Constraints of Compound Systems: Prerequisites for Thermodynamic Modeling Based on Shannon Entropy

Abstract

Thermodynamic modeling of extensive systems usually implicitly assumes the additivity of entropy. Furthermore, if this modeling is based on the concept of Shannon entropy, additivity of the latter function must also be guaranteed. In this case, the constituents of a thermodynamic system are treated as subsystems of a compound system, and the Shannon entropy of the compound system must be subjected to constrained maximization. The scope of this paper is to clarify prerequisites for applying the concept of Shannon entropy and the maximum entropy principle to thermodynamic modeling of extensive systems. This is accomplished by investigating how the constraints of the compound system have to depend on mean values of the subsystems in order to ensure additivity. Two examples illustrate the basic ideas behind this approach, comprising the ideal gas model and condensed phase lattice systems as limiting cases of fluid phases. The paper is the first step towards developing a new approach for modeling interacting systems using the concept of Shannon entropy.

Discrete Modeling Approach as a Basis of Excess Gibbs-Energy Models for Chemical Engineering Applications

“…With this construction in mind, the Shannon entropy of the simple cubic lattice can be written as where p b · c · d represents the probability of the neighborhood into which site A is placed, and p a | b · c · d is the probability that a is inserted into the b · c · d neighborhood. After this insertion, the resulting a · b · c · d cluster represents a statistically independent subsystem of the lattice. Consequently, the total entropy of the system is calculated by summations over all respective a , b , c , and d .…”

“…Discrete modeling was introduced as a novel approach that uses the concept of Shannon entropy to develop thermodynamic models that can describe fluid-phase behavior. − Focusing on strongly interacting condensed-phase systems, a preceding paper addressed the application of discrete modeling to lattice systems of finite size, which were represented by a distribution of discrete local compositions, synonymous with discrete energy classes. This first step used the same, coarse-grained level of discretization as is used in classical statistical thermodynamics in terms of its partition functions, yet it avoids a priori averaging of local compositions and limitation to systems of infinite size .…”

“…Strongly interacting molecules positioned on the sites of a lattice are not statistically independent, and thus the molecules themselves cannot be seen as independent subsystems in the context of information theory. 2 To account for this, various approximations have been proposed in the previous literature.…”

Cluster-Based Thermodynamics of Interacting Dice in a Lattice

“…As a continuation of our previous work [ 38 , 39 , 40 , 41 ], in this paper, the basic cluster approach proposed by Vinograd [ 36 , 37 ] is extended to two-component systems of six-sided dice in a simple cubic lattice.…”

Discrete Modeling of Lattice Systems: The Concept of Shannon Entropy Applied to Strongly Interacting Systems

“…The method of discrete modeling was introduced in two previous papers as a novel approach to incorporate a more detailed picture of molecular conditions in thermodynamic models. , As a proof of concept, the thermal and caloric equations of state, the heat capacity, and the Maxwell–Boltzmann distribution of energies for ideal gas were properly derived on the basis of the discretized states of individual molecules.…”