Elements of Physical Biology

2 Emerson, Conduct of Life, Everyman’s Library Edition, 1915, p. 157. Compare also Lee Wilson Dodd’s lines: « , . . . Nor do the stars retrace their glistening snail marks of slow destiny.” 8 J. Perrin, Traité de Chimie Physique, 1903, vol. 1, p. 142. 4 Petronievics, Science Progress, 1919, p. 406. 5 Emerson, Conduct of Life, p. 158. 6 For this reason the characterization of the trend of evolution given by Petronievics, loc. cit., is inadequate.

undoubtedly contain an anthropomorphic element.’ At best they give every opportunity for divergence of opinion as to what constitutes a “higher form.’”’ If, on the other hand it is stated that evolution proceeds from simpler to more complex forms, or from less specialized to more specialized forms, then the direction of evolution is but poorly defined, for the rule is at best one with many exceptions. It should be particularly noted that all these efforts to specify the direction of evolution attempt to do so in terms of a single component of the evolving system. Such definitions of the direction of evolution are foredoomed to failure. It is the system as a whole that evolves, and we can hope to establish a definition of the direction of evolution only in terms of the system as a whole. Evidently, we must seek a more precise indication of the direction of evolution if our definition is to be truly expedient. We must analyze further the contents of our mind when it contemplates the concept of evolution. We return to our examples of the pendulum, or of the earth in its orbit. When frictional resistances are neglibible, or are disregarded, the periodic series of events in the system may be history, but seems hardly worthy of the name evolution. In actual fact the motion of the pendulum bob gradually dies down, owing to friction and other dissipative forces. The motion is not strictly periodic. The pendulum does not, actually, count out similar seconds, unidentified, but marks, by its greater amplitude, an earlier vibration as distinguished from a later. So also, the earth in its motion is slightly delayed by frictional forces introduced by the tides; it slows down a little asthe centuries pass. The strictly periodic process is changed into one in which successive days differ by a trifle in length. The process has a definite direction in time.

We feel justified in speaking of the system as “evolving.’”? Now the thing to mark is that what has imparted to the process its directed character is fi is frictional resistance, dissipative forces, typical irreversible effects, to speak in the language of the ROTI S ie 7™“Kvolution is thus almost synonymous with progress, though the latter term is usually confined to processes of development in the moral, as distinguished from the physical world. Further, this idea, as Mr. Spence remarks, has rather a subjective value in existence, as judged by our feelings”’ Gineyal, Brit., 9th edition, vol. 8, p. 751). Coinpare also Bertrand Russell, Our Knowledge ae the Eternal World, 1914, p. 12. ‘A process which led from amoeba to man appeared to the philosopher to be obviously a progress—though whether the amoeba would agree with this opinion is not known.”

Again, consider a typical example of what we are all agreed to speak of as evolution: the history of the earth and its living inhabitants. The readjustments, the re-adaptations of life-forms which have here taken place, were undoubtedly due in part to changes in external conditions, such as climate, geographic distribution of land and sea, etc. In part, also, such changes have gone on and are going on before our eyes independently of any external changes, and under approximately constant conditions. Organic evolution being a slow process, it takes a certain time, when equilibrium or near-equilibrium is disturbed, for a new equilibrium or near-equilibrium to become established. There is therefore a tendency for internal readjustments or changes to lag behind the external changes by which they are conditioned. As a special case, if an external change is followed by constant external conditions, internal changes may continue to proceed under constant external conditions.

Now such internal changes in a material system, which lag behind the determining external changes, or which go on under constant external conditions, are typically irreversible processes.® 8 A process is said to take place reversibly, if the direction of the change is reversed by a suitable alteration, Towever small, of the (generalized) force applied to produce the change. For example, if two equal weights are suspended from the ends of a string passed over a simple pulley, then, the weights being initially at rest, any weight however small, added on one side of the system will produce motion downward on that side, provided there is no friction at the pulley and no stiffness in the string. If, on the contrary, a weight, however small, is lifted off from the same side of the system, motion will be initiated in the opposite direction. Note that if there is friction at the pulley, these statements are no longer true. It will now require a weight of definite size, perhaps a decigram, or a milligram, to start or reverse the motion.

