Stable equilibrium

“ Hence, if the surfaces of discontinuity are constrained to pass through corresponding fixed lines in the surfaces of discontinuity belonging to the varied state just mentioned, there must be a state of stable equilibrium for the system thus constrained which will differ infinitely little from the given state of the system, the stability of which is in question, and will have the same entropy, quantities of components, and exterior, but less energy. ”

“ While thus ruling the oscillations of value, they themselves obey a superior force, which makes value gravitate towards Cost of Production, and which would settle it and keep it there, if fresh disturbing influences were not continually arising to make it again deviate. To pursue the same strain of metaphor, demand and supply always rush to an equilibrium, but the condition of stable equilibrium is when things exchange for each other according to their cost of production, or, in the expression we have used, when things are at their Natural Value. ”

“ Here we have a number of units severally placed in unstable equilibrium, and in such relative positions that each, while falling into a state of stable equilibrium, gives an impulse to the next sufficient to make the next, also, fall from unstable to stable equilibrium. Now since, among mingled compound molecules, no one can undergo change in the arrangement of its parts without a molecular motion that must cause some disturbance all round ”

“ The absolutely stable state of equilibrium of the system is therefore characterized through the maximum of :
If the system consist of numerous phases, then, because , in accordance with , is linear and homogeneous in , and , the quantity referring to the whole system is the sum of the quantities referring to the individual phases. If the expression for is known as a function of the independent variables for each phase of the system, then, from equation , all questions concerning the conditions of stable equilibrium may be answered. ”

“ When the return and equivalence are suspended, there is a necessary continuance of the movement, in the tendency toward a stable equilibrium of another kind, which would come about when all the force had been transferred. For instance: You give good schools for less than their market value ”

“ If both the masses are infinitely large, or if one which contains all the components of the system is infinitely large, and will be constant, and the condition reduces to
The equilibrium will therefore be stable or unstable according as the surface of tension is less or greater than a hemisphere. ”

“ If the system is not unstable, the equilibrium is either neutral or stable. Of course we can determine which of these is the case by reference to the imaginary system, since the determination depends upon states of equilibrium, in regard to which the real and imaginary systems do not differ. ”

“ Let us first examine the case in which is positive and the equilibrium is stable. As does not vanish at the point in question, there is a definite isopiestic passing through that point, on one side of which the pressures are greater, and on the other less, than on the line itself. ”

“ And here we have contemplated, under its several aspects, the correlative general fact that, during these extensive molecular re-arrangements, there are evolved large amounts of energy. In the one case the components of organic matter are regarded as falling from positions of unstable equilibrium to positions of stable equilibrium; and in the other case they are regarded as giving out in their falls certain momenta—momenta that may be manifested as heat, light, electricity, nerve-force, or mechanical motion, according as the conditions determine. ”

“ Imagine an energy system of prodigious complexity and extreme delicacy of organisation which has an indefinitely large number of stable poises. Imagine it thrown from one poise to another with great facility, each poise being the resultant of all the energies of the system. Suppose now that the partial return of a situation which has formerly caused it to assume a stable poise, throws it into an unstable condition, from which it most easily returns to equilibrium by reassuming the former poise. ”

“ There may, however, be equilibrium more or less stable. I believe all attempts hitherto made have failed to account for the fixity of certain, nay, of very many, types throughout geological time; but the mere consideration that one may be in a more stable state of equilibrium than another so far explains it. ”

“ As these false underpinnings are given up, the equilibrium is temporarily experienced as unstable. However, nothing is less stable than an upside-down orientation. Nor can anything that holds it upside down be conducive to increased stability. ”

“ The phrase unstable equilibrium is one used in mechanics to express a balance of forces of such kind, that the interference of any further force, however minute, will destroy the arrangement previously 359subsisting ”

“ If the slightest cause of disturbance arise, the equilibrium is destroyed, and the ring would inevitably fall in upon the planet. Such causes of derangement are incessantly present, so that unstable equilibrium cannot be an appropriate explanation of the phenomena. ”

“ If there is just as much of the wood or ice above the water as is equal to the bulk of the man, then the man, in stepping upon it, will sink it just to the water's edge."
"But perhaps one man would be heavier than another man," said Marco.
"Yes," replied Forester; "but then he would be larger, and so, according to the principle, he would make more wood sink before the equilibrium was reached."
"What is equilibrium?" asked Marco.
"Equilibrium is an equality between two forces," replied Forester. ”

“ If equilibrium be disturbed at one point, it is reestablished only by propagating its own disturbance to all the internal equilibria; and, in order that it may itself endure, it must not be disturbed by a perturbation of reaction from within; the internal equilibria must be restored at the expense of the forces developed by the nutrition, and must be fixed by modifications of structure. ”

“ And the fact which we have here particularly to observe, is, that as a corollary from the general law of equilibration above set forth, the evolution of every aggregate must go on until this equilibrium mobile is established; since, as we have seen, an excess of force which the aggregate possesses in any direction, must eventually be expended in overcoming resistances to change in that 448direction: leaving behind only those movements which compensate each other, and so form a moving equilibrium. ”

“ We may, by parallel reasoning, reach the conclusion that, even apart from the action of any external force, the equilibrium of a homogeneous aggregate must be destroyed by the unequal actions of its parts on each other. ”

