Function Gi Research Articles

Information theory and variational properties of a strongly inhomogeneous collisionless plasma

All solutions describing the electric field of a collisionless Maxwellian or not-Maxwellian onedimensional plasma at stationary equilibrium in a box can be obtained from the requirement that the electrostatic energy has an extremum value under the constraint that I = − ∑i∫δξiGi(n) dx be given, where the Gi(n) are any functions of the density n(x), each Gi(n(x)) being defined for x in the interval Δxi delimited by the zeros of dn/dx or by the walls of the box. The functions Gi may be chosen in an arbitrary way, which is in agreement with the occurrence of an arbitrary function in the most general solution of the relevant mathematical equations. The quantity I describes the “information” which is necessary for a complete description of the electric properties of the physical system. Moreover, it satisfies the following remarkable properties: 1. a) it takes an extremum value at equilibrium with respect to arbitrary variations of the electric field, which, however, should be fixed at the boundaries of each Δxi, under the constraint that the electrostatic energy has a given value; 2. b) in the limit in which thr Penrose stability criterion holds everywhere in the box (linear theory) or can be considered to hold as a local criterion (not too inhomogeneous plasmas) this extremum value is a maximum when the plasma is stable, while if I is minimum the plasma is unstable; 3. c) the extremum value of I with respect to arbitrary variations of n(x), under the constraint that the total number of particles is fixed, corresponds to a completely homogeneous plasma and is connected to the stability problem in the same way as in b). On the ground of these results and according to information theory the quantity I can be assumed as a definition of an entropy which describes the irreversible behaviour of the collisionless plasma, also when situations very far from the statistical equilibrium are considered. Moreover, using this definition of the entropy a quantity F can be introduced the extremum values of which for a fixed volume of the box and fixed kinetic energy of the plasma characterize the equilibrium configurations and which plays formally the same role as the Helmholtz free energy in thermodynamics.

An Extremum Result

The main object of this paper is to prove the following:Theorem. Let f1, … ,fk be linearly independent continuous functions on a compact space. Then for 1 ≤ s ≤ k there exist real numbers aij, 1 ≤ i ≤ s, 1 ≤ j ≤ k, with {aij, 1 ≤ i, j ≤ s} n-singular, and a discrete probability measure ε*on, such that(a) the functions gi = Σj=1kaijfj 1 ≤ i ≤ s, are orthonormal (ε*) to the fj for s < j ≤ k;(b)The result in the case s = k was first proved in (2). The result when s < k, which because of the orthogonality condition of (a) is more general than that when s = k, was proved in (1) under a restriction which will be discussed in § 3. The present proof does not require this ad hoc restriction, and is more direct in approach than the method of (2) (although involving as much technical detail as the latter in the case when the latter applies).

Vertices of plane curves

PROOF. Apply Theorem J for the case w==r. We obtain for gr an expression which differs from the one just written only in the fact that the terms Brtr+ifr+i+Brtr+2fr+2+ • • • +Br,nfn are missing from its numerator. But the coefficients J3r,, = 0 when r<s. Hence the two expressions are equal. REMARK. The generalization of the method to orthonormalization with respect to a general norming or weight function p is obvious. One applies the process described to the functions (p)fi and obtains functions g% (=linear combinations of the (p)fi) which are orthonormal with respect to the weight function unity. Dividing through by the common factor (p) one forms functions gi(p)~~* which are orthonormal with respect to p.

Solution of the problem of Plateau

ion being made, in case the representation g is improper of the first kind, of the values of 0, at most denumerably infinite in number, where gi(0) is discontinuous. (19.9) rests on the fact (whose proof is trivial) that if fi(m)(t) tends uniformly to the continuous fi(t) when m-* oo, then if tm -t as m-> oo, we have lim fi(m) (t) =fi(t) for m-> oo . The assertion is now easily proved that if (19.10) Xi= =WFi(w) This content downloaded from 157.55.39.128 on Mon, 18 Jul 2016 05:43:48 UTC All use subject to http://about.jstor.org/terms 304 JESSE DOUGLAS [January are the harmonic functions determined by gi(O), then (19.1 1) Z1F 2 (w) = 0, t= 1 so that the surface (19.10) is minimal. For consider (18.2) without the factor w: (in)' 1 2eiG (19.12) F1 (w) = 2 i9 2gi(m) 6)dO. Since all the polygons P(m) are contained in a finite region of space, the functions gi(m)(0) are uniformly bounded; and if w is any fixed point interior to the unit circle, the denominator (eiO -W)2 remains superior in absolute value to a fixed positive quantity when ei0 describes C. Therefore the integrand in (19.12) remains uniformly bounded during the limit process (19.9); consequently the limit of the integral is equal to the integral of the limit: (19 .13) lim Fi (w) = Fi (w) . M-n+00 It is evident that in case g is improper of the first kind this result is not affected by the circumstance that the points of discontinuity of g,(G) are not considered in the limit relation (19.9), since these points, being at most denumerably infinite in number, form a set of zero measure. The result (19.11) now follows from (19.13) and the subsistence of (19.5) for every m. 20. The minimal surface is bounded by r. To show that the minimal surface whose existence is proved in the preceding section is bounded by r, we must prove that the representation (19.8) of r is proper. That it cannot be improper of the second kind is proved in ?18, which, being based on the relation (19.11), applies here with full validity. We cannot however apply the argument of ?17 to prove that (19.8) cannot be improper of the first kind. For although we would still have for a g of this kind A (g) = + so, it would not be true in the case of a general Jordan contour that A (g) sometimes takes finite values. We therefore use the following argument, based on the relation (19.11), to obtain the desired result. Suppose that under g the point P of C corresponds to the arc Q'Q of r. Since r is a Jordan curve, Q' and Q are distinct: and if ai denote the co6rdinates of Q', bi of Q, the distance Q'Q or I with (20.1) 12 = (b.ai)2 i-a1 is not equal to zero. This content downloaded from 157.55.39.128 on Mon, 18 Jul 2016 05:43:48 UTC All use subject to http://about.jstor.org/terms 1931] THE PROBLEM OF PLATEAU 305 There is no loss of generality in supposing P to be at w = 1, for this may be achieved by a rotation of the unit circle, which changes nothing essential.