Discrete configuration space

Usually the space over which the objective function is minimized is not defined as the p-dimensional space of p continuously variable parameters. Instead it is a discrete configuration space of very high dimensionality. In general the number of elements in the configuration space is exceptionally large so that they cannot be fully explored with a reasonable computation time. 

Because MC is a numerical technique to calculate multidimensional integrals like the one over configuration space in Eq. (5.1), we begin by discretizing configuration space and rewrite the integral as... 

Discrete Configuration Spaces of Generalized Simplicial Complexes... 

We can connect the discrete configuration spaces to Horn complexes by noticing that for any generalized simphcial complex A and any positive integer n, we have... 

A real-valued function defined on the discrete configuration space T (k) is called a configurational fiunction. Also the Hamiltonian of a system is a configurational function ... 

Note that we have employed discrete notation, although rigorously this sum is an integral in configuration space. The choice is for clarity in the following derivations, and our conclusions will be unaffected by the notation. With minor manipulation, (3.59) gives this important result... 

The MC method considers the configuration space of a model and generates a discrete-time random walk through configuration space following a master equation41,51... 

Let us consider a stochastic system described by a generic variable C. This variable may stand for the position of a bead in an optical trap, the velocity field of a fluid, the current passing through a resistance, of the number of native contacts in a protein. A trajectory or path V in configurational space is described by a discrete sequence of configurations in phase space. 

This would require an appropriate discretization of the configuration space, which is in most cases not feasible computationally. 

The combination of all the local densities of states, g(E L,xj), represents a lumped picture of the pocket. This description is intermediate between the overall density of states for the whole pocket g(E Lmax) and the exhaustive description of every microscopic detail of the energy landscape within the pocket one can achieve in the discrete case. With this information, it is possible21 to construct a transition matrix M(T) in the lumped configuration space that allows the simulation of the evolution of the system for temperature T, and thus yields estimates for Teq(R) and tcsc(R)-... 

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fiirth), we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. 

The auxiliary conditions which must be satisfied by a solution of the amplitude equation in order that it be an acceptable wave function are given in Section 9c. These conditions must hold throughout configuration space, that is, for all values between — oo and + oo for each of the ZN Cartesian coordinates of the system. Just as for the one-dimensional case, it is found that acceptable solutions exist only for certain values of the energy parameter W. These values may form a discrete set, a continuous set, or both. 

Weight associated with a discrete configuration (on a lattice) Weight associated with a configuration in continuous space Component of ry... 

Here we review well-known principles of quantum statistical mechanics as necessary to develop a path-integral representation of the partition function. The equations of quantum statistical mechanics are, like so many equations, easy to write down and difficult to implement (at least, for interesting systems). Our purpose here is not to solve these equations but rather to write them down as integrals over configuration space. These integrals can be seen to have a form that is isomorphic to the discretized path-integral representation of the kernel developed in the previous section. 

The electronic—of N electrons of the mass m, and the charge —e which positions in the spin-configurational space are determined by the corresponding radii vectors rj, T2, , where each r, i = 1,2,. .N belongs to the real three-dimensional space and the spins ai, 02, , On where each cr, = 1, 2,. .., N takes the value from Z2 = 1/2, the discrete two-dimensional spin space... 

Now let us consider the discrete version of minimizing the action . We work on the time interval [0, r]. Consider the v + 1 points qa to qv in configuration space... 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

In the classical case, instead of discrete distribution w ( ) we have a continuous probability function p (p,q) that is probability to have a subsystem with given momentum p and co-ordinate in the configuration space ... 

We consider the discrete state space Q s Q ,np,..., Qvn-2 of tho rotational isomeric configurations of a chain of N bonds having v states accessible to each bond. The stochastic process of v xv " transitions between those configurations is the object of study. In the following, the terms states and system will be used interchangeably for configurations and chain , respectively. 

A standard concept in algebraic topology is that of a configuration space. Here we consider the discrete version. 

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