“ They are regarded as satisfying a certain well-defined class of conditions of equilibrium, but as not satisfying at all certain other conditions which would be necessary for complete equilibrium, in consequence of which they are subject to gradual changes, which ultimately determine their rupture. ”

“ In the cases studied, are found the two fundamentally important relations expressed by the law of equilibrium: the equilibrium constant is a measure of the stability of a certain system and, in a way, of its reactivity at a given temperature ”

“ It is required to determine the conditions of stability for a system in equilibrium, consisting of two different homogeneous masses meeting at a surface of discontinuity, the perimeter of which is invariable, as well as the exterior of the whole system, which is also impermeable to heat. ”

“ The equilibrium which is due to such passive properties is thus widely distinguished from that caused by the balance of the active tendencies of the system, where an external influence, or a change in the initial state, infinitesimal in amount, is sufficient to produce change either in the positive or negative direction. ”

“ Respecting the structural state simultaneously reached, it must obviously be one presenting an arrangement of forces that counterbalance all the forces to which the aggregate is subject. So long as there remains a residual force in any direction—be it excess of a force exercised by the aggregate on its environment, or of a force exercised by its environment on the aggregate, equilibrium does not exist ”

“ Every change is towards a balance of forces; and of necessity can never cease until a balance of forces is reached. When treating of equilibration under its general aspects (First Principles, Part II., Chap. xxii.) , we saw that every aggregate having compound movements tends continually towards a moving equilibrium; since any unequilibrated force to which such an aggregate is subject, if not of a kind to overthrow it altogether, must continue modifying its state until an equilibrium is brought about. ”

“ It is a corollary that when the environment is changed, the equilibrium of functions is disturbed, and there must follow one of two results—either the equilibrium is overthrown or it is re-adjusted: there is a re-equilibration. ”

“ The equations which express the condition of equilibrium, as also its statement in words, are to be interpreted in accordance with the general usage in respect to differential equations, that is, infinitesimals of higher orders than the first relatively to those which express the amount of change of the system are to be neglected. But to distinguish the different kinds of equilibrium in respect to stability, we must have regard to the absolute values of the variations. ”

“ Hence, for a state of equilibrium which is on the limit between stability and instability, it is necessary that the equation shall be satisfied, when the relations between , and are determined from the fundamental equations on the supposition that the conditions of equilibrium relating to temperature and the potentials remain satisfied. ”

“ Such a system is formed by ice, water, and vapour, in which the three portions, each in itself homogeneous, can be mechanically separated from one another. When equilibrium exists between different, physically distinct parts, it is known as heterogeneous equilibrium. It is, now, with heterogeneous equilibria, with the conditions under which a heterogeneous system can exist, that we shall deal here. ”

“ Equilibration is the final result of these trans­for­ma­tions which an evolving aggregate undergoes. The changes go on until there is reached an equilibrium between the forces which all parts of the aggregate are exposed to and the forces these parts oppose to them. Equilibration may pass through a transition stage of balanced motions (as in a planetary system) or of {142} balanced functions (as in a living body) on the way to ultimate equilibrium; but the state of rest in inorganic bodies, or death in organic bodies, is the necessary limit of the changes constituting evolution. ”

“ If this force is such that in expending itself it disturbs beyond rectification the balance of the organic processes, then the aggregate is disintegrated or decomposed. But if it does not overthrow that moving equilibrium constituting the life of the aggregate, then the aggregate continues in that modified form produced by the expenditure of the force. Thus, by direct equilibration, continually furthered by indirect equilibration, there must arise this distinction between the outer part adapted to meet outer forces, and the inner part adapted to meet inner forces. ”

“ For in every case the balance arrived at is relative, and not absolute—is a cessation of the motion of some particular body in relation to a certain point or points, involving neither the disappearance of the relative motion lost, which is simply transformed into other motions, nor a diminution of the body’s motions with respect to other points. Thus understanding equilibration, it manifestly includes that equilibrium mobile, which at first sight seems of another nature. ”

“ In the first place, if the system is in a state in which the entropy is greater than in any other state of the ame energy, it is evidently in equilibrium, as any change of state must involve either a decrease of entropy or an increase of energy, which are alike impossible for an isolated system. ”

“ Either complete equilibrium must in itself be an impossibility, or the changes of energy introduce themselves in the circular process before that equilibrium which is in itself possible has appeared.—But it would be madness to ascribe a feeling of self-preservation to existence! ”

“ The most important truth which can be called in to aid us in this difficulty and apparent confusion of phenomena is that of the perpetual competitive operation of both the forces of evolution and of dissolution. Both these influences are at all times and in all kinds of aggregates simultaneously at work. The history of every aggregate is that of its struggle with these opposite contending influences. The final equilibration implies this. It is the establishment of equilibrium between just these forces. In the evolution of a star the forces of dissolution are mostly within the aggregate. ”

“ Whence we found it to follow that the final structural arrangements, must be such as will meet all the forces acting on the aggregate, by equivalent antagonist forces. What is the implication in the case of organic aggregates; the equilibrium of which is a moving one? We have seen that the maintenance of such a moving equilibrium, requires the habitual genesis of internal forces corresponding in number, directions, and amounts to the external incident forces—as many inner functions, single or combined, as there are single or combined outer actions to be met. ”