Hamiltonian field-theoretic

Continuum models go one step frirtlier and drop the notion of particles altogether. Two classes of models shall be discussed field theoretical models that describe the equilibrium properties in temis of spatially varying fields of mesoscopic quantities (e.g., density or composition of a mixture) and effective interface models that describe the state of the system only in temis of the position of mterfaces. Sometimes these models can be derived from a mesoscopic model (e.g., the Edwards Hamiltonian for polymeric systems) but often the Hamiltonians are based on general symmetry considerations (e.g., Landau-Ginzburg models). These models are well suited to examine the generic universal features of mesoscopic behaviour. 

This expression has a formal character and has to be complemented with a prescription for its evaluation. A priori, we can vary the values of the fields independently at each point in space and then we deal with uncountably many degrees of freedom in the system, in contrast with the usual statistical thermodynamics as seen above. Another difference with the standard statistical mechanics is that the effective Hamiltonian has to be created from the basic phenomena that we want to investigate. However, a description in terms of fields seems quite natural since the average of fields gives us the actual distributions of particles at the interface, which are precisely the quantities that we want to calculate. In a field-theoretical approach we are closer to the problem under consideration than in the standard approach and then we may expect that a simple Hamiltonian is sufficient to retain the main features of the charged interface. A priori, we have no insurance that it... 

Note that another consisitent approach to the problem of mobile particle accumulation is based on the field-theoretical formalism [15, 37, 51]. However by two reasons this approach is not useful for the study of immobile particle aggregation (i) the smallness of the parameter U(t) = n(t)vq [Pg.414]

Finally, for completeness in Appendix A 7.1 we consider the formal relation of the continuous chain model to a field theoretic Hamiltonian, used to describe critical phenomena in ferrornagnets. It was this relation discovered by de Genries [dG72] and extended by Des Cloizeaux [Clo75, which initiated the application of the renormalization group to polymer solutions and led to the embedding into the larger realm of critical phenomena. 

It is easily seen that for such trial functions the minimization of the Hamiltonian K, Eq. (5.1), may be replaced by the minimization of a specified nonlinear functional 6(0) of the molecular states 0 alone. In the following we refer to either formulation as seems convenient. This argument also enables one to connect these field theoretical models with the earlier suggestion of mine that molecular structure states can be associated with those solutions of the Schrodinger equation for the full molecular Hamiltonian ft that satisfy certain subsidiary conditions3,35), if the latter are associated with the nonlinearity in the functional 6(0). As we shall see, it may happen that 6(0) has two degenerate minima and it is in this sense that the dynamics gives rise to a double-well structure. 

The preceding discussion has been completely based on the Heisenberg representation. The foundations of DFT, on the other hand, are usually formulated within the framework of the Schrodinger picture, so that one might ask in how far this field theoretical procedure can be useful. It is, however, possible to go over to an appropriately chosen Schrodinger representation as long as one does not try to eliminate the quantised photon fields (compare Sections 7d, lOg of Ref. [34]). The Hamiltonian then reads... 

A general quantum field-theoretical framework for treating all the different problems outlined above may be specified in the interaction representation. Let the total system under consideration be described by the time-dependent Hamiltonian... 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

The same field-theoretical representation may be obtained starting from the quite different type of lattice model known as one of the basic models in the theory of magnetic systems. We present it here since for what follows it serves to introduce different types of disorder in the polymer system. Let us consider a simple (hyper) cubic lattice of dimension d, and to each site prescribe a m-component vector S r) with a fixed length (for convenience one usually sets 15 = m). Imposing a pair interax tion with the energy proportional to the scalar products between pairs of spins, this defines the Stanley model (also known as the 0(m) symmetric model). The Hamiltonian of this model reads [77] ... 

The effective field-theoretical Hamiltonian of the model (14), obtained via special Stratonovich-Hubbard transformation, passing from the discrete S3 tem to the continuous field theory, is the same as (13). The relevant global properties of a microscopic model is represented by the structure of the effective Hamiltonian. In this case, is a squared bare mass proportional to the temperature distance to the critical point, o > 0 is a bare coupling constant. Note that the effective Hamiltonian (13) preserves the 0(m) symmetry of the Stanley model (14). As far as the Stanley model is in this sense equivalent to the 0(m) symmetric theory, the analytic continuation m —> 0 of this model again leads to the polymer limit. 

By the Hubbard-Stratonovich transformation we have rewritten the partition function of the interacting multi-chain systems in terms of noninteracting chains in complex fluctuating fields, il7 -i- W and il7 - W. hi field theoretical polymer simulations, one samples the fields U and W via computer simulation using the above Hamiltonian (cf. Sect. 4.4). 

We should note that the Monte-Carlo simulation with tw = 0 effectively samples the EP Hamiltonian. This version of field-theoretic Monte Carlo is equivalent to the real Langevin method (EPD), and can be used as an alternative. Monte Carlo methods are more versatile than Langevin methods, because an almost unlimited number of moves can be invented and implemented. In our applications, the W and tw-moves simply consisted of random increments of the local field values, within ranges that were chosen such that the Metropolis acceptance rate was roughly 35%. In principle, much more sophisticated moves are conceivable, e.g., collective moves or combined EPD/Monte Carlo moves (hybrid moves [84]). On the other hand, EPD is clearly superior to Monte Carlo when dynamic properties are studied. This will be discussed in the next section. 

I will not dwell on the field theoretical details in the following, but we have to take note of the starting point, the QED Hamiltonian... 

From Field-Theoretic Hamiltonians to Particle-Based Models Soft-Core Models... 

Single-Chain-in-Mean-Field Simulations and Grid-Based Monte Carlo Simulation of the Field-Theoretic Hamiltonian... 

From Field-Theoretic Hamiltonians to Particie-Based Modeis Soft-Core Modeis 219... 

The interactions between a localized vibrational state with the host elastic continuum can be described by the perturbation theory of Fano or in an equivalent way, with the second quantization field theoretical method of Anderson. The Anderson-Fano Hamiltonian appropriate for xenon hydrate is ... 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

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