Evolutionary entropy and the Second Law of Thermodynamics
aa r X i v : . [ q - b i o . P E ] M a y Evolutionary entropy and the Second Law of ThermodynamicsLloyd A. Demetrius Christian Wolf
Dept. of Organismic and Department ofEvolutionary Biology MathematicsHarvard University The City Colleges of New YorkCambridge, Mass. 02138, U.S.A. New York, 10031. U.S.A.May 20, 2020 bstract
The dynamics of molecular collisions in a macroscopic body are encoded by theparameter
Thermodynamic entropy — a statistical measure of the number of molec-ular configurations that correspond to a given macrostate. Directionality in the flowof energy in macroscopic bodies is described by the Second Law of Thermodynam-ics: In isolated systems, that is systems closed to the input of energy and matter,thermodynamic entropy increases.The dynamics of the lower level interactions in populations of replicating organismsis encoded by the parameter
Evolutionary entropy , a statistical measure which describesthe number and diversity of metabolic cycles in a population of replicating organisms.Directionality in the transformation of energy in populations of organisms is describedby the Fundamental Theorem of Evolution: In systems open to the input of energy andmatter, Evolutionary entropy increases, when the energy source is scarce and diverse,and decreases when the energy source is abundant and singular.This article shows that when ρ →
0, and N → ∞ , where ρ is the production rate ofthe external energy source, and N denote the number of replicating units, evolution-ary entropy, an organized state of energy; and thermodynamic entropy, a randomizedstate of energy, coincide. Accordingly, the Fundamental Theorem of Evolution, is ageneralization of the Second Law of Thermodynamics. Introduction
Most changes in the macroscopic properties of physical systems — the universe of solids,liquids and gases, and the macroscopic behavior of biological systems — the world of macro-molecules and cells, seem to be highly irregular, chaotic and unpredictable. There existhowever, processes in both physical and biological systems which manifest predictable irre-versible changes.These changes are characterized by the flow and transformation of energy. In physical sys-tems, the irreversibility is generated by the dynamics of molecular collisions in aggregates ofinanimate matter. This process is encoded by the parameter, thermodynamic entropy, a sta-tistical measure of the number of molecular configurations that correspond to a macroscopicstate. Directionality in the flow of energy is described by the Second Law of Thermodynam-ics: Thermodynamic entropy increases in isolated systems, that is, systems that are closedto the input of energy and matter [2], [3].Irreversibility in biological systems is depicted in terms of the transformation of energy froman external source to a population of replicating organisms. The transformation is drivenby a variation-selection process which is encoded by the statistical parameter, evolutionaryentropy, a measure of the number and diversity of metabolic cycles in a population of repli-cating organisms [9], [10]. Directionality in this process of energy transformation is expressedby the Fundamental Theorem of Evolution: Evolutionary entropy increases when the energysource is limited and diverse, and decreases when the energy source is abundant and singular.Thermodynamic entropy and evolutionary entropy are statistical measures of microscopicorganization in aggregates of inanimate matter, and populations of replicating organisms,respectively.Thermodynamic entropy describes the extent to which energy is spread and shared amongthe molecules that comprise the macroscopic body. A solid has small entropy since themolecules occupy fixed positions within the macroscopic body. A gas has large entropy sincethe molecules are free to move around. Thermodynamic entropy can be considered a measureof positional disorder, a randomized state of energy.Evolutionary entropy describes the multiplicity of cyclic pathways of energy flow betweenthe different metabolic states in a population of replicating organisms. A semelparous pop-ulation (annual plants) consists of organisms that reproduce at a single stage in their life3ycle. Semelparous populations have entropy zero. The energy transformation from birthto reproduction proceeds by a single pathway. An iteroparous population (perennial plants)consists of organisms that can reproduce at several distinct stages in their life-cycle. En-ergy transformation in perennials is described by a multiplicity of reproductive stages. Aniteroparous population has positive entropy. Evolutionary entropy can be considered a mea-sure of temporal order, and organized state of energy.In this article we will describe an analytic relation between these two statistical measuresof positional disorder, and temporal organization, respectively. Our analysis is based on theobservation that a population of replicating organisms is an open system which maintainsits viability by appropriating energy from an external energy source and transforming thischemical energy into biomass. We will show that when ρ → N → ∞ , where ρ isthe production rate of the energy source, and N the population size, then the Evolutionaryentropy and Thermodynamic entropy coincide.We will appeal to the relation between Thermodynamic entropy and Evolutionary entropyto show that the Fundamental Theorem of evolution is a generalization of the Second Lawof Thermodynamics.The relation between the Second Law of Thermodynamics and the processes which underliethe development and the evolution of living organisms, has been addressed by scientists fromdiverse disciplines, e.g., [12], [13], [18], [21], [25].Prominent investigations in the Physical and Chemical Sciences include Schr¨odinger [21], whointroduced the term negative entropy to describe the process whereby organisms maintaintheir ordered states by appropriating chemical energy from the environment, and Prigogine[18], who proposed the notion of entropy production to characterize the mechanism biologicalsystems exploit to maintain coherent behavior of steady states far from thermodynamicequilibrium.Investigators from Evolutionary genetics, have addressed the relation between the SecondLaw and the evolution of living organisms by proposing various measures of Darwinian fitnessas analogues of the Boltzmann entropy. The proposition of the Malthusian parameter as ameasure of Darwinian fitness, see Fisher [13], is one of the most influential efforts to relatethe Second Law with evolutionary dynamics. In Section 6 we reevaluate these investigationswithin the framework of evolutionary entropy and its relation with the Second Law.4his article is organized as follows: Section 2 provides a succinct account of the processwhich underlies energy flow in aggregates of inanimate matter. Section 3 gives an accountof energy transformation in populations of organisms, and the origin of the concept Evolu-tionary entropy. Section 4 presents the analysis of a mathematical model which forms theframework for the Fundamental Theorem of evolution. Finally, the relation between Thermo-dynamic entropy and Evolutionary entropy is analyzed in Section 5. This relation providesthe analytical basis for the claim (Section 6) that the Second Law of Thermodynamics is thelimiting case of the Fundamental Theorem of Evolution. The main example of directionality in physical systems is the flow of heat between bodiesat different temperatures. Empirical studies and experimental observations led Clausius tothe proposition that the flow of energy as heat from a body at higher temperature to one ata lower temperature is a universal phenomenon. The flow is spontaneous in the sense thatthe process does not require the action of some kind of work, mechanical or electrical. Therealization that the reverse process, namely the transfer of heat from a cold body to a hotterbody is not spontaneous, constitutes an asymmetry in Nature.
