# Miscellania about Entropy, Energy, and Available Free Energy

## Abstract

**:**

## 1. Introduction

- Thermodynamics of irreversible processes with the objective to determine the fields of mass density, velocity and temperature of a body for given initial and boundary data. More often than not the actual determination of the fields requires complex numerical schemes, and it is rarely done without severe simplifying assumptions.
- Stability analysis, i.e. identification of available free energies which assume extrema at the end of possibly strongly irreversible processes under special boundary conditions–adiabatic ones, or isothermal, or isobaric, or non-moving boundaries.
- Thermostatics which describes quasi-static or reversible processes which are so slow that in a reasonable approximation they may be considered sequences of equilibria. This branch is largely adequate for the treatment of even fast moving engines; it is the subject of most of engineering thermodynamics.

## 2. Available Free Energies

#### 2.1. First and Second Laws

_{0}on the surface:

#### 2.2. Available Free Energies

_{0}; thermodynamicists love to say that the body is immersed in a heat bath of temperature T

_{0.}Indeed, in that case the heating $\dot{Q}$ may be eliminated between the first and second laws and we obtain:

_{ij}on the surface element dA moving with the velocity v

_{i}:

_{i}is the outer unit normal of the element dA.

- i)
- The surface is at rest, i.e. the volume is constant.
- ii)
- Only part of the surface is at rest. On the remaining part the stress is isotropic, i.e. t
_{ij}= -p_{0}δ_{ij}, with a constant and uniform surface pressure p_{0}(the best-known visualization of this circumstance is a vertical cylinder closed off at the top by a horizontal piston whose weight determines p_{0}). - iii)
- A part of the surface is fixed, another part is free of stresses, and a third part $\partial {V}_{0}$--horizontal (say), i.e. normal to the 3-direction – moves with the uniform vertical velocity $\frac{\mathrm{d}L}{\mathrm{d}t}$ under a constant load ${P}_{0}={\displaystyle \underset{\partial {V}_{0}}{\int}{t}_{3j}{n}_{j}\mathrm{d}A}$ (the common visualization is a vertical elastic rod of length L, fixed at the bottom and loaded by P
_{0}on the upper surface).

_{1}and (5)

_{2,3}are not the free energy and the free enthalpies, respectively, of equilibrium thermodynamics (also known as the Helmholtz and Gibbs free energies, respectively). Indeed, T

_{0}, p

_{0}, and P

_{0}are temperature, pressure, or load on the surface of the body. Inside the body the temperature and the stress may be strongly non-uniform and time-dependent fields. This point is conceptually important, – particularly with respect to temperature – but in the sequel of this article it plays no role, since we shall consider bodies with constant and uniform temperature.

_{0}V or P

_{0}L in (5)

_{2,3}– or the corresponding terms for different boundary conditions – may be interpreted as the energies E

_{L}of the loading device. We join them to U+E

_{pot}+E

_{kin}and define E = U+E

_{pot}+E

_{kin}+E

_{L}of body and loading device and write:

_{0}S → minimum in equilibrium

_{0}is so small that the entropic term in (6) can be neglected, the availability tends to a minimum, because the energy does. On the other hand, if T

_{0}is big, so that the energetic term in (6) may be neglected in comparison with the entropic one, the availability tends to a minimum, because the entropy tends to a maximum.

_{0}it is neither the energy that becomes minimal in equilibrium nor the entropy which becomes maximal; the two tendencies compete and often they find a compromise and minimize A, the availability.

_{0}, E, and S, the surface temperature is best understood: It determines the mean kinetic energy of the molecules of the body at the surface and may be considered as a measure for the intensity of their thermal motion. If the truth were known, the energy is the most mysterious one among the three quantities, but we have gotten used to it, particularly to the kinetic energy and the gravitational potential energy. Entropy still needs a definition in molecular terms and will then be crystal-clear.

