Generic models

See examples in Generic.mw

For mathematical details see Random walk on a lattice: Basic formulas

To cite this code use: A Zhugayevych et al, JPCC 117, 4920 (2013) [pdf]

Notations

• k k-vector
• d::posint spatial dimension
• o::1..d enumerates dimensions
• M::Matrix(d) matrix of Cartesian translation vectors as columns (default is identity matrix)
• n::posint number of sites in unit cell
• m::1..n enumerates sites in unit cell
• mt=[m,t1,...,td] enumerates sites in translationally invariant system, here t denotes integer translations
• mtsid indexing table mts[mtsid[mt]]=mt
• n2::posint size of the extended cell consisting of all sites coupled to unit cell starting from n sites belonging to the unit cell
• sites::Array(n2,0..d) such that sites[i,0]=mts[i][1] and sites[i,o]=mts[i][1+o]
• replicas::Vector(n) whose i-th entry is list of all replicas of i-th site including i
• Points/points::Matrix(d,n) whose columns are Cartesian/lattice coordinates
• W::Matrix(n,n2) transition rate matrix (its diagonal elements will be set to zero if appropriate)
• Wk::Matrix(n) Fourier transform of W
• W0,Wp,Wpp are Wk, its derivative diff(Wk(k),I*k), and second derivative all at k=0 and with zero diagonal
• U::Vector decay rates vector
• H::Matrix(n,n2) single-cell real-space Hamiltonian
• Hk::Matrix(n) Fourier-transform of H

Procedures

Builders

• BuildW(n,genfun,genopt,{sym::boolean})::W build transition rates matrix (symmetric if sym), genfun determines generating function which uses genopt,_rest and can be one of
  • "Matrix" use Matrix
  • "RandomMatrix" use RandomMatrix
  • "Generate" use RandomTools[Generate]
  • "Sample" use Statistics[Sample]
  • "GamEps" use Gam[i,j](Sample(genopt[1]))*exp(Eps[i](Sample(genopt[2]))) (returns W,Gam,Eps)
• BuildA(W,/U,$)::Matrix(n) build generator of Markov chain such that P(t)=exp(A*t)
• BuildH(tbH,k::{name,indexable},{symmetric,printout},$)::H,Hk,mts,mtsid build translationally invariant Hamiltonian matrix, here tbH::table of translationally unique Hamiltonian elements tbH[m0,m,t1,...,td]

Solvers

• GreensF(s,W,/U,$)::Matrix compute Green's function G=(s-BuildA(W,U))^(-1) (datatype is derived from s,W)
• StatSol(W,/i0,simplifyf,{printout},$) find stationary solution, with respect to site i0::nonnegint:=0 (must be significant site to avoid the loss of precision), if zero i0 is set to a site with the maximum outflow
• ZeroExpan(W,/i0,simplifyf,{printout},$) expand the resolvent at zero
• Velocity(sites,W,/M,replicas,i0,simplifyf,{printout},$) velocity for infinite periodic lattice
• DiffusionTensor(sites,W,/M,replicas,i0,simplifyf,{printout},$) diffusion tensor
• DiffusionLength(sites,W,U,points,/M,replicas,{printout},$) mean square displacement

Analyzers

• PathExpansion(H,E,oi,of,vmin,Nmax,/subsys,{Lmax,neglect,printout,digits,nprint},$)
::table([seq(L=list([v,o1,o2,...,oL]),L=0..Lmax2)]) perform path expansion of sub-Hamiltonian in perturbation theory over nondiagonal matrix elements by expanding Hrenorm[oi,of]=H[oi,of]+H[oi,env].(E-Henv)^(-1).H[env,of] in powers of transfer integrals
  • E::numeric perturbation eigenenergy
  • oi,of::posint indices of a matrix element to be expanded
  • subsys::list(posint):=[oi,of] list of indices of the subsystem, other indices are considered as environment env
  • vmin::numeric expansion threshold in value of tau-matrix elements
  • neglect::numeric:=1e-3 threshold for perturbation terms vmin2=neglect*vmin
  • Nmax::posint maximum value of terms to keep at each iteration
  • Lmax::posint:=99 maximum order of expansion
  • Lmax2 internal variable defining maximum nonzero term of expansion
  • nprint::posint:=9 number of largest terms to print