Clausius appealed to mathematical arguments to show that this asymmetry entails the ex-istence of a property of matter which he called entropy.
The Clausius entropy admits ananalytic description, namely(1) dS c = dQT Here, dQ is the small amount of heat added to a body with temperature T . The quantity dS c represents the increase in entropy.Material aggregates such as a gas are composed of molecules which move according to thelaws of classical mechanics. The issue of reconciling the time symmetric description ofmolecular dynamics with the time asymmetric description of the evolution of a macroscopicsystem emerged as a fundamental problem as soon as Clausius’ discovery was recognized.The problem was ultimately resolved by Boltzmann.5he model proposed by Boltzmann was based on the microscopic representation of theClausius entropy. Boltzmann’s analysis was based on the quantity(2) S = k b log W. Here W denotes the number of microscopic configurations compatible with a given macrostate,and k b , a constant, called Boltzmann’s constant.If N denotes the number of molecules in the macroscopic body, then the number of config-urations associated with a given macroscopic state is given by(3) W = (cid:18) N ! n ! n ! n ! . . . (cid:19) . This follows from the fact that W is the number of ways of distributing the N molecules sothat n are in state 1, n molecules in state 2 and so on.The Boltzmann entropy of a solid and a gas are illustrated in Fig. 1 (a) and Fig. 1. (b),respectively. Solid GasFig. 1. (a), 1 (b)The molecules in a solid occupy a fixed position in the macroscopic body. The entropy S ascomputed in Eq. (2) is small. The molecules in the gas are free to move around, the entropyS is large. The evolutionary dynamics of the Boltzmann entropy was analyzed under the followingassumptions:(a) The molecules of the gas, which is assumed to be confined in a container, move ran-domly. 6b) The velocities of the molecules are randomly distributed .(c) The molecules collide with each other and velocities change after collision.Boltzmann showed that if the number of molecules in the system is immensely large, andthe system evolves in an isolated environment, the quantity S will increase to an equilibriumstate.In systems which are isolated, that is closed to the input of energy and matter, the analysisof the dynamics of the interacting molecules shows that S the degree of energy spreadingand sharing among the microscopic storage modes of the system, satisfies the principle(4) ∆ S ≥ . Since the Boltzmann entropy, S , is identical to the Clausius entropy defined in (1), therelation (4) can be interpreted as a molecular dynamical explanation of the asymmetry ofthe flow of heat energy.The evolution of an isolated macroscopic system evolving in time, is exemplified by themacroscopic diversity profile of a fluid in the three frames in Fig. 2. The dots in the figurerepresent the density variable of the fluid at different times during the evolution of thesystem. The evolution can be considered as the flow of heat energy. The left half of thesystem in Fig. 2 (a) is hotter than the right half. In Fig. 2 (c) the temperature is uniform.(a) (b) (c)Fig. (2) The macroscopic density profile of an isolated system at three different times. Heat is the simplest and most frequently used medium by which energy is transformed inaggregates of inanimate matter: solids, liquid and gases.. However, there is a fundamental7estriction which exists on the conversion of thermal energy into work. This restriction isexpressed by the relation(5) ω = q (cid:18) T − T T (cid:19) . Here ω is the maximal work derived and q is the absorbed heat, and T and T are theabsolute temperatures of the material bodies between which the heat passes.The relation (5) has significant implications in the study of energy transformations in bi-ological systems: There is almost no temperature differential between the cells in a tissueor between the tissues in an organism. This implies that in living matter, thermal energycannot be effectively transformed into work.Living organisms are essentially isothermal chemical machines, Lehninger (1965) [15]. Thechemical components of these machines are not in thermodynamic equilibrium but in adynamic steady state:The critical parameter in the dynamics of energy transformation in living organisms is nottemperature, the mean kinetic energy of the molecules in a macroscopic body, but cycle time,the mean turn over time of the metabolic entities that comprise the population. Living organisms differ from material aggregates not only in terms of their isothermal char-acter, but also in terms of their interaction with the external environment.Organisms maintain their integrity by appropriating resources from the external environmentand transforming this chemical energy into metabolic energy and biomass. Organisms canbe classified in terms of various states: age; size; metabolic energy. A population, thefundamental unit of the evolutionary process, can be represented as a network or a directedgraph, Fig. (3). 8
Fig. (3) Population as a directed graph.The nodes of the graph correspond to the different states; the links between the nodesdescribe the interaction between the individuals that belong to the different states. Thegraph represents the transfer of energy between the individuals that define the differentstates.Evolutionary entropy, H , a concept which has its mathematical roots and its biologicalrationale in the ergodic theory of dynamical systems [10], can be expressed as(6) H = e S e T .
