Langevin simulation

Studying fluctuations beyond the Gaussian approximation is difficult. Special types of fluctuations, e.g., capillary waves of interfaces, can sometimes be described analytically within suitable approximations [58]. The only truly general methods are however computer simulations. Here we shall discuss two different approaches to simulating field theories for polymers Langevin simulations and Monte Carlo simulations. 

As long as one is mainly interested in composition fluctuations (EP approximation, see Sect. 3), the problem can be treated by simulation of a real Langevin process. The correct ensemble is reproduced by the dynamic equation 

The choice of M r,r ) determines the dynamic properties of the system. For example, M r,r ) = 5(r - r )/rkRT corresponds to a non-conserved field, while field conservation can be enforced by using a kinetic coefficient of the form = VrA(r- r )Vr/. Different forms for the Onsager coeffi- 

The approach is commonly referred to as external potential dynamics (EPD). A related approach was originally introduced by Maurits and Fraaije [31]. However, these authors do not determine U W) exactly, but only approximately by solving separate Langevin equations for real fields Wa and Wb. This amounts to introducing a separate Langevin equation for a real field iU (i.e., an imaginary U in our notation) in addition to Eq. 97. 

In our case, the complex Langevin equations that simulate the Hamiltonian Eq. 11 read 

---

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

Langevin simulations of time-dependent Ginzburg-Landau models have also been performed to study other dynamical aspects of amphiphilic systems [223,224]. An attractive alternative approach is that of the Lattice-Boltzmann models, which take proper account of the hydrodynamics of the system. They have been used recently to study quenches from a disordered phase in a lamellar phase [225,226]. 

The periodicity in the NMF indicates the presence of non-equlibrium processes with periodic solutions and is related to the number of states and the transition rates [46, 67]. It is important to mention that the Langevin simulations are very sensitive to the model used and the identity with the experimental data set is indicative for a driven conformational loop which is coupled to the catalytic step generating the substrate product conversion with photon emission (Fig. 4.22). 

The Langevin simulations give the results shown in Fig. 17 for the time... 

Figure 17. Populations arising from B-state excitation from Langevin simulations versus time for X state (---), A state (----), A state (-----X state (.X and dissociated atoms...

Size distributions of (he agglomerates generated by cluster-duster aggregation compu-tations may approach a self-preserving form.This is found both forthe Langevin simulations and for l andom walk on a lattice (Meakin. 1986). Direct calculations of the self-preserving distribution,s are made in the sections that follow. 

Compared with simulations of the two state model we find perfect agreement. Fitting the time T on the excitation loop as well as the two excitation rates tq and ri to the FHN system, also leads to a reasonable agreement between theory and Langevin simulations of the FHN system. 

B. Martire and R. G. Gilbert, Chem. Phys., 56, 241 (1981). Langevin Simulation of Picosecond-Resolved Electronic Spectra in Solution. 

Fig. 2 Averaged densities across the order-disorder transition in a two-dimensional ternary system with A, B homopolymers and A-B copolymers (20% homopolymer volume fraction), as obtained from Complex Langevin simulation runs...

Examples of anisotropy parameters are shown in Fig. 9 (left). They are clearly suited to characterize the phase transition between the disordered phase at low and the lamellar phase at high xN. For comparison, the same system was also examined with complex Langevin simulations. The results are shown in Fig. 9 (right). The transition points are the same. 

Fig. 9 Anisotropy parameters 2 (solid) and 5 4 (dashed) vs. xN for different homopolymer volume fractions

Fig. 10 Phase diagram from simulations for the same system as above. The circles show the results from Monte Carlo simulations, the squares those from the complex Langevin simulations. The dotted lines correspond to the mean-field prediction from Fig. 5. From [80]...

A treatment that studies the case of unphysical dynamics of the solvent particles has been carried out in Ref. 31, also using a microscopic approach. This was done in order to discuss if it is justified to replace an MD algorithm in which the solvent particles obey strictly Newton s equations of motion by a Langevin simulation in which every solvent particle is artificially coupled to a weak friction and a weak heat bath (this latter method has some... 

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