Thermodynamic state space

For systems having not more than two degrees of freedom, we can define a State Space using one or two independent variables and one dependent variable as axes, completely analogous to our Variable Space for functions ( 2.2.2). Just as functions are represented by lines or surfaces in Variable Space, equilibrium systems are represented by lines or surfaces in State Space. 

V and T. Only equilibrium states are defined by equations of state. Thus the surface 

Clearly, Metastable Equilibrium Surfaces are completely analogous to Stable Equilibrium Surfaces. There is, however, one important difference between the two surfaces. Every metastable system, and corresponding metastable equilibrium surface, has at least one degree of freedom more than the corresponding stable system. To amplify what we mean by this, we will introduce the concept of constraints. In this discussion, we follow the ideas of Reiss (1965) very closely, though not his usage of some terms. 

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Another aspect of lattice models concerns the determination of phase behavior. As far as continuous models were concerned we emphasized already that an investigation of phase transitions in such models usually requires a mechanical representation of the relevant thermod3marnic potential in terms of one or more elements of the micrascopic stre.ss teusor. The existence of sucli a mechanical representation was linked inevitably to symmetry considerations in Section 1.6, where it was also pointed out that such a mechanical expression may not exist at all. In this case a determination of the thermodynamic potential requires thermodynamic integration along some suitable path in thermodynamic state space, which may turn out to be computationally demanding. 

The link to the molecular level of description is provided by statistical thermodynamics whore our focus in Chapter 2 will be on specialized statistical physical ensembles designed spc cifically few capturing features that make confined fluids distinct among other soft condensed matter systems. We develop statistical thermodynamics from a quantum-mechanical femndation, which has at its core the existence of a discrete spectrum of energj eigenstates of the Hamiltonian operator. However, we quickly turn to the classic limit of (quantum) statistical thermodynamics. The classic limit provides an adequate framework for the subsequent discussion because of the region of thermodynamic state space in which most confined fluids exist. 

In thermodynamic-scaling Monte Carlo (ThScMC) one estimates relative free energies like Eq. (2.7), not, however, simply between two states but among arbitrarily many states over a substantial region of the thermodynamic state space, within a single MC sampling run. 

The second condition may sometimes be satisfied automatically by the first. This will occur if the targeted states are chosen sufficiently dense in thermodynamic state space that there is strong overlap of the rN subregions corresponding to adjacent states. However, the relative breadths of the relevant distributions become narrower as the system size N increases, and the second condition becomes more demanding. 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodynamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic description. The transition from quantum to cla.s.sic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

In connection with the thermodynamic state of water in SAH, it is appropriate to consider one more question, i.e., their ability to accumulate water vapor contained in the atmosphere and in the space of soil pores. It is clear that this possibility is determined by the chemical potential balance of water in the gel and in the gaseous phase. In particular, in the case of saturated water vapor, the equilibrium swelling degree of SAH in contact with vapor should be the same as that of the gel immersed in water. However, even at a relative humidity of 99%, which corresponds to pF 4.13, SAH practically do not swell (w 3-3.5 g g1). In any case, the absorbed water will be unavailable for plants. Therefore, the only real possibility for SAH to absorb water is its preliminary condensation which can be attained through the presence of temperature gradients. 

It is a remarkable fact that properties (13.4a-c) are necessary and sufficient to give Euclidean geometry. In other words, if any rule can be found that associates a number (say, (X Y)) with each pair of abstract objects ( vectors X), Y)) of the manifold in a way that satisfies (13.4a-c), then the manifold is isomorphic to a corresponding Euclidean vector space. We introduced a rather unconventional rule for the scalar products (X Y) [recognizing that (13.4a-c) are guaranteed by the laws of thermodynamics] to construct the abstract Euclidean metric space Ms for an equilibrium state of a system S, characterized by a metric matrix M. This geometry allows the thermodynamic state description to be considerably simplified, as demonstrated in Chapters 9-12. 

Everything that we have done so far in this example is completely standard. The next step in which we identify the slow fundamental thermodynamic relation in the state space M2 (i.e., we illustrate the point (IV) (see (120))) is new. Having found the slow manifold Msiow in an analysis of the time evolution in Mi, we now find it from a thermodynamic potential. We look for the thermodynamic potential ip(q,p, e, //. q, e, v) so that the manifold Msiow arises as a solution to... 

The small number of variables needed for thermodynamic state description is certainly surprising from a microscopic molecular dynamic viewpoint. For the complete molecular-level description of an arbitrary state (phase-space configuration) of the order of 1023 particles, we should expect to require an enormously complex nonequilibrium function independent variables (i.e., positions rt and velocities r,-), time evolution until equilibrium is achieved, we find that a vastly simpler description is possible for the resulting equilibrium state state properties R, R2.i.e., for a pure substance,... 

It is necessary to consider the mechanics of a continuous medium to determine the thermodynamic state of a fluid. The properties of a fluid can be determined that are at rest relative to a reference frame or moving along with the fluid. Every nonequilibrium intensive parameter in a fluid changes in time and in space. 

Liouvillean space the expectation values become simple matrix elements, no longer a trace, and may even be viewed as thermal vacuum expectation values with respect to the given initial thermodynamic state. We can now restate Eq. (32) as a stationary action principle in this superspace ... 

State space, configuration space, phase space. An abstract space spanned by coordinate axes, one for each thermodynamic coordinate, on which a given point represents the numerical value of that coordinate. A hyperspace is then formed by a mutually orthogonal disposition of these axes about a common origin. 

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