#### 2.3. S = k ln W

_{1}, N

_{2},…N

_{P}}, where N

_{i}(i=1,2,…P) is the number of atoms on position i. By the rules of combinatorics the number of realizations of that distribution is:

^{−12}s by order of magnitude at normal temperatures. And the assumption is that each and every realization occurs just as frequently as any other one. This is known as the a priori assumption of equal probability and it seems to be the only reasonable unbiased assumption given the randomness of thermal motion. It follows that a distribution with few realizations occurs less often than a distribution with more realizations, and most often is the equi-distribution ${N}_{i}=\frac{N}{P}$, because – rather obviously, by (7)

_{1}– that is the distribution with most realizations. For example, let us start with a distribution in which all atoms are stacked – nice and orderly – on point 3 (say). In this case the number of realizations is equal to 1 and the entropy is zero by (7). But the thermal motion will quickly mess up this orderly stack and lead to a distribution with more realizations and – eventually, as equilibrium is approached – to the equi-distribution with most realizations. Such is the nature of entropic growth. By (7) and ${N}_{i}{}^{E}=\frac{N}{P}$ we obtain the equilibrium entropy as (here and frequently in the sequel we make use of the Stirling formula to replace factorials by the logarithmic function):

^{E}= NklnP

_{1}, ${\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.}$ is the smallest volume that can accommodate a single occupiable point. We might say that ${\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.}$ quantizes the volume and again it was Boltzmann who introduced this notion without, however, taking it seriously. In fact Boltzmann [3] considered “…. it needless to emphasize that (in writing P = αV) we are not concerned here with a real physical problem, ….. the assumption is nothing more than an auxiliary tool.”

_{0}, the boundary temperature):

#### 2.4. Boltzmann´s Controversies

## 3. Applications of the Availability Concept to Physics

#### 3.1. Planetary Atmospheres

_{0}S(H). If we do the calculation (the detailed calculation has been presented in [5]) for a particular case, we obtain E(H) and S(H) as shown in Figure 1: the energy is minimal at H = 0–as expected – and grows monotonically and the entropy tends to infinity as well as H grows. However, E(H) grows less steeply than S(H) so that the available free energy E(H)–T

_{0}S(H) has its minimum for H → ∞, cf. the fat curve in Figure 1. In other words, the entropy prevails and a planet with a stable atmosphere is not a valid proposition.

#### 3.2. Osmosis. Pfeffer Tube

^{2}as cross-section of the tube, and 20 °C – the graph A(H) = E(H)-T

_{0}S(H) implies that now neither energy nor entropy prevail. Both tendencies – the decay of energy and the growth of entropy – reach a compromise where A(H) has a minimum for H

^{E}a little less than 10 m. Thus a little less than half of the water is sucked into the tube (the calculation is again part of [5]). Of course, with more salt the tube may suck in all the water, so that there is no compromise and the entropy prevails.

#### 3.3. Rubber Molecule

#### 3.3.1. Entropy and gravitational potential energy of a rubber molecule

_{i}(i = 1,2,...P) is the number of links pointing into direction i. P is the total number of directions and it is considered finite. As was explained in Section 2.3 this entropy – or its counterpart (7) for a gas of atoms – makes good sense, if there is thermal motion of the links and if all realizations of a distribution {N

_{i}} are equally probable.

#### 3.3.2. A one-dimensional model

_{+}and N

_{-}. In that case we obtain for the entropy with q = N

_{-}/N:

^{E}is close to zero, while energy prevails for small T and the chain becomes straight with r

^{E}close to Nb.

^{E}that is essentially equivalent to (18), i.e. the M-and T-dependence of r

^{E}is quite similar. For more information about rubber – and the chain model – I refer the readers to the book [6].

## 4. Extrapolation to Genetics

#### 4.1. A Population of Cells and Its Entropy

_{A},N

_{a}} – with N

_{A}+N

_{a}= 2N – and we assume that every one of the $\frac{(2N)!}{{N}_{A}!{N}_{a}!}$ realizations of a distribution is equally probable in the course of the mutational process. That assumption is tantamount to saying that a distribution with many realizations is more probable than one with less realizations. In analogy to thermodynamics we may thus define an entropy of the population by [the factor k in (7) is now dropped, because it would be futile here]:

_{A}= N

_{a}= N, cf. Figure 5a.

