Macrostate labels

The generic case is a subsystem with phase function x(T) that can be exchanged with a reservoir that imposes a thermodynamic force Xr. (The circumflex denoting a function of phase space will usually be dropped, since the argument T distinguishes the function from the macrostate label x.) This case includes the standard equilibrium systems as well as nonequilibrium systems in steady flux. The probability of a state T is the exponential of the associated entropy, which is the total entropy. However, as usual it is assumed (it can be shown) [9] that the... 

Since the notation does not always make it apparent, we should note that a and c play operationally similar roles as macrostate labels Together they identify the distinct sets of equilibrium macroscopic properties emerging from the equilibrium distribution, Eq. (1). For some purposes we will find it useful to concatenate the two labels into a single grand macrostate label... 

Throughout this section we use the notation introduced in Eq. (7) in which the phase label is absorbed into an extended macrostate label CC. The path defined by the set of macrostates f may (and sometimes will) extend from one phase to another. 

Figure 1. Temperature-dependent macrostate dissection of a two-dimensional potential-energy landscape, (a) Potential V as a function of two coordinates, (b) Gibbs-Boltzmann distribution p at low (left), medium (middle), and high (right) temperatures, (c) Corresponding p at each temperature constructed from solutions to the characteristic packet equations, (d) Characteristic packet solution parameters R° and 0 for each macrostate (labeled with indices a, / , 6, y, and s). (e) Trajectory diagram of macrostate conformational free energies Fa as a function of temperature. (Reproduced from Church et al. [17] with permission obtained.)...

Macrostates are collections of microstates [9], which is to say that they are volumes of phase space on which certain phase functions have specified values. The current macrostate of the system gives its structure. Examples are the position or velocity of a Brownian particle, the moments of energy or density, their rates of change, the progress of a chemical reaction, a reaction rate, and so on. Let x label the macrostates of interest, and let x(r) be the associated phase function. The first entropy of the macrostate is... 

When there are more macroscopic observables B,C,... the process can be continued. The end result is a collection of coarse-grained observables //, A, 5, C, ..., which all commute with one another. The Hilbert space H is decomposed in linear subspaces that are common eigenspaces of these observables. We shall call these subspaces phase cells and indicate them with a single label J. They correspond to definite values of the coarse-grained variables, which we shall now denote by Ej, AJy BJy Cj,. These phase cells are the macrostates. 

A macrostate of a macromolecule can be described with the help of the end-to-end distance R. To give a more detailed description of the macromolecule, one should use a method introduced by the pioneering work reported by Kargin and Slonimskii (1948) and by Rouse (1953), whereby the macromolecule is divided into N subchains of length M/N. One can consider the ends of the macromolecule and the points, at which the subchains join to form the entire chain, as a particles (the beads), labelled 0 to N respectively, and their positions will be represented by r°, r1,..., rN. 

Figure 2 Protonation microequilibria of cetirizine. Arrows are labeled by microconstants measured experimentallyProtonation microstates are labeled by their relative contributions (probabilities) to the respective parent macrostate... 

Now we ask the following question for each state how many microstates give rise to the same macrostate To keep track of the microstates we label the atoms 1-4. Although they have different numbered labels, since the atoms are all the same, there is externally no difference between them. For states A and B, only one microstate results in the specified macrostate—atoms 1-4 on the left side or the right side, respectively. 

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