The quantity e S is called population entropy and e T is the mean cycle time [9]. These quantitiescan be formally described by considering the set of nodes of the graph, denoted by X =(1 , , . . . , d ) . We now fix an arbitrary vertex a ∈ X and consider the set X ∗ of all directedpaths which start at a , end at a , and do not visit a in the middle. An element ˜ a in X ∗ iswritten as a → β → β → · · · → β n − → a which we denote by(7) ˜ a = [ aβ β · · · β n − a ] . We define φ ˜ a = φ aβ φ β β · · · φ β n − a . The quantities e S and e T are given by(8) e S = − X ˜ a ∈ X ∗ φ ˜ a log φ ˜ a e T = X ˜ a ∈ X ∗ | ˜ a | φ ˜ a , where | ˜ a | is the length of the path ˜ a .In the case of a population in which the individuals are divided into age–classes, the popu-lation entropy e S is given by(10) e S = − X j p j log p j Here p j is the probability that the mother of a randomly chosen new born belongs to theage–class ( j ). The life-cycle of a population of annual plants and the population of perennialplants are described in Fig. 4(a) and Fig. 4(b) respectively. Fig. 4(a)
Fig. 4(b)The population entropy of annuals is e S = 0. The energy flow in the population is describedby a unique cycle. The perennials, as shown in Fig. 4(b), reproduce at several distinctstages in the life cycle. Energy transformation within the population is described by severaldistinct metabolic cycles.The recurrence time in an age–structured population is the generation time, the mean ageof mothers at the birth of their offspring. This is given by(11) e T = X j jp j Organisms are metabolic entities endowed with a genome which, together with the envi-ronment, regulates individual behavior. The flow of energy in a population of replicatingorganisms is modulated by adaptive, dynamical processes. These processes, which have no10ounterpart in physical systems, derive from the intrinsic instability of organic molecules,such as DNA, RNA and proteins, and competition between the organisms for an externalenergy source. These processes are:(i)
Mutation:
Random changes in DNA; the genetic endowment of the organisms.The effect of mutation on the genome will be the emergence of a population consistingof two types of organisms: an ancestral type of large size, a variant of small size withthe genetic endowment of the mutant.(ii)
Selection:
Competition between the ancestral type with entropy H and the varianttype with entropy H ∗ will result in a change in the genetic and phenotypic compositionof the population.The directional change, ∆ H in the evolutionary entropy is described by the relation, [8], [9](12) (cid:16) − Φ + γM (cid:17) ∆ H > . The parameters Φ and γ , are macroscopic parameters; functions of the microlocal variableswhich describe the interaction between the organisms. These parameters are correlated withthe resource endowment of the environment.The parameter Φ corresponds to the resource amplitude:(i) Φ <
0: Scarcity; Φ >
0, abundance.The parameter γ corresponds to the resource composition.(ii) γ <
0: singular; γ > M denotes the population size, whereas ∆ H = H ∗ − H , H and H ∗ denotesthe evolutionary entropy of the incumbent and the variant population respectively.The relation (12) is a derivative of the Entropic Selection Principle: The outcome of competi-tion between an incumbent population and a variant is contingent on the resource endowmentand is predicted by Evolutionary entropy.The principle can be graphically exemplified by the profiles of a variant and and incumbentpopulation. Fig. (5) and (6) describe the changes in profile due to the introduction of a11utant which ultimately replaces the ancestral type. Fig. (5) describes the situation wherethe resource endowment is limited and diverse. Fig. (6) represents the condition where theresource endowment is abundant and singular.(a) (b) (c) (d)Fig. (5) Evolution under limited, diverse resources: ∆ H >
H <
The analytic basis for the directionality principle described in [9] will be reviewed in thissection. We refer to [8] and [9] for the detailed description of the concepts and mathematicalarguments that underlie Eq. (12).The Darwinian theory of evolution provides a necessary and sufficient mechanism for theadaptation of a population to its environment. The critical elements of the theory areinherent in three principles, see Levins and Lewontin [16] and Demetrius and Gundlach [11].(i) Physiological, behavioral and morphological traits vary among the members of a pop-ulation (
Variation )(ii) The phenotypic traits are partly heritable: Descendants in a lineage will have traitssimilar to their ancestors (
Heritability ).(iii) Different variants have different capacities to appropriate resources from the externalenvironment and to convert these resources into metabolic energy, and the demographiccurrency of survivorship and reproduction (
Natural Selection )These principles entail that the frequency of different types in a population will change dueto the continued generation of new variants, and selection against those types who are lesseffective in dealing with the exigences of the environment.The dynamical system which characterizes these three principles can be described as a two-step process: 13a)
Mutation;
Random changes in the genetic endowment of a small subset of theincumbent population.These changes will result in the incidence of two types — the incumbent, endowed withthe ancestral genotype, and the variant with a mutant inheritance.(b)
Selection:
Competition between the incumbent and the variant types for the resourcesof the external environment.Directionality Theory is the study of the dynamical system generated by the process ofmutation and natural selection. The analysis assumes that the fundamental unit of the evo-lutionary process is a population. This biological object is an aggregate of replicating units.The evolutionary argument applies to units at different hierarchical levels: macromolecules,cells, organisms.Formally, a population can be described by a directed graph, denoted by G , see Fig. (3)The nodes of the graph represent the replicators. The links between the nodes describe theinteraction between the different agents.At steady state, the population process, depicted as a directed graph, can be represented asan abstract dynamical system, defined in terms of the following elements:(i) Ω: The set of genealogies, that is the set of sequences generated by the interactionbetween the elements(ii) The probability measure µ . This parameter describes the frequency distribution of thegenealogies.(iii) The shift operator σ defined by σ : ( x k ) k ( x k +1 ) k . The element x = ( · · · x − x − x x x · · · ) is a genealogy.The probability measure µ described in (ii) is the steady state, the Gibbs measure inducedby the potential function φ : Ω → R . 14 .1 Dynamical systems and Evolutionary Entropy We consider the population as an abstract dynamical system of the form (Ω , µ, σ ), where σ denote the shift map on Ω. Two measure preserving transformations (Ω , µ, σ ), (Ω ∗ , µ ∗ , σ ∗ )are said to be isomorphic if there is a one–to–one correspondence between all (but a setof measure zero) of the points in each measure space, so that corresponding points aretransformed in the same way.The dynamical entropy, the Kolmogorov–Sinai entropy, constitutes an isomorphism invariantof measure preserving transformations. This mathematical object is thus a fundamentalstatistical invariant of the dynamical system.The work initiated in Demetrius [10] exploited the notion of isomorphism invariant of measurepreserving transformations to develop an intrinsic and fundamental property of the popula-tion dynamics. This quantity is called evolutionary entropy. The term ”entropy” refers tothe mathematical origin of the concept: in particular its significance in the classification ofmeasure preserving transformations. The term ”evolutionary” reflects the biological rootsof the concept. The statistical measure predicts the outcome of competition between anincumbent population and a variant type, and hence constitutes a quantitative measure ofDarwinian fitness [10].The concept was originally introduced in the study of the evolutionary dynamics of demo-graphic networks. These systems can also be described in terms of a directed graph. Thenodes of the graph correspond to age classes, the interaction between the nodes representsthe flow of energy from one age class to another. This energy flow is in terms of survivor-ship from one age-class to the next; and reproduction, an energy flow from the age-class ofreproductives to the age class of newborns. These processes can be represented in termsof an abstract dynamical system (Ω , µ, φ ). The function φ : Ω → R which describes thesurvivorship and reproduction schedule is locally constant [10]. The evolutionary entropy H , in these classes of models can be expressed in the form(13) H = e S e T .
