Spatial finite-size scaling

We will assume that the ground-state energy of the Hamiltonian (20) has a critical exponent a 2 (for example, a short-range potential V(r) and 2 < 6 / 3). The main hypothesis of the spatial finite-size scaling (SFSS) ansatz, which makes the different values of a compatible, is that the coefficient aR, analytical for finite values of R, has to develop a singularity at the exact critical value Xc when R > oo as... 

The time dependence of is essentially nonlinear at aU hydration levels studied. Translational motion of water molecules in such complex system as low-hydrated biomolecules is determined by the following factors restriction of the motions in the direction normal to the protein surface restriction of the motion due to the finite size of a biomolecule spatial disorder due to fractal-like structure of diffusion pathway temporal disorder due to the presence of the strongly attractive sites on the surface. Relative importance of these factors depends on the time and length scales considered, on the properties of a biomolecule and on the hydration level. In pores, MSD of molecules normally to the pore wall (axis) nonlinearly increases at short times and achieves saturation at longer times. As a result, the time dependence of the total MSD is... 

The extension of Gillespie s algorithm to spatially distributed systems is straightforward. A lattice is used to represent binding sites of adsorbates, which correspond to local minima of the potential energy surface. The discrete nature of KMC coupled with possible separation of time scales of various processes could render KMC inefficient. The work of Bortz et al. on the n-fold or continuous time MC CTMC) method can lead to computational speedup of the KMC method, which, however, has been underutilized most probably because of its difficult implementation. This method classifies all atoms in a finite number of classes according to their transition probability. Probabilities are computed a priori and each event is successful, in contrast to the Metropolis method (and other null event algorithms) whose fraction of unsuccessful (null) events increases drastically at low temperatures and for stiff problems. In conjunction with efficient search within a class and dynamic variation of atom coordi-nates, " the CPU time can be practically independent of lattice size. After each event, the time is incremented by a continuous amount. 

The exponential dispersion is only valid until the distance between the particles is smaller than the characteristic length scale of the flow. If initially there is a finite distance, So, between the two particles, 0 < o -C L, the exponential growth S(t) So exp(Aii) saturates at around ts ln(L/ o). If the advection takes place within a bounded domain, then after this time the distance between the two particles fluctuates chaotically and is comparable to the size of the domain. In the case of an open unbounded system, e.g. in a spatially periodic velocity field, the dispersion becomes diffusive and 5(t) t1/2 for large t. 

In a spatially homogeneous system the magnitude of the fluctuations scales as the square root of the system size (or total number of particles) (cf. (22)). This trivial scaling does not provide complete information on the actual amplitude of the fluctuations for a given finite system since the strength of the fluctuations also depends on the system s dynamical state. It is well known that fluctuations can grow to macroscopic size near critical points (critical opalescence is a familiar manifestation of this growth) or near bifurcation points in systems far from equilibrium [22]. 

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