of the statistical mechanics of lattice versions. It introduces the reader to the most parts in statistical mechanics and the speculation of section transitions. the advance is equipped on a company mathematical and actual foundation. quantity 1 comprises an account of mean-field and cluster edition tools effectively utilized in many purposes in solid-state physics and theoretical chemistry in addition to an account of tangible effects for the Ising and six-vertex versions and people derivable by means of transformation equipment. quantity 2 comprises large remedies of scaling concept, algebraic and real-space renormalization tools and the eight-vertex version. it is also an account of sequence tools and a therapy of dimer assemblies.
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Extra info for Statistical Mechanics of Lattice Systems: Volume 1: Closed-Form and Exact Solutions
Example text
71) is true for any la ttice dimension. However, for two- or three-dimensional lattices, m is discontinuous at J{ = 0 not only at T = 0 but for a range 0 :::; T < Tc , where Tc is a critical temperature. 36). However, in the one-dimensional system we must replace t he volume v per atom by the lattice spacing, which is denoted by b. 100) m0 being the moment of an Ising dipole and J-Lo the permeability of free space. 100) reduces to = T - 1 exp(21IT) . 101) For higher-dimensional lattices, Xr has a singularity on the J{ = 0 axis for a non-zero critical value of T , which is associated with the onset of long-range order.
25) in two stages, the first over the values of u corresponding to fixed values of Q1, ... 2 D istributions in General 37 second over all values of the latter. However the first summation, for which all extensive variables are fixed , is precisely that required to obtain a canonical partition. Hence we have Z11 (T, 6 , .. ,~1 , Qn- 11+1> .. · , Qn) = ~L ~ { exp ( {Ql, ... ,Q~} t ~;(JdT) x Z(T, Ql, ... , Qn) }. 31) Now suppose that E = Ekin +Ec, as assumed in Sect. 2. 32) exp[-H'Ic(O'c)/T]' /\~~) where it is assumed that Q; are dependent on the configurational microstate.
1, ... ,n. , + 1, giving a constant-field grand distribution. 4 Restricted Distributions for Lattice Models Suppose that the centre of each molecule (or atom) occupies a lattice site and that the N sites are equivalent and form a regular lattice occupying a volume V . Consider the following conditions (i) The volume per lattice site v 0 = VjN is constant. 3 Particular Distributions 41 (ii) Every lattice site is occupied by the centre of a molecule of one of the "' components. 55) N. The volume vo must be replaced by a length per site and an area per site for one- and two-dimensional systems respectively.
Statistical Mechanics of Lattice Systems: Volume 1: Closed-Form and Exact Solutions by Dr. David A. Lavis, Professor George M. Bell (auth.)
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