Here e S is a measure of the uncertainty in the age of the model of a randomly chosen newbornand e T is generation time. 15here exist many biological systems in which the individual elements can be parameterizedin terms of a finite number of classes. These include the demographic models we will discussin this article. In these systems, the phase space Ω can be modeled by a subshift of finitetype with a general transition matrix A and a potential function φ : Ω → R that is locallyconstant. This locally constant condition excludes many phenomena of scientific interest,for example bioenergetic processes which exist in metabolic reactions, and the interactionwhich describe the exchange of non-material resources in social networks. We now develop a general theory for the evolutionary entropy for continuous potentials,to accommodate these processes. We restrict our representation to the introduction of therelevant objects and results, see e.g. [1],[4],[14],[17],[19],[24].We will describe as before, the population as a mathematical object – a directed graphas depicted in Fig. (4). The nodes of the graph represent the states, namely groups ofindividuals of a given age or size, as in the analysis of demographic networks [10], or groupsdefined in terms of their behavior or social norms, as in the analysis of social networks . Thelinks between the nodes describe the transfer of energy between the various states.Let X = { , . . . , d } and Y = ∞ Y n = −∞ X n where X n = X . Let A = ( a ij ) ≤ i,j ≤ d be a d × d -matrix with entrees in { , } . Let(14) Ω = Ω A = { x ∈ Y : a x i x i +1 = 1 } and let σ : Ω → Ω denote the shift operator on Ω. We say (Ω , σ ) is a subshift of finite typewith transition matrix A . Let φ : Ω → R be a continuous potential.Our goal is to establish a formula for the Evolutionary entropy H evol ( φ ) (which we will simplydenote by H ) in terms of a quantity that determines the diversity in pathways of energy flowdenoted by e S , and the mean cycle time e T . 16et M denote the set of all σ -invariant Borel probability measures on Ω, and let M E ⊂ M denote the subset of ergodic measures, see e.g. [24] for the definitions.Given n ≥ n -tuple τ = τ · · · τ n − ∈ X n is Ω-admissible provided that A ab = 1for all pairs of consecutive elements ab in τ . We denote by L n Ω the set of all A -admissibletuples of length n . Given τ ∈ L n Ω we denote by[ τ ] = { x ∈ Ω : x = τ , . . . , x n − = τ n − } the cylinder of length n generated by τ . Next we introduce the growth rate parameter of the potential φ . Given n ≥ n -th partition function Z n ( φ ) at φ by(15) Z n ( φ ) = X τ ∈L n Ω exp sup x ∈ [ τ ] S n φ ( x ) ! , where(16) S n φ ( x ) = n − X k =0 φ ( σ k ( x ))denotes statistical sum of length n of x . We define the growth parameter of φ (with respectto the shift map σ ) by(17) r ( φ ) = lim n →∞ n log Z n ( φ ) = inf (cid:26) n log Z n ( φ ) : n ≥ (cid:27) . Moreover, h top ( σ ) = r (0) denotes the topological entropy of σ . Recall that h top ( σ ) = log λ where λ is the spectral radius of the transition matrix A . Given µ ∈ M we denote by h µ ( σ )the Kolmogorov-Sinai entropy of the measure µ given by(18) h µ ( σ ) = lim n →∞ − n X τ ∈L n Ω µ ([ τ ]) log µ ([ τ ]) , Note that in the context of dynamical systems r ( φ ) is refered to as the topological pressure P top ( φ ) ofthe potential φ [17, 19, 24]. µ ([ τ ]) = 0 are omitted from the sum. The growth rate parameter satisfiesthe well-known variational principle (see e.g. [24]):(19) r ( φ ) = sup µ ∈ M (cid:18) h µ ( σ ) + Z φ dµ (cid:19) . If µ ∈ M achieves the supremum in (19), we call µ an equilibrium state of φ . We denote theset of equilibrium states of φ by ES ( φ ). Recall that ES ( φ ) is nonempty. We define the evolutionary entropy of φ by(20) H = H ( φ ) = sup { h µ ( σ ) : µ ∈ ES ( φ ) } . We observe that the set(21) R φ = {∫ φ dµ : µ ∈ ES ( φ ) } is a closed interval [ a φ , b φ ]. For µ ∈ ES ( φ ) it follows from (19) that(22) H = h µ ( σ )if and only if R φ dµ = a φ . In particular, the supremum in equation (20) is a maximum.Further, by using ergodic decompositions combined with a convexity argument we concludethat there exists at least one µ ∈ ES ( φ ) ∩ M E with H = h µ ( σ ). It turns out that for a”large” set of potentials φ the set R φ is a singleton. For example, if φ is H¨older continuousthen ES ( φ ) and R φ are singletons, see [5].Let (Ω , σ ) be a transitive subshift of finite type, and let φ : Ω → R be a continuous potential.Then the evolutionary entropy H of φ is given by the formula(23) H = e S e T , where e S is the limit of entropies f S n of countable Bernoulli shifts (see e.g. [20]) and T denotesthe mean cycle time of the system. In particular, if φ is locally constant, then e S is the entropyof a countable (finite or infinite) Bernoulli shift.18e briefly discuss the main differences between the theory of locally constant and continuouspotentials. The locally constant case has been successfully applied to situations where thepopulation can be partitioned into homogenous groups whose statistical properties are en-coded by a small number of parameters. A prototype of such an example is the demographicmodel which we will discuss in the Section 4.2. In this model, the partition is obtained bydividing the population into age-groups. We then associate with each of these groups