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Geometry and Dynamics of Populations

Published online by Cambridge University Press:  14 March 2022

Melvin Avrami
Melvin Avrami*
Affiliation:
School of Mines, Columbia University
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Extract

We wish here to consider the theory of a population or system made up of individuals whose number and size change with time. As usual, the description of these changes will be referred to as the kinetics, whereas the description of the special circumstances under which unchanging (equilibrium) conditions subsist will be called the statics of the population. A third category, the conditions for a steady state, i.e., when the variables inside the system do not change, but linked variables outside (or at the boundary) do, will be described by what we shall call the rheostatics of the system. All three, statics, rheostatics, and kinetics are subdivisions of the general dynamics of the system.

Type
Research Article
Information
Philosophy of Science , Volume 8 , Issue 1 , January 1941 , pp. 115 - 132
Copyright
Copyright © The Philosophy of Science Association 1941

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References

1 Number and size are merely important representatives of a large group of extensive (i.e., algebraically or vectorially additive) variables which may be observable in a population. Other such quantities are masses, momenta, kinetic energies, etc. The following considerations, though specifically dealing with numbers and volumes, may be generalized to take account of any number of extensive and intensive variables (see footnote 6).

2 From the Greek rheo-statikos meaning “flow—causing to stand.” An alternative would be rheostasis meaning flow—standing still.“

3 “Phase” is used here in its usual physico-chemical sense, i.e., as a piece of matter having a certain internal molecular or atomic arrangement. Detailed applications to phase dynamics will be found in a series of papers in Jour. Chem. Phys. (a) 7, 1103 (1939), (b) 8, 212 (1940), (c) 9, Jan. (1941); referred to as K. P. C. I, II, III respectively.

4 In some instances, the critical condition may be the attainment of a degree of internal order, concentration, etc. The above formulation is, however, sufficiently general for our purposes here.

5 Quantities entirely analogous to N, V, and F may, of course, be introduced for the nucleus population.

6 See 3(c). Here too, shape and orientation are only special instances of a group of intensive variables which may be observable in the population. Other examples would be color, temperature, pressure, i.e., in general including the environment parameters.

7 The minus sign in equation (6‘) is due to the fact that the rate of change of v with the time of birth t’ of the grain is negative.

8 If to every v at t, corresponds a unique t' (i.e., f(v, t, t')dv infinitely sharp) then

It is worth noting that, while the corresponding condition (6') for the actual grains is only an approximation, (6) will often be exactly valid; not perhaps for living grains with their complex internal structure but for the phase grains mentioned above.

9 This is readily seen to give, upon integration over any macroscopic unit volume, the previously defined extended volume density; for details of this and the following see 3(c).

10 These restrictions imply that (13') becomes increasingly inaccurate towards the end of transformation.

11 In obtaining the indicated result for (A7), a useful formula is 159c, p. 44, given by L. B. W. Jolley in his Summation of Series, Chapman & Hall, London (1915).