Claude Garrod

Reduction of the N‐Particle Variational Problem

A variational method is presented which is applicable to N‐particle boson or fermion systems with two‐body interactions. For these systems the energy may be expressed in terms of the two‐particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a2+a1+a1′a2′| Ψ). In order to have the variational equation: δE/δΓ = 0 yield the correct ground‐state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N‐particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two‐particle and one‐particle density matrices of an N‐particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ...

The variational approach to the two−body density matrix

A variational method for the two−body density matrix is developed for practical calculations of the properties of many−fermion systems with two−body interactions. In this method the energy E = JHijkl ρijkl is minimized using the two−body density matrix elements ρijkl = 〈ψ‖a+ja+iakal‖ψ〉 as variational parameters. The approximation consists in satisfying only a subset of necessary conditions—the nonnegativity of the following matrices: the two−body density matrix, the ’’two−hole matrix’’ Qijkl = 〈Ψ‖ajaia+ka+l‖Ψ〉 and the particle−hole matrix Gijkl = 〈Ψ‖ (a+iaj−ρij)+ (a+kal−ρk) ‖Ψ〉. The idea of the method was introduced earlier; here some further physical interpretation is given and a numerical procedure for calculations within a small single−particle model space is described. The method is illustrated on the ground state of Be atom using 1s, 2s, 2p orbitals.

The variational calculation of reduced density matrices

Abstract We discuss the computational problem encountered in making direct variational calculations of the reduced density matrices of many-particle systems. The problem is one of minimizing a linear function within a convex domain defined by a finite set of nonlinear constraints. Two different algorithms are presented for which working programs have been written.

The application of group theory to generate new representability conditions for rotationally invariant density matrices

New linear conditions are derived which must be satisfied by a two‐body density matrix. In the derivation, the ideas of Davidson and McRae are extended so that full use is taken of the symmetries of the system. The coefficients of the linear form are determined by means of reduction of a chosen group in a physically meaningful chain of its subgroups.

Mapping of crystal growth onto the 6-vertex model

Abstract A simple kinetic TLK crystal model is studied, where growth and evaporation take place by attachment and detachment of atoms at kink sites along steps. Assuming the steps to run up and to the right exclusively, the model is mapped onto the 6-vertex model, where the step lines correspond to the lines obtained by marking the arrows pointing down and to the left. (Since vertex 2 is absent, the model is actually a 5-vertex model). The growth process is described by an exact master equation, which is then solved for small-size versions of the model; the growth rate is calculated in terms of the kinetic coefficients. The limit for large sizes is discussed.

Rigorous Statistical Mechanics for Nonuniform Systems

The thermodynamic limit of a classical system with many‐body interactions and under the influence of an external potential is investigated for the canonical ensemble. By scaling the external potential to a sequence of domains which are isotropic expansions of an initial domain confining the system, it is shown that the canonical free energy per particle has an infinite system limit. Moreover, by restricting consideration to internal interactions which have the property that separated groups of particles have negligible mutual attractive energy as the system becomes infinite, it is proven that the free energy per particle limit is precisely the free energy per particle obtained by minimizing the integral ∫[φρ + f(ρ, β)] with respect to all properly normalized functions ρ(r). φ is the external potential; f(ρ, β) is the free energy per unit volume for a uniform system of density ρ and inverse temperature β. The only technical complication is the above‐mentioned restriction on the allowed internal interaction...

The density of a nonuniform system in the thermodynamic limit

We discuss the existence, continuity, and other properties of the canonical and grand canonical density distributions in the thermodynamic limit for nonuniform classical mechanical systems. For an external potential φ defined on a domain Λ the free energy per unit volume for fixed temperature is given by F(ρ0,φ) = min ∫Λ[ρ(x)φ(x) + f(ρ(x))]dx/V(Λ) where the minimum is over all density distributions satisfying the restriction of fixed average density ρ0, and f(ρ0,β) is the free energy per unit volume in the thermodynamic limit when φ = 0. We prove that if φ is not constant over any region of finite volume then the density distribution which minimizes is unique, and also that the density is the functional derivative of F(ρ0,φ) with respect to φ. We also show that the density distribution of an infinite nonuniform system is the limit of density distributions associated with finite systems of increasing size.

A stochastic model of three-dimensional crystal growth

This paper describes a stochastic model of crystallization from a gas or dilute solution. The model is limited to a crystal of rectangular symmetry whose surface has nonzero Miller indices. By a mapping into the modified KDP model, the kinetic growth coefficient can be given approximately as an analytic function of the Miller indices of the surface. Numerical simulations indicate that the aproximation is accurate within a few percent at all surface orientations.

Symmetry properties of nonlinear barrier coefficients

This paper concerns the properties of a symmetric barrier between two reservoirs. The barrier can passK conserved quantities. The current of theith quantity is assumed to satisfy the nonlinear relationJi=AijΔβj+BijklΔβjΔβkΔβl where the Δβis are the affinity differences across the barrier andAij andBijkl are functions of the average affinities of the reserviors. It is shown thatBijkl is symmetric in all indices.

Growth modes and models for smooth and rough surfaces

A brief survey of general growth modes and models for smooth vs. rough surfaces is given, with special reference to the growth rate in the different regimes (continuous growth, layer growth, step flow, spiral growth, dendritic growth). In particular, two models, mapping surfaces onto the 6-vertex model, are studied. Under one mapping the surface steps are identified with the lines occurring in the line representation; the other mapping is that of van Beijeren. Master equations are written for finite-size versions of the models, and solved analytically (for extremely small sizes), or by Monte Carlo simulation, using Glaubers rules for the growth, evaporation and diffusion probabilities. For the first model the steady growth rate is shown to be smaller than the initial growth rate for an equiprobable ensemble of configurations. In the second model both the growth mode and the surface structure depend explicitly on temperature T and supersaturation Δμ. The surface is rough at all temperatures, but the nature of the roughness changes: at high T, the roughness is logarithmic if Δμ is small, but a cross over takes place to a power-law behaviour when Δμ increases.