theprobability of survivorship, and with the reproductive classes, the mean number of offspringsproduced by individuals in the group.However, in more heterogeneous situations the application of locally constant potentialsmay lead to less accurate and in some cases misleading predictions. This is for examplethe case in the evolution of social behavior of humans. Social preferences and dispositionsare continuous variables and individuals often invoke their memory of past interactions indeciding whether to cooperate or defect in social encounters [6]. These situations can not beaccurately modeled by partitioning.While the two mathematical approaches, the use of locally constant and continuous poten-tials, both originate in the mathematical thermodynamic formalism [19], they are based ondifferent methods. Namely, in the locally constant case, the input is a finite set of parameterswhich yield an explicit formula for the Evolutionary entropy. In contrast, for continuous po-tentials, the Evolutionary entropy is implicitly defined and thus there is no explicit formula.It should be noted however that it has been recently established that for a large class ofcontinuous potentials the Evolutionary entropy is computable in the sense of computableanalysis [7], i.e., it can be computed by a Turing machine (a computer program for ourpurposes) at any pre-described accuracy. Demographic networks whose behavior can be analyzed in terms of locally constant potentialsconstitute a well studied example where the evolutionary entropy concept can be explicitlycharacterized.The network can be described by the directed graph given by Fig. (7)19 m b b b b d m m m d Fig. (7)The graph represents a population whose members are classified in terms of age classes.The parameters ( b i ) denote the survivorship from age class ( i ) to ( i + 1). The parameters( m i ) denote the mean number offspring produced by individuals in age class ( i ).The interaction matrix is given by A = m m . . . . . . m d b . . . . . . ... . . . b d − The potential φ : Ω → R will be locally constant and described by φ ( x ) = log a x x .Hence Evolutionary entropy, H evol ( φ ), can be obtained by evaluating the entropy H of theMarkov chain. P = p p . . . . . . p d . . . . . .
00 1 . . . . . . ... . . . where p j = l j m j λ j λ is the dominant eigenvalue of the matrix P . The function l j is given by l j = ( j = 1 b . . . b j − j > H of the Markov chain is(24) H = e S e T , where e S = − d X j =1 p j log p j and e T = d X j =1 jp j The quantity e S is the variability in the age at which individuals reproduce and die and thequantity e T denotes the mean cycle time. We note that (24) is the special case of (23) forlocally constant potentials. The mathematical model of the Darwinian process considers factors such as the resourceabundance and the resource variation as elements of the evolutionary process. The modelpostulates mutation, selection and inheritance as the principles underlying the dynamics ofevolution.The theory distinguishes between the incumbent population, described by a dynamical sys-tem (Ω , µ, φ ) and a variant of small size described by a system (Ω , µ ( δ ) , φ ( δ )). Here φ ( δ ) isa small perturbation of φ of the form(25) φ ( δ ) = φ + δψ, where R φ dµ = R ψ dµ . The Entropic Selection Principle is concerned with the dynamics of the competition betweenthe Incumbent population (Ω , µ, φ ), and the Variant (Ω , µ ( δ ) , φ ( δ ). The analysis of the We note that this condition does not restrict the generality of the approach since it can be achieved byre-normalizing ψ . , µ, φ ) are (see Section 4.2):(1) The Evolutionary entropy H = e S e T . (2) The growth rate r ( φ ) = lim n →∞ n log Z n ( φ ).(3) The Demographic Index Φ = lim n →∞ n E n ( S n φ ).(4) The Demographic Variance σ = lim n →∞ n V ar n ( S n φ ) . (5) The Correlation index κ = lim n →∞ E n [ S n φ − E n [ S n φ ]] .The Evolutionary entropy H , the growth rate r , and the reproductive potential are relatedby the identity(26) r = H + Φ . (II) The Variant population (Ω , µ ( δ ) , φ ( δ ) is derived from a mutation. This is representedin terms of a perturbation of the function φ . We denote the parameters that characterizethe variant population by ( r ∗ , H ∗ , Φ ∗ , σ ∗ ) where r ∗ = r ( δ ) , H ∗ = H ( δ ) , Φ ∗ = Φ( δ ) and σ ∗ = σ ( δ ). Further, we define(27) ∆ r = r ∗ − r, ∆ H = H ∗ − H, ∆ σ = σ ∗ − σ . We have, by using perturbation methods in [9], the following relations:(28) ∆ r = Φ δ, ∆ H = − σ δ, ∆ σ = γδ, where γ = 2 σ + κ . The condition for the increase in frequency and ultimate fixation of the variant populationis evaluated by considering the stochastic dynamics of the frequency of the variant.The continuous time diffusion approximation is used in order to consider the function(29) p ( t ) = N ∗ ( t ) N ( t ) + N ∗ ( t )Here N ( t ) denote the population size of the resident population and N ∗ ( t ) the populationsize of the invader. 