#### 4.2. Selective “Energy”

_{a}. We represent E by a graph between N

_{a}= 0 and N

_{a}= 2N of the form shown in Figure 5b. We call E the selective energy. Its values may be visualized as lying in a potential well whose deepest point, i.e. the selectively most preferred state, occurs at N

_{a}= 2N, and the highest point, i.e. the least preferred state, lies at N

_{a}= 0. The shortfall of E from 2NΔ measures the selective advantage of the population, cf. Figure 5b.

#### 4.3. Selective Free Energy

_{a}= 2N, the selective energy must be dragged up against its tendency to decay. Under those circumstances we do not expect the entropy to become maximal, nor the selective energy to become minimal. The entropy will grow as far as it can under the constraint of an increasing selective energy.

_{C}= exp[S

_{C}] is the maximum number of realization to which W can rise under the constraint):

_{C}becomes maximal, because the entropy S does, as if there were no selection. On the other hand, if β is large, S

_{C}becomes maximal because E is minimal, as if there were no mutation. Therefore

^{1}/β may be considered as a measure for the intensity of mutation.

^{1}/β as characterizing the mutational intensity and instead of maximizing S

_{C}we minimize the selective free energy:

_{a}is shown in Figure 6.

## 5. Extrapolation to Sociology

#### 5.1. A Population of Hawks and Doves. Entropy

_{H}hawks and N

_{D}= N-N

_{H}doves. The number of realizations of the distribution {N

_{H},N

_{D}} is equal to $\frac{N!}{{N}_{H}!{N}_{D}!}$ and therefore the entropy of the population is:

_{H}is the fraction of hawks among the birds. Figure 7a shows the graph of the function S(z

_{H}) which has a maximum at z

_{H}=

^{1}/

_{2}.

_{H},N

_{D}} changes stochastically to one with more realizations until in the end, when all birds have given birth, the equi-distribution would statistically prevail, i.e. entropy would be maximal – unless there are “energetic” features.

#### 5.2. Contest. “Social Energy”

^{H}= z

_{H}g

^{HH}+(1-z

_{H}) g

^{HD}

g

^{D}= z

_{H}g

^{DH}+ (1-z

_{H}) g

^{DD}respectively

_{H}g

^{H}+ (1-z

_{H}) g

^{D}or, by (5.3)

g = z

_{H}

^{2}(g

^{HH}+ g

^{DD}− g

^{HD}− g

^{DH}) + z

_{H}(g

^{HD}+ g

^{DH}-2 g

^{DD}) + g

^{DD}

_{H}

^{2}2.3 - z

_{H}2.6 + 0.3

_{H}= 0 and z

_{H}= 1, cf. Figure 7b (dashed). We assume that – within the time of a generation – the contest strategy will force the population toward those maxima of gain, because higher gain may mean a higher chance of survival.

_{H}) upside down and obtain a “double-well potential” E(z

_{H}) = -Ng(z

_{H}), cf. Figure 7b (solid), with two minima at z

_{H}= 0 and z

_{H}= 1. We call E the social energy of the population and we may say that in the long run the contest favours pure populations, i.e. either hawks or doves, because their social energies are minimal. Actually we may suppose that the population splits apart into spatially separate colonies of pure hawks or pure doves, so that no interspecies conflict occurs. That should be the final state, if the entropic trend for mixing hawks and doves were ignored.

#### 5.3. Entropic Growth and Energetic Decay Combined

_{H},N

_{D}} cannot rise to the entropic maximum, because that trend competes with the trend toward an energetic minimum. For example, consider a population with N

_{H}smaller but nearly equal to N, its entropy increase is at first strong enough to pull the population out of the energetic minimum at N

_{H}= N, since, after all, the slope of entropy is infinite while the slope of energy is finite. So, there is an initial growth of entropy and energy, the latter being forced against its natural tendency. Therefore, as has been explained in Section 4.3, the entropy grows only as far as it can given the constraint of the concomitant growth of energy (the present case is richer than the one in Section 4, since the energy is not a linear function of z

_{H}, nor even a monotonic one). And, once again, the maximization under a constraint is done by use of a Lagrange multiplier. As in (21) we maximize the constrained entropy:

_{C}is largely due to a minimum of energy. And if β is small, the maximum of S

_{C}is due to a maximum of entropy. We may say that β measures the relative importance of energy and entropy in the approach to constrained equilibrium. Or, from what has been said,

^{1}/

_{β}may be taken as characterizing the rate of reproduction.

