Noises in thin metallic fills, semiconductors, nerve tissues, and many other media, have measured spectral densities proportional tof^\theta - 2}withfthe frequency and\thetaa constant0 \leq \theta < 2. The energy of these "f^{\theta-2}noises" behaves more "erratically" in time, than expected from functions subject to the Wiener-Khinchin spectral theory. Moreover, blind extrapolation of the "f^{\theta -2}law" tof=0incorrectly suggests, when0 \leq 1, that the total energy is infinite ("infrared catastrophe"). The problems thus raised are of the greatest theoretical interest, and of the greatest practical importance in the design of electronic devices. The present paper reinterprets these spectral measurements without paradox, by introducing a concept to be called "conditional spectrum." Examples are given of functions ruled by chance, that have the observed "erratic" behavior and conditional spectral density. A conditional spectrum is obtained when a procedure, meant to measure a sample Wiener-Khinchin spectrum, is applied to a sample conditioned to be nonconstant. The conditional spectrum is defined, not only for nonconstant samples from all random functions of the Wiener-Khinchin theory, but also for nonconstant samples from certain nonstationary random functions, and for nonconstant samples from a new generalization of random functions, called "sporadic functions." The simplest sporadic functions, having af^{-2}conditional spectral density, is a "direct current" with a single discontinuity uniformly distributed over- \infty < t < \infty. The otherf^{\theta -2}noises to be described partake both of direct current and of white noise (the\theta =2limit off^{\theta - 2}noise), and continuously span the gap between these limits. In many cases, their noise energy can be said to be proportional to the square of their "dc" component. Empirical studies will be suggested, and the descriptive value of the concepts of dc component and of spectrum will be discussed.

"Boltzmann's problem" of statistical thermodynamics, is that of eliminating the paradoxical incompatibility of structure, existing between the irreversibility of the classical phenomenological thermodynamics, and the reversibility of any purely kinetic model, one could ever think of for these phenomena. One finds that, in order to construct kinetic "analogs" to the laws of phenomenological thermodynamics, the dynamics of large assemblies of molecules (Liouville theorem, etc,\cdots) must be completed by some hypotheses of randomness. Once established, this randomness can be followed up in its development, with no new conceptual paradox (although with great technical difficulty: there is a great amount of current work on this topic). But the introduction of randomness still raises entirely uncleared problems. Since, therefore, the kinetic foundations of thermodynamics are not sufficient in the absence of further hypotheses of randomness, are they still quite necessary in the presence of such hypotheses? Or else, could not one "short-circuit" the atoms, by centering upon any elements of randomness, for example those introduced by the process of observation? Our aim is to show (partly after Szilard) that a substantial part of the results, usually obtained through kinetic arguments, could be obtained by postulating from the outset a statistical distribution for the properties of a system, and following up with a purely phenomenological argument. The spirit of the theory is extremely close to that of the conventional (Copenhagen) approach to quantum theory, and the results are quite parallel, although the mathematics is quite different. Randomness is introduced by following the modern statistical theory of the estimation of non directly observable intensive variables of state, such as the temperature. The discussion of the methodological foundations of modern statistics can thus be translated into a full-fledged, and possibly significant , counterpart of the discussion of the kinetic foundations of thermodynamics. Statistics is thus provided with a particularly concrete example for some of its more involved methods; thermodynamics appears clarified in its classical aspects, and is further completed with an apparently new uncertainty relationship. It may also be of interest to the communication engineer to have a unified treatment of the foundations of fluctuation phenomena, and of methods of fighting noise : a discussion of entropy and information performed in this spirit, will be given in Part II of the paper.

The purpose of the investigation to be reported here differs in one essential respect from the purpose of most statistical studies on problems of communication. We wish to state this difference from the beginning, in order to avoid misunderstanding of the methods used later on. Most communication problems are, broadly speaking, engineering problems; that is, they deal with the construction and evaluation of new methods of communication, using one's knowledge of "Nature," acquired elsewhere. Our problem is, broadly speaking, physical; we seek to improve our knowledge, or at least our description, of "Nature" by using models in which some statistical concepts, developed in the study of recent communication problems, are used side by side with such concepts of physics as were already used to solve older engineering problems. This difference of purpose will also explain an unessential character of this paper: the calculations are much simpler than in other papers of the symposium, because here one could choose the problems leading to simple theories by the present methods (but not by previous ones), whereas in other papers the problems were imposed.