Maximum Entropy - Minimum Energy

It is often stated that the maximum entropy principle implies at the same time a minimum energy. This is basically correct however, the arguments are often not clearly presented. 

Question 3.1. If we have dU = TdS +. .., how can the energy U tend to approach a minimum, when the entropy S wants to approach a maximum  

We start with a formal derivation of the equivalency of the minimum principle of energy and maximum principle of entropy and illustrate subsequently the principle with a few examples in back-breaking work. A more detailed treatment based on the concavity of entropy, dealing with thermodynamic potentials, can be found in the literature [16]. 

The last equality in Eq. (3.30) follows from Eq. (3.29). Forming the second derivative in Eq. (3.30) results in 

The mathematical tools to get these equations properly are detailed in Sect. 1.11.4. Assuming a positive temperature T U, X), it follows that the energy is a minimum. A comparison of the condition of maximum entropy and minimum energy reveals that both conditions are different only by the factor —T U, X). 

---

Thus, the aforementioned directionality arises from the combination of (A. 11) and (A. 13). All closed systems tend to the state of maximum entropy (minimum free energy), that is, the equilibrium. 

The second law also describes the equilibrium state of a system as one of maximum entropy and minimum free energy. For a system at constant temperature and pressure the equilibrium condition requires that the change in free energy is zero ... 

As we all know from thermodynamics, closed systems in equilibrium have minimum free energy and maximum entropy. If such a system were brought out of equilibrium, i.e. to a state with lower entropy and higher free energy, it would automatically decay to the state of equilibrium, and it would lose all information about its previous states. A system s tendency to return to equilibrium is given by its free energy. An example is a batch reaction that is run to completion. 

That this should be so is a corollary of the Second Law of Thermodynamics which is concerned essentially with probabilities, and with the tendency for ordered systems to become disordered a measure of the degree of disorder of a system being provided by its entropy, S. In seeking their most stable condition, systems tend towards minimum energy (actually enthalpy, H) and maximum entropy (disorder or randomness), a measure of their relative stability must thus embrace a compromise between H and S, and is provided by the Gibb s free energy, G, which is defined by,... 

The fact that each of the two variables, S and U may be expressed as a function of the other, indicates that the extremum principle could likewise be stated in terms of either entropy or energy. The alternative to the maximum entropy principle is a minimum energy principle, valid at constant entropy, as graphically illustrated for a one-component system in figure 1, below. 

Figure 1 Diagram to demonstrate that an equilibrium state, identified at point A, may be described as either a maximum entropy or a minimum energy state.

The surface BCDE represents a segment of the surface defined by the fundamental equation characteristic of a composite system with coordinate axes corresponding to the extensive parameters of all the subsystems. The plane Uo is a plane of constant internal energy that intersects the fundamental surface to produce a curve with extremum at A, corresponding to maximum entropy. Likewise So is a plane of constant entropy that produces a curve with extremum A that corresponds to minimum energy at equilibrium for the system of constant entropy. This relationship between maximum entropy... 

An extremum principle minimizes or maximizes a fundamental equation subject to certain constraints. For example, the principle of maximum entropy (dS)v = 0 and, (d2S)rj < 0, and the principle of minimum internal energy (dU)s = 0 and (d2U)s>0, are the fundamental principles of equilibrium, and can be associated with thermodynamic stability. The conditions of equilibrium can be established in terms of extensive parameters U and. S, or in terms of intensive parameters. Consider a composite system with two simple subsystems of A and B having a single species. Then the condition of equilibrium is... 

It reflects the following results of our theory (a) For each value Si for the entropy, stable equilibrium state Ag is the state of minimum energy (b) for each value of energy, stable equilibrium state Ag is the state of maximum entropy (c) because each stable equilibrium state is unique, the temperature (3E/9S)n g is uniquely defined at each point of EgAgAg and (d) the ground state... 

Nearly two hundred years ago Maupertius tried to show that the principle of least action was one which best exhibited the wisdom of the Creator, and ever since that time the fact that a great many natural processes exhibit maximum or minimum qualities has attracted the attention of natural philosophers. In dealing with the available energy of chemical and physical phenomena, for example, the chemist seeks to find those conditions which make the entropy a maximum, or the free energy a minimum, while if the problems are treated by the methods of energetics, Hamilton s principle ... 

Particularly related to thermodynamics is the second law. The second law states that at constant energy the entropy tends to reach a maximum in equilibrium. The reciprocal statement is that at constant enfropy the energy tends to reach a minimum in equilibrium. Often in thermodynamics the principle of minimum energy is deduced from the principle of maximum entropy via a thermodynamic process that is not conclusive, in that as the initial assumption of maximum entropy is violated. These methods originate to Gibbs. Other proofs run via graphical illustrations. 

The progress of a chemical reaction comprises variation of the positions of the atomic nuclei along a multidimensional potential hyperface. The path with minimum energy with respect to the other degrees of freedom is called the reaction coordinate. If the potential along this coordinate exhibits a maximum, it is called the transition state and its difference to the initial state is denoted as activation energy E. From the partition functions of the initial and transition states, the activation entropy AS is derived. Within the framework of transition state theory (TST) [1], the rate constant for the reaction is then given by... 

Gibbs stated the extremum principle in two versions In an isolated system the entropy tends to a maximum at constant energy, or alternatively, the energy tends to a minimum at constant entropy. 

The compromise between the minimum energy situation in which all counterions are as near as possible to the charged surface and the co-ions expelled away and the maximum entropy situation where all ions are homogeneously distributed in solution results in an ion distribution of minimum Gibbs energy. According to these principles. 

A further requirement, that of recovery of shape after deformation, is also provided by high molecular mobility and the fact that long-chain polymers have a preference to exist as a randomly coiled chain, which represents a minimum energy-state condition for the polymer. A move away from this state, such as occurs during mechanical deformation of the polymer, is only achieved by the input of energy, which creates a state of higher order in the thermodynamic sense, i.e., a reduction in the entropy (disorder) of the system. The return to maximum entropy is one of the key elements of a high elastic recovery [4, 5]. 

---