Macrostate probability, defined

We overcome this problem by using Metropolis Monte Carlo (MMC) sampling [42] within each macrostate. In contrast with Monte Carlo integration, which uses a fixed kernel to select each sample point Q(i), in MMC a trial sample conformation Q(iy is selected with a transition distribution centered about the previous accepted sample conformation 2, 1). We accept CI(,) with probability pMMC, which is determined as in standard MMC procedure except that the potential is modified to account for the restriction of sampling to the macrostate region defined by w ... 

This is a law about the equilibrium state, when macroscopic change has ceased it is the state, according to the law, of maximum entropy. It is not really a law about nonequilibrium per se, not in any quantitative sense, although the law does introduce the notion of a nonequilibrium state constrained with respect to structure. By implication, entropy is perfectly well defined in such a nonequilibrium macrostate (otherwise, how could it increase ), and this constrained entropy is less than the equilibrium entropy. Entropy itself is left undefined by the Second Law, and it was only later that Boltzmann provided the physical interpretation of entropy as the number of molecular configurations in a macrostate. This gave birth to his probability distribution and hence to equilibrium statistical mechanics. 

Remark. Instability and bistability are defined as properties of the macroscopic equation. The effect of the fluctuations is merely to make the system decide to go to one or the other macroscopically stable point. Similarly the Taylor instability and the Benard cells are consequences of the macroscopic hydrodynamic equations. ) Fluctuations merely make the choice between different, equally possible macrostates, and, in these examples, determine the location of the vortices or of the cells in space. (In practice they are often overruled by extraneous influences, such as the presence of a boundary.) Statements that fluctuations shift or destroy the bistability are obscure, because on the mesoscopic level there is no sharp separation between stable and unstable systems. Some authors call a mesostate (i.e., a probability distribution P) bistable when P has two maxima, however flat. This does not correspond to any observable fact, however, unless the maxima are well-separated peaks, which can each be related to separate macrostates, as in (1.1). 

It would be hard to write a definitive account of the development of extended sampling methods we will not attempt to do so. The seminal ideas are probably correctly attributed to Torrie and Valleau [22], who coined the terminology umbrella sampling. The huge literature of subsequent advances and rediscoveries may be rationalized a little by dividing it into two, according to how the macrostates to be weighted are defined. 

Again, the term macrostate refers to the thermodynamic state of the composite system, defined by the variables N, E, and V2, E2. A more probable macrostate will be one that corresponds to more possible microstates... 

Where k is Boltzmann constant and Q the weight of configuration (thermodynamic probability). Q is defined as the number of microstates forming a macrostate. With... 

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