22et f ( N, t ) and f ∗ ( N ∗ , t ) denote the density of the processes N ( t ) and N ∗ ( t ). The evolutionof the density is given by the solution of the Fokker–Planck equation(30) ∂f∂t = − r ∂ ( f N ) ∂N + σ ∂ ( f N ) ∂N and(31) ∂f ∗ ∂t = − r ∗ ∂ ( f ∗ N ∗ ) ∂N ∗ + σ ∗ ∂ ( f ∗ N ∗ ) ∂N ∗ We can now invoke the constraint, total population size,(32) M = N ( t ) + N ∗ ( t ) M constant, to derive a Fokker –Planck equation for the probability density function ψ ( p, t )of the stochastic process which describes the change in frequency of the invading population.We have(33) ∂ψ∂t = − ∂ [ α ( p ) ψ ] ∂p + 12 ∂ ( β ( p ) ψ ] ∂p , where(34) α ( p ) = − p (1 − p )[∆ r − M ∆ σ ]and(35) β ( p ) = p (1 − p ) M [ σ p + σ ∗ (1 − p )]The analysis of (33) shows that the outcome of competition between the invading type andthe resident population is determined by the selective advantage s , given by(36) s = ∆ r − M ∆ σ The perturbation relation given by (28) entails the following implications.Φ < ⇒ ∆ r · ∆ H > > ⇒ ∆ r · ∆ H < γ < ⇒ ∆ σ ∆ H > γ > ⇒ ∆ σ · ∆ H < s ∗ = − (Φ − γ/M )∆ H , γ , and the change ∆ H is given by(38) ( − Φ + γ/M )∆ H > γ and determined by H .are summarized in Table (1) and (2).Constraints Φ , γ OutcomeΦ < γ > H > > γ < H < Table 1
Relation between macroscopic parameters Φ , γ and Selection Outcome, ∆ H Constraint Population Size Outcome
M > Φ γ ∆ H > < , γ < M < Φ γ ∆ H < M > Φ γ ∆ H < > , γ > M < Φ γ ∆ H > Table 2
Relation between macroscopic parameters Φ , γ and Selection Outcome ∆ H Statistical Thermodynamics is concerned with understanding the macroscopic properties ofmatter in terms of the molecular constituencies. The fundamental concepts in this disciplineare temperature (i.e. the mean kinetic energy of the molecules), and the thermodynamicentropy, i.e., the number of molecular configurations which are associated with a givenmacrostate. 24volutionary dynamics is concerned with understanding the flow and transformation of en-ergy in populations of replicating organisms in terms of the dynamical behavior and thebirth and death rates of the individuel organisms.The fundamental parameters in this theory are the mean cycle time, i.e. the generationtime, and the Evolutionary entropy, i.e., the number of replicating cycles generated by theinteraction between the individuals.We show that these two theories are isomorphic in the sense that there exists a correspon-dence between the macroscopic parameters that define the theories.We will furthermore show in Section 6 that the correspondence between the classes of vari-ables is analytic. We will explore this analyticity to show that the Fundamental Theoremof Evolution, the directionality principle for evolutionary entropy, is a generalization of theSecond Law of Thermodynamics, the directionality principle for thermodynamic entropy.
We consider a gas as a system consisting of N interacting molecules. Let X = { , . . . , d } denote the phase space. Further let M denote the set of probability measures on X . Considera potential function φ : X → R as representing the potential energy. The mean energy ofthe system in state µ = ( µ i ) is given by(39) Φ = d X i =1 µ i log φ ( x i ) = µ (log φ ) . The Entropy S ( µ ) is given by(40) S ( µ ) = − X µ i log µ i . The quantity Z is determined by the variational principle(41) log Z = sup µ ∈ M [ µ (log φ ) + S ( µ )] . Moreover, the maximum in (41) is attained by a unique measure ˆ µ , that is, log Z = ˆ µ (log φ )+ S (ˆ µ ). The physical interpretation ascribed to the variational principle can be discerned byexpressing the potential function in the form(42) φ ( x i ) = exp( − βE i ) , β = kT . The quantity T is the temperature and k is the Boltzmann constant. Theexpression for the distribution ˆ µ = (ˆ µ i ) now becomes(43) ˆ µ i = exp( − βE i ) P exp( − βE i ) . The variational principle then asserts that the distribution ˆ µ maximizes S − EkT . Equivalentlyit minimizes free Energy F which is given by(44) F = E − k S T. We consider an age-structured population whose dynamics is given by(45) ˜ u ( t + 1) = A ˜ u ( t ) , where ˜ u ( t ) denotes the age-distribution and A the population matrix(46) A = m m . . . . . . m d b . . . . . . ... . . . b d − Let Ω denote the phase space, i.e., the set genealogies generated by the graph associatedwith the matrix A . Let e M denote the space of shift-invariant probability measures on Ω.Consider the function(47) φ ( x ) = log a x ,x . The mean energy is given by(48) Φ( µ ) = Z φ dµ. The entropy H µ ( σ ) is the Kolmogorov-Sinai entropy of the measure µ , see (18) for thedefinition. With these definitions the following variational principle holds:(49) log λ = sup µ ∈ e M (cid:20) H µ ( σ ) + Z φ dµ (cid:21) . µ ∈ e M . Hence(50) log λ = H + Φ and log λ = Φ + e S e T , where(51) e S = − X p j log p j , Φ = P p j log φ j e T and e T = X jφ j . We can use the expressions given in (44) and (50) to derive a formal relation between thetwo classes of macroscopic parameters. This correspondence is given in Table 1.Thermodynamic variable Evolutionary ParametersFree Energy F Growth rate ˜ r Inverse Temperature T Generation Time ˜ T Mean Energy E Reproductive Potential ΦThermodynamic Entropy S Population Entropy e S Table 3
Relation between the macroscopic parameters
The dynamics of molecular collisions in a macroscopic body are encoded by thermodynamicentropy. The dynamics of the lower level interactions in a population of replicating organ-isms are encoded by Evolutionary entropy. These two parameters, as shown in Section 5,are formal analogues. We will now show that they are analytically related. This relationwill be the cornerstone for the analytical fact that the Fundamental Law of Evolution is ageneralization of the Second Law of Thermodynamics.