In the first instance the change is reversible, in the second it is said to be irreversible. Note that in this example the circumstance that imparts to the process an irreversible character is the presence of friction, which causes the dissipation of energy, that is to say, its conversion into heat at the temperature of the surroundings. Again, consider a vessel containing water at a temperature T; in a room at temperature T2. If T, is ever so slightly greater than T2, heat passes from the vessel to the surroundings, and vice versa. When, therefore, T; and T, are very nearly equal, the passage of heat from the vessel to the surroundings is essentially reversible. If there is a material difference between T,; and T:, the heat transference is irreversible. For example, if the vessel is at 50°C. and the room at 20°C., heat will pass from the vessel to the room. And the direction of this heat transfer will remain unchanged if the temperature of the vessel is

We are thus led, from two slightly different points of view to the following definition of evolution: Evolution is the history of a system undergoing irreversible changes. Scope of Definition. It is worth while at this point to consider briefly what kind of history this definition excludes and what it includes. It has already been noted that we have excluded certain purely mechanical systems of periodic habit, such as the frictionless pendulum and the planet circling in its orbit through empty space, in absence of tidal effects.° It is not the case, however, that all purely mechanical systems are excluded, that is to say, all systems in which all energy is either kinetic or potential (configurational), all forces either inertia forces or positional forces. If our knowledge of such a system is statistical in character, if we know only averages of certain of the variables defining the state of the system, it may happen that certain changes therein appear to us irreversible, and would accordingly be classed, by our definition, among processes of evolution.!° This leads to the seemingly embarrassing conclusion that a process is or is not a process of evolution, according to the

reduced 1, 2 or even 10°. Not until the vessel is cooled by more than 30° will the stream of heat be reversed. The passage of heat in such case, from a body at one temperature to another at essentially lower temperature, is irreversible in this sense, the sense in which the term is employed by the physicist in discussions of this kind. In the case in which internal readjustments lag behind changes in external conditions, there is necessarily a finite difference between the applied (generalized) force, and the opposing resistance. Such processes are, therefore, of necessity, irreversible.

From the examples given, it will be seen that during a reversible change a system is at all times (very nearly) in equilibrium. It can therefore be said that a reversible change is one in which the system passes through a continuous succession of equilibria. In fact, the change is strictly reversible only if the difference in the applied (generalized) force and the resistance is infinitesimal, and the change is infinitely slow. * Such tidal effects act as brakes and destroy the exact periodicity of the motion.

9 The irreversibility also of those changes occurring in a system whose internal adjustments lag behind changes in the applied forces, may be apparent, and may disappear when detailed knowledge of the individual parts of the system takes the place of statistical data. The reason for this is that when the reaction or readjustment is expressed in a statistical way, an average of individual reactions may show a lag, although each individual reaction itself may be immediate,

nature and extent of our knowledge regarding the system. So, for example, the establishment of thermal equilibrium in a body of gas initially at non-uniform temperature is evolution if we merely know its total mass, composition, volume, pressure and initial temperature distribution. But should we be informed of the exact initial state of each molecule, then the process by which thermal equilibrium is established (if this does occur) would be classed, together with the journey of the earth in its orbit, among the cases excluded, as mere history, from our definition.

This, upon reflection is neither as strange, nor as embarrassing as it may at first sight appear. For problems of evolution are in large measure problems of probabilities, statistical problems. Incidentally, this reflection disposes of the rather foolish objection sometimes raised against the theory of evolution, that it ascribes the course of events in an evolving system to chance. When we describe a phenomenon as being governed by chance, we do not, of course, mean that there are no definite causes (determining factors) at work; we merely state in these terms that the causes are complex and not known to us in detail.

Practically there is no cause for embarrassment, since we never do know material systems in sufficient detail to compute their state at every instant from the initial state, except in terms of averages. In principle, however, it is necessary to make the admission that, in the last analysis, whether we class the history of a system as evolution or not must depend on the extent and detail of our knowledge of that system.!! It will thus be seen that the line of division between reversible (purely mechanical) and irreversible (dissipative) processes is not

11 To be quite exact, evolution, according to this, should be defined in terms of a point of view, say about as follows: Evolution is the history of a system, regarded as a progressive change or development, to which its unidirectional character is imparted by irreversible changes going on in the system. That a point of view is involved is also implied in the following definition given by Kar! Pearson: ‘““A causal description of the appearance of successive stages in the history of a system forms a theory of the evolution of that system.

