Fast microscopic variable

The most obvious variables of interest are the slow macroscopic variables represented by the operators A,B,C,... Because the statistical mechanics provides the macroscopic properties from the microscopic world, we also need to deal with fast microscopic variables most often the flux variables. Expressions (91) and (92) show that, at short time, the dynamics is dominated by the first few terms of the expansion... 

In Section 2.2 we mentioned the impossibility to strictly substantiate the equilibrium descriptions for all cases of life and the need to apply equilibrium approximations in some situations. The vivid examples of the cases, where the strongly nonequilibrium distributions of microscopic variables are established in the studied system and the principal difficulties of its description with the help of intensive macroscopic parameters occur, are fast changes in the states at explosions, hydraulic shocks, short circuits in electric circuits, maintenance of different potentials (chemical, electric, gravity, temperature pressure, etc.) in some spatial regions or components of physicochemical composition interaction with nonequilibrium and sharply nonstationary state environment. 

Obviously, the above algorithms are not suitable when transients of the finer scale model are involved (Raimondeau and Vlachos, 2000), as, for example, during startup, shut down, or at a short time after perturbations in macroscopic variables have occurred. The third coupling algorithm attempts fully dynamic, simultaneous solution of the two models where one passes information back and forth at each time step. This method is computationally more intensive, since it involves continuous calls of the microscopic code but eliminates the need for a priori development of accurate surfaces. As a result, it does not suffer from the problem of accuracy as this is taken care of on-the-fly. In dynamic simulation, one could take advantage of the fast relaxation of a finer (microscopic) model. What the separation of time scales between finer and coarser scale models implies is that in each (macroscopic) time step of the coarse model, one could solve the fine scale model for short (microscopic) time intervals only and pass the information into the coarse model. These ideas have been discussed for model systems in Gear and Kevrekidis (2003), Vanden-Eijnden (2003), and Weinan et al. (2003) but have not been implemented yet in realistic MC simulations. The term projective method was introduced for a specific implementation of this approach (Gear and Kevrekidis, 2003). 

In Section II we compare particle mechanics in the slow and fast variable timescale regimes. We start the discussion by showing the following. For damped macroscopic particles, the potential energy function whose minima locate the particle s points of static equilibrium also produces the forces which drive its dynamics. For damped microscopic particles, in contrast, the potential that determines the particle s statics may or may not produce the forces that drive its dynamics. 

Thermodynamics is the study of equilibrium at a macroscopic level. When a system is in mechanical equilibrium, there is no net force imbalance that causes motion. Complete thermodynamic equilibrium is more extensive and requires not only mechanical equilibrium but also thermal, phase, and chemical equihbrium. We can use classical thermodynamics to analyze chemically reacting and nonequiUbrium flows, such as those in fuel cells, but are restricted to only the quasi-equilibrium beginning and end states of the process, with no details of the reaction itself. Thermodynamics can tell us the potential for reaction and direction of spontaneous reaction, but not how fast the reaction will occur. Classical thermodynamics also assumes a continuous fluid, meaning that there are enough molecules of a substance to yield accurate values of thermodynamic variables like pressure and temperature. As such, classical thermodynamics is generally inappropriate for use with microscopic-level molecular charge transfer processes and electrochemical reactions. 

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