Energy transformation in inanimate matter is determined by the Second Law of Thermody-namics. The Law asserts that Thermodynamic entropy S increases.27e write(52) ∆ S > , where(53) S = k b log W. In equation (53) the quantity S describes the extent to which energy is spread and sharedamong the microscopic energy modes of the system whereas the parameter W denotes thenumber of molecular configurations that are compatible with the macrostate of the system.The validity of inequality (52) requires that the system is isolated and closed to the input ofenergy and matter. The evolution of energy in living matter is determined by the Entropic Selection Principle.This principle applies to systems which are open to the input of energy and matter.Recall that by (38) the Evolutionary entropy H evolves according to the rule(54) ( − Φ + γ/M )∆ H > . Now let R denote the Resource endowment and assume that R evolves according to thedifferential equation(55) d R ( t ) = ρ R ( t ) dt + β dt, where ρ denotes the production rate of the external resource. We now assume that theResource process and the population process are in dynamical equilibrium. Here we use(56) adr ( δ ) dδ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ =0 = Φ , adσ ( δ ) dδ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ =0 = γ and write ρ = a Φ, β = aγ . 28f we assume that there is no exchange in energy and matter between the population andthe external environment, we have(57) ρ = 0 , β = c, which implies that(58) Φ = 0 and γ = k where k is a numerical constant.The Laws describing the changes in evolutionary entropy will in view of (54) become(59) ∆ H > , where(60) H = e S e T .
Since(61) ∆ H · ∆ e S > e S > . Irreversibility in the Second Law of Thermodynamics is given in terms of the function∆
S > S denotes the thermodynamic entropy.29rreversibility in the Fundamental Principle of Evolution is given by( − Φ + γ/M )∆ H > . We have shown that when the system is transformed from a process open to input of energyand matter to an isolated system, (38) reduces to the condition∆ e S > S and evolutionary entropy H .We first recall that(63) H = e S e T , where(64) e S = − X k p k log p k , and(65) e T = X k k p k . Let N denote the population size, which is assumed to be large. We also assume that thetotal number of replicative cycles is also of the order N . Hence n k , the number of cycles oflength k , is of the order n k = p k N We write(66) p k = n k N , where(67) N = X n k . Therefore, p k is the probability that a randomly chosen cycle in the network of interactivemicrostates has length k . 30herefore,(68) e S = − X k n k N log n k N .
Hence(69) e S = − N X n k log (cid:16) n k N (cid:17) , and(70) e S = − N X k n k (log n k − log N ) . We conclude that(71) N e S = − X k n k log n k + X k n k log N, which implies(72) N e S = N log N − X k n k log n k . Recall that by Stirling’s formula we have(73) log N ! = N log N − N. Combining (72) and (73) yields(74) N e S = log (cid:20) N ! n ! n . . . (cid:21) We conclude that(75) e S = 1 N log (cid:20) N ! n ! n ! . . . (cid:21) Now let W denote the number of microscopic states which are compatible with a givenmacrostate. The thermodynamic entropy S is given by(76) S = k b log W, where(77) W = (cid:20) N ! n ! n ! . . . (cid:21) . Hence(78) S = k B ˜ S We conclude that as ρ →
0, the thermodynamic entropy and the evolutionary entropycoincide. 31he relation between the two measures of organization implies that the Fundamental The-orem of evolution is a generalization of the Second Law of thermodynamics.The relation between the parameters which are involved in energy transformation in physicaland biological systems is described in Table (3).
Table 4
Relation between the parameters in ThermodynamicTheory and Evolutionary Theory.
Parameter Thermodynamic Theory Evolutionary Theory
Organizing Variable Temperature Cycle TimeFitness Parameter Thermodynamic Entropy S Evolutionary Entropy H Selection Principle ∆
S > − Φ + γ/M )∆ H > The two classes of entities that constitute the natural world - the aggregates of inanimatematter, and the populations of living organisms, both manifest a hierarchical structure withordering in terms of time and energy scales. The aggregates in the physical world - theensemble of solids, liquids and gases, range from the submicroscopic to galactic. The elementsin the living world, the integrated assembly of DNA, RNA and proteins, scale from viruses, touni-cells, to multi-cells and to communities of plants and animals. Complex human societies,organized by both genes and culture, are at the top of this hierarchy.The various states of organization in aggregates of inanimate matter, and in populationsof cells and higher organisms are the outcome of the transfer and transformation of en-ergy. Energy is a collective concept which can exist in many forms: The most commonlyencountered forms are heat (thermal energy), motion (kinetic or mechanical energy), light(electromagnetic energy) and metabolism (chemical energy).The laws which pertain to energy transformation in inanimate matter are based on theempirical observation that in isolated systems, that is, systems closed to the input of energyand matter, there is the tendency for energy to disperse and spread within the enclosure [2],[3].The molecular dynamics explanation of this principle can be formulated in terms of theanalytical rule ∆ S ≥ . S , a measure of positional disorder, is given by S = k B log W, where W denotes the number of microscopic configurations consistent with a given macrostateand k B is the Boltzmann constant.Energy transformations in populations of metabolic and replicating entities: macromolecules,cells, higher organisms, occur under constraints which are quite distinct from the typicalsituations observed in aggregates of inanimate matter. There are three characteristic aspects:(i) Openness: Living organisms are maintained by a continuous exchange of energy andmatter with the external environment.(ii) Isothermal condition: The low temperature differential between the organelles in a cellindicate that the cells do not act as heat engines. Living organism are isothermalchemical machines.(iii) Size: The number of molecules in a cell, and the number of cells in a population areof magnitude much smaller than the number of molecules in a gas.These constraints entail that evolutionary selection, the process that drives the transfer andtransformation of energy in populations of replicating organisms, will necessarily have a dif-ferent character from thermodynamic selection, the process describing energy transformationin aggregates of inanimate matter.The mathematical analysis of thermodynamic processes shows that thermodynamic entropy,a measure of positional disorder, will be replaced by evolutionary entropy, a measure oftemporal organization.Evolutionary entropy describes the rate at which the population appropriates chemical en-ergy from the external environment and converts this energy into biological work. Evolu-tionary entropy, H is given by(79) H = e S e T .