“If the theory be so satisfactory that it resumes in some simple statement the whole range of organic change, we term it the law of evolution,’ (Grammar 30 very sharply drawn. Furthermore, the cases excluded are, in point of fact, ideal cases. Real processes are always irreversible. Hence, after all, history, real history, is always evolution, and, though in principle the two concepts may be distinct, in practice they coincide in scope. This is the gain: Having analyzed the submerged implications of the term evolution as commonly used, so as to bring them into the focus of our consciousness, and having recognized that evolution, so understood, is the history of a system in the course of irreversible transformation; we at once recognize also that the law of evolution is the law of irreversible transformations; that the direction of eyolution (which, we saw, had baffled description or definition in ordinary biological terms), is the direction of irreversible transformations. And this direction the physicist can define or describe in exact terms. For an isolated system, it is the direction of increasing entropy.” The law of evolution is, in this sense, the second law of thermodynamics.'®

12 More generally, it is the direction of decreasing thermodynamic potential, this potential being variously defined, according to the conditions of transformation. As dar a the second law of the theory of energy is now generally / regarded as essentially a statistical law. So viewed, the second law of energy “+ becomes a principle stated wholly in terms of the theory of probability. It is the law that the physical world tends, in each of its parts, to pass from certain less probable to certain more probable configurations of its moving particles. As thus stated the second principle . . . . becomes a law of evolution” (Josiah Royce, Science, 1914, vol. xxxix, p. 551.)

“Un systéme isolé ne passe jamais deux fois par le méme état. “Le second principe affirme un ordre nécéssaire dans la succession de deux phenoménes, sans retour possible aux états déjA traversés. C’est pourquoi j’ai cru expressif d’ appeler ce principe un principe d’évolution. Il se trouve qu’en proposant ce nom je suis fidéle a la pensée de Clausius, car le mot é&rporn, d’od il a tiré entropie, signifie précisement évolution.”’ (J.Perrin, Traité de Chimie Physique, 1903, vol. 1, pp. 142-143.)

“Tl est hautement improbable qu’un systéme isolé passe deux fois par le méme état; cela est d’autant plus improbable que la complication du systéme est plus grande, et pratiquement il serait insensé de se placer dans cette hypothése d’un retour al’état initial.’ (J. Perrin, loc. cit., p. 146). Simple Mechanical Example. It will be desirable, at this point, to consider, by the aid of a simple example, the manner in which some of the facts considered in the preceding pages find expression in the analytical formulation of the behavior of mechanical systems. Take the example of the simple pendulum. For small vibrations the restoring force, tending to draw back the bob to its lowest position,

is easily shown to be mg where z is the horizontal displacement, m the mass of the bob, and g the acceleration of gravity. This force is expended upon two items, first, in overcoming the inertia of the bob, and producing an acceleration a. The force so expended is measured by ma. Second, a part of the force mg is expended in overcoming the resistance of the air. If v is the velocity of the bob, this part of the force is measured (for ordinary velocities) by ky, where k is a constant depending on the shape of the bob, etc. We have then

or, since the velocity v is the rate of change of x with time, i.e., sre : GI OB oe and ais the rate of change of = with time, i.e. a = aie re Now there are certain general characteristics to be observed in this equation, characteristics which are typical of the equations of motion of mechanical systems. The equation contains the first and | second derivatives of x with regard to ¢ and no higher derivatives, | The first derivative is introduced by the frictional force, and disappears if this force is zero, Le., if the coefficient k in (2) is zero.

Now in this simplified form the equation has the following peculiarity: It is indifferent to the sign of ¢. For, in differentiating twice in succession with regard to—t, the positive sign of the second derivative is restored. This is the analytical symptom, as it were, of the reversibility of the process.4 It should be noted that this peculiarity : F Pe ine A disappears at once if the frictional term k Fi is present, for a single differentiation with regard to—¢ yields a result with sign opposite to that of differentiation with regard to ¢. The presence of a frictional force, therefore, imparts to the process an irreversible character, | it establishes a distinction between ¢ and —¢; it singles out one direc- | tion in time as a peculiar direction, the forward direction, the direc- ' tion of progression.

Now, in point of fact, in the equations of motion of all real systems dx the frictional term (viscosity term) k — or its equivalent is present, : : . : d though it may be small as compared with the inertia term ma ; The reversible system, in which this term is wholly absent (zero) ¥Y isan ideal case, it represents a limit towards which real systems may approach; an abstraction. another such ideal limiting case, another abstraction, which is of much interest because certain important classes of real systems

Cw approach it very closely. This is the case in which the inertia term al 2 . 0 m _ is negligible, so that in the case of the pendulum, qf for example, the equation representing the history of the system reduces to dx x ta (4) _/ The history of such inertia-free systems is typically of the irreversible kind. They have, furthermore, a property illustrated by certain features in the equation (4) above: ae may put it, the velocity vanishes with the displacement from equilib- > rium. Moreover, differentiation of (4) gives

It will be observed that if x is zero, then also is zero, or, aS we from which it is seen that the acceleration also vanishes with the velocity.!* This implies that when the system is in its equilibrium position, it is also actually at rest, unlike the pendulum, which swings twice through its equilibrium position in each vibration. The former property, the vanishing of the accelerations with the velocities, so that the equilibrium position is necessarily the position of rest,|

is characteristic of an important class of systems, including those with which we shall here be chiefly concerned. Another important characteristic of such systems, which is also exemplified by equation (4), is that the velocity is uniquely determined for every value of x. This is not the case in the motion represented by (3). This latter equation gives, upon integration, We have, then, at the one extreme the ‘‘purely mechanical” system free from frictional (viscosity) effects, and, in its most typical form, periodic in habit.

