Hi, although latex export works correctly in some cases, the following case (see attached file) does not work and reports an (unhelpful) error message and then aborts the export. It would be helpful if it could at least report the troubling line number, but a correct export would be most desirable naturally. wait, how do i attach a file? the only thing I can think of is export to html and then cut and paste below, or you can email me and get the error-causing file. Here is the html:
Thermodynamic Lattice Models
J. Brandon Keith, Jiao Y. Y. Lin, Lek-Heng Lim, …,Matt Redmond, Richard B. McClurg, and Brent Fultz
Models
Ising:
S = +1, -1
Ising with field:
Lattice gas:
Spin glass:
S = +1, 0, -1
Polymer folding:
Heisenberg (dipole):
Multipole:
Vibrating particle:
W is similar to above except the Ds (Wigner functions) are replaced by Cs (spherical harmonics) with additional r^l factors (where r goes from 0 to infinity)
*Partition function is integral over phase space of exponential of this
Multiple Hypergeometric Function (MHF) formulation
*Write partition function as
where {g_ij} are a collection of indices for 2xNxM sums, K is J/kT, and i and j index a given atom i with j nearest neighbors
*denom is 1 for Ising models and
*for the Heisenberg model this is a large, coupled Multiple Hypergeometric Function (MHF)
*for the Ising model the coupled exponential terms are separable resulting in a large polynomial of sinh(K), cosh(K)
*No phase transition unless take thermodynamic limit! (i.e. N -> infinity)
*Because MHF has no standard rules for evaluation, it is solved numerically using Monte Carlo where N is as large as possible...
Decoupling the MHF
*From the formula below, it is apparent monomials contributing to Z with "(1-1)" factors go to zero
*By studying the types of g's that create nonzero monomials, one may distinguish 8 different cases which interact with one another in the x and y direction
These interactions can be written as a matrix...i.e. in y-direction:
Groebner deformation
Each atom or spin is a cross product of these interaction matrices, or, in 2d, an 8x8x8 hypermatrix:
The system becomes a grid of hypermatrices (i.e. 2x2 system on right), where adjacent hypermatrices matrix multiply in the usual way with their neighbors.
note one index per atom
Interacting hypermatrices seems to be the most natural formulation of the problem--no indefinitely large matrices to diagonalize (like transfer matrix approach)
The above 8x8x8 tensor (for 2D Ising) can be reduced to a 2x2x2x2 tensor:
Different types of eigendecompositions. A local one in the traditional sense of an eigendecomposition has no solution, somewhat like certain matrices can only be put into Jordan canonical form rather than an eigen form. This can be proved.
A nonlocal decomposition works. For the 2D Ising, one can tensorially multiply
where I have written it as nested 2d slices (i.e. two of the dimensions are on the outside, and two on the inside of each "inner" matrix). Using these tensors one can easily construct an indefinitely long row of spins, such as the two-spin row above (acually a 2x2x2x2x2x2 matrix), or the three-spin row below:
Although these are written as tensors, just remove the extra parentheses and matrix multiply them together to get the product of two rows, or three rows, etc....you get the idea. To diagonalize them all, start with the base tensor and diagonalize that, then use it to form these product tensors (which act like matrices during row multiplication).
Q: monte carlo more or less "works" already--MultipleHypergeometric Functions (MHFs) would only "speed things up"
A: MHFs would completely change the field, fromprotein folding to microstructural modeling. Why? Once one can write down an analytical expression for a partition function in a 3d grid, one can take a real system, define an arbitrarily-fine grid, and conduct structural simulations in the lattice gas pattern with almost arbitrary forcefields, adding constraints for particle density, or only allowing new particles to enter from system boundaries, etc.
--> Large-scale distributed computing endeavors such as folding@home and docking@home could be done on laptops. --> Microstructural material modeling, as just another of billions of examples, would be more wide-spread since *vast* regions of material could be modeled on the atomistic scale.
-->The monte carlo numerical technique would essentially cease to exist. The only gains afforded by md would be kinetics, which I believe could also be obtained through methods of non-equilibrium statistical mechanics, so even those might become almost extinct. Stat mech textbooks would have to be rewritten.
*Solving 3D spin glass problem was shown to be NP-hard
--> can finally prove P = NP (famous computer science prob)
On the computational complexity of Ising spin glass models
out simulations utilizing finite lattices, e.g. Barahona ..... Np-hard if the existence of a polynomial algorithm for its solution implies the existence ..... We have classified the spin glass models into hard and easy ones. ...www.iop.org/EJ/article/0305-4470/15/10/028/jav15i10p3241.pdf