The quantity e S denotes the number of bioenergetic cycles in a population of metabolic andreplicating enteties. The quantity e T denotes the mean cycle time.33n the evolutionary process, directional changes in evolutionary entropy, will be contingenton the external resource constraint. These changes are expressed by(80) (cid:16) − Φ + γM (cid:17) ∆ H > . The quantities Φ and γ are correlated with the resource endowment, its amplitude and itsvariability, respectively.The relation (80), the kernel of the Fundamental Theorem of Evolution, entails that:(i) Evolutionary entropy increases when the resource endowment is scarce and diverse.(ii) Evolutionary entropy decreases when the resource endowment is abundant and singu-lar.The directionality Principle, as given in (80), is applicable to the energy transformation atvarious hierarchical levels.(1) Molecular: The principle has provided an explanation for the changes in sequence lengthobserved in experimental studies of the evolution of the Q β virus. These studies showthat sequence length increases when the resource is scarce, and decreases when the resourceabundant, see, e.g. [23].(2) Demographic: The evolution of life history: The principle elucidates the increase initeroparity when the resources are scarce, and the shift to semelparity when they are abun-dant, see [22], [26].(3) Social: The evolution of cooperation: Cooperation in a social network refers to theinteraction between social agents to achieve a particular goal. This activity may involvecosts and benefits. The principle predicts altruistic behavior when the resources are scarceand diverse, and selfish behavior when the resources are abundant and singular [11].This article has shown that the Fundamental Theorem of Evolution is the natural general-ization of the Second Law of Thermodynamics.Both Laws are concerned with Energy and its Transformation. The Laws, however havedifferent domains of validity. This fact derives from the different constraints that regulateenergy transformation in inorganic matter and living organisms.34 eferences [1] Aaronson, J: An introduction to infinite ergodic theory,
Mathematical Surveys andMonographs , AMS, 1997.[2] Atkins, P.: Conjuring the universe: The origins of the laws of nature , Oxford UniversityPress, 2018.[3] Berry F.S.:
Three laws of nature , Yale Univ. Press, 2019.[4] Bowen, R.:
Equilibrium states and the ergodic theory of Anosov diffeomorphisms , Sec-ond revised edition. With a preface by David Ruelle. Edited by Jean–Rene Chazottes.Lecture Notes in Mathematics, 470. Springer Verlag, Berlin 2008, viii+75 pp.[5] Bowen, R.:
Some systems with unique equilibrium states , Math. Systems Theory ,(1974/1975), 193–202.[6] Bowles, S and Ginitis, H., A cooperative species: Human reciprocity and its evolution ,Princeton University Press, 2011.[7] Burr, M. and Wolf, C.,
Computability at zero temperature , preprint.[8] Demetrius, L.:
Directionality principle in thermodynamics and evolution,
Proc. Natl.Acad. Sci. (1997), 3491–3498.[9] Demetrius, L.: Boltzmann, Darwin and Directionality theory,
Physics Reports (2013), 1–85.[10] Demetrius, L.:
Demographic parameters and natural selection,
Proc. Natl. Acad. Sci. (1974), 4645–4647.[11] Demetrius, L., Gundlach, V.: Directionality Theory and the Entropic Principle of Nat-ural Selection , Entropy , (2014), 5428–5522.[12] Eigen, M.: Steps towards life: A perspective on evolution , Oxford University Press,1996.[13] Fisher, R.A.:
The genetical theory of natural selection , The Clarendon Press, 1930.3514] Kitchens, B.: Symbolic Dynamics: One–sided, two sided and countable state Markovshifts, Springer Verlag, Berlin Heidelberg 1998.[15] Lehninger, A.: Bioenergetics, W.A. Benjamin, 1965.[16] Levins, L. and Lewontin, R.: The Dialectical Biologist, 1985[17] Mauldin, D. and M. Urbanski: Graph directed Markow Systems: Geometry and Dy-namics of Limit Sets, Cambrigde: Cambridge University Press, 2003.[18] Prigogine, I., Nicolis, G. and Babloyantz, A.:
The thermodynamics of evolution , PhysicsToday (1972), 11,23.[19] Ruelle, D.: Thermodynamic Formalism, Cambridge University Press, Cambridge, 2004.[20] Sarig, O.: Thermodynamic formalism for countable Markov shifts, Proc. of Symposiain Pure Math. (2015), 81–117.[21] Schr¨odinger, E.: What is life? , Cambridge University Press, 1944.[22] Stearns, S.:
The Evolution of Life Histories,
Chapman & Hall, 1992.[23] Spiegelman, S :
An approach to the experimental analysis of precellular evolution,
Quar-terly Reviews of Biophysics (1971), 213–253.[24] Walters, P: An introduction to ergodic theory , Graduate Texts in Mathematics 79,Springer, 1981.[25] Wicken, J. S.:
Evolution, Thermodynamics and Information , Oxford University Press,1987.[26] Ziehe, M. and Demetrius L.,
Directionality theory: an empirical study of an entropicprinciple in life-history evolution , Proc. Biol. Sci.272