As an intermediate link we have systems exhibiting both inertia and frictional effects. Their action may resemble that of a pendulum swinging in air; typically the history of such a system exhibits the phenomenon of damped oscillations, a periodicity over which there is superimposed the dying away of the motion. The damping is introduced by the frictional effects. At the other extreme we have inertia-free, or, as we might say, completely damped systems, typically irreversible in their history.

The system and processes with which we shall largely be concerned here seem to belong essentially to this third type, as will be seen in the development of the theme. so that for every value of x there are two possible values of 16 Compare E. Buckingham, Theory of Thermodynamics, 1900, p. 33. 16 This does not, however, preclude the possibility of oscillations. More will be said on this point later, Supposons que nous voulions placer un grain d’avoine au milieu d’un tas de blé: cella sera facile; supposons que nous voulions ensuite l’y retrouver et l’en retirer; nous ne pourrons pas y parvenir. Tous les phénoménes irréversibles, d’aprés certains physiciens, seraient construits sur ce modéle.— H. Poincaré.

One point, to which allusion has been made incidentally, calls for comment. Many processes which, viewed in the gross, present the appearance of typically dissipative, irreversible phenomena, have long been suspected, and have in recent years been fully demonstrated to be, in fact, of the reversible type, ‘‘purely mechanical”’ processes, the details of which are merely hidden from our view owing to the diminutive dimensions (and correspondingly immense number) of the units at play. So, for example, consider the case of two vessels A and B at equal pressures, communicating by a tube that can be closed by means of a turncock. Let the vessel A contain 1 gram of nitrogen gas, and let B contain 1 gram of oxygen gas, the communication between A and B being closed. It is a matter of common knowledge that if the stopcock is now opened, the gas from the A will flow over into the vessel B and vice versa, and in a short time an equilibrium is reached in which each vessel contains 0.5 gram of each gas. Now, in point of fact, the molecules of the gas behave (approximately) like a number of elastic spheres, their equations of motion contain no dissipative term, but are of the type (3) (Chapter II). We should therefore expect the system to exhibit periodic motion, we should expect that after a certain lapse of time the initial condition should return, and that all the nitrogen should once more be contained in the vessel A all the oxygen in B. In actuality, such a thing is never cheered How is this discrepancy to be explained?

Let us replace the two vessels and the gas molecules by some simple analogues of dimensions readily accessible to our senses, and let us watch a process analogous to the diffusion of the gas from one vessel into the other. We provide ourselves with two boxes or urns.' In one of these, A we place 50 black balls; in the other B, we place 50 white balls. We shuffle both boxes thoroughly, and then draw blindly a ball from A, and one from B, and we return them to opposite urns. We continue this as long as desired. The more lightly drawn curve in figure 1 shows the graphic record of an actual series of drafts of this kind. The stair-case-like line shows how in successive drafts the number of black balls in box A gradually diminished until at last there remained about 25, onehalf of the original number, in box A. But note that there are fluctuations, sometimes the box contains 26, 27, 28, then again

Fig. 1. Grapx or Mopeu Process ILLUSTRATING THE STATISTICAL MEANING or IRREVERSIBILITY The more lightly drawn curve records the number of black tickets remaining in urn A after successive drafts. The heavier curve records the previous and ensuing history of 50 tickets found in urn A at the end of the fiftieth draft, (Reproduced from A. J. Lotka, Two Models in Statistical Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 122.) 27, 26, etc. of the original balls. It is nowise impossible that, if we continue the drafts for a long time, some time or other all the original 50 black balls will be back in box A; but it 7s highly improbable that this should happen within any reasonable time. Curiously enough, the urn model is competent to illustrate also this highly improbable course of events. For this purpose, instead of starting with 50 black and 50 white balls, we start with the balls, or in this case more conveniently tickets, all white, and numbered from 1 to 50 in urn A, and from 51 to 100 in urn B. After a suit-