#### 5.4. Constrained Equilibria for Different Resource Values

_{C}in the form:

_{H}= 0. Therefore the maximum of S

_{C}lies a little distance to the left of the entropic maximum at z

_{H}= ½

_{.}For any given z

_{H}other than the maximizer, the constrained entropy S

_{C}does not have its equilibrium value so that there will be a slow drift toward the maximum in time.

_{C}and they correspond to dove-rich and hawk-rich populations. The entropy has given up its decisive role and the social energy dominates. With this we may envisage a situation in which the population is split into dove-rich and hawk-rich colonies and where the population as a whole has a constrained entropy lying on the concavifying straight line which is represented by the common tangent of the concave parts of S

_{C}(z

_{H}) cf. Figure 9. In that case the value of S

_{C}is higher than the value of the homogeneous population. We have:

## References

- Clausius, R. Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie. Poggendorffs Annalen der Physik
**1865**, 125, 353–400. [Google Scholar] [CrossRef] - Loschmidt, J. Über den Zustand des Wärmegleichgewichts eines Systems von Körpern mit Rücksicht auf die Schwerkraft. Sitzungsberichte der Akademie der Wissenschaften Wien, Abteilung 2
**1876**. [Google Scholar] - Boltzmann, L. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte der Akademie der Wissenschaften Wien, Abteilung 2
**1872**, 66, 275–370. [Google Scholar] - Gibbs, J.W. On the equilibrium of heterogeneous substances. Trans. Conn. Acad. Sci.
**1876**(part 1),**1878**(part 2). [CrossRef] - Müller, I.; Weiss, W. Entropy and Energy–A Universal Competition; Springer: Heidelberg, Germany, 2005. [Google Scholar]
- Müller, I. Thermodynamics and evolutionary genetics. Continuum. Mech. Thermodyn.
**2010**, 22, 189–201. [Google Scholar] [CrossRef] - Müller, I.; Strehlow, P. Rubber and Rubber Balloons (Lecture Notes in Physics 637); Springer: Heidelberg, Germany, 2004. [Google Scholar]
- Maynard-Smith, J.; Price, G.R. The logic of animal conflict. Nature
**1973**, 246, 15–18. [Google Scholar] [CrossRef] - Straffin, P.D. Game Theory and Strategy; The Mathematical Association of America: Washington, DC, USA, 1993. [Google Scholar]
- Müller, I. A History of Thermodynamics–the Doctrine of Energy and Entropy; Springer: Heidelberg, Germany, 2007. [Google Scholar]

**Figure 1.**Energy, entropy and available free energy of a planetary atmosphere in a spherical shell of thickness H (M

_{A}-mass of the atmosphere). Continuous graphs refer to β = 1, dashed ones to β = 8.

**Figure 3.**Energy, entropy and available free energy as functions of H in the Pfeffer tube. (data given in the text).

**Figure 5.**a. Entropy as a function of $\frac{{N}_{a}}{2N}=q$, b. “Potential well” for the population.

© 2010 by the authors; licensee MDPI, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Müller, I.
Miscellania about Entropy, Energy, and Available Free Energy. *Symmetry* **2010**, *2*, 916-934.
https://doi.org/10.3390/sym2020916

**AMA Style**

Müller I.
Miscellania about Entropy, Energy, and Available Free Energy. *Symmetry*. 2010; 2(2):916-934.
https://doi.org/10.3390/sym2020916

**Chicago/Turabian Style**

Müller, Ingo.
2010. "Miscellania about Entropy, Energy, and Available Free Energy" *Symmetry* 2, no. 2: 916-934.
https://doi.org/10.3390/sym2020916