For pt.I see Found. Phys., vol.9, no.11-12, p819-29 (1979). A general assumption made in a previous paper is modified by the elimination of the metrical entropy in favour of empirical entropy. Without reference to absolute temperature or metrical entropy, one then arrives at a generic form of the canonical probability in phase in which temperature does not occur. It is argued that this is as it should be if one looks upon the canonical ensemble as merely being the representative of a system in thermal equilibrium with its surroundings. One motivation for this work is to take a view of 'statistical mechanics' which does away with probabilistic notions altogether.
For pt.I see Found. Phys., vol.9, no.11-12, p819-29 (1979). A general assumption made in a previous paper is modified by the elimination of the metrical entropy in favour of empirical entropy. Without reference to absolute temperature or metrical entropy, one then arrives at a generic form of the canonical probability in phase in which temperature does not occur. It is argued that this is as it should be if one looks upon the canonical ensemble as merely being the representative of a system in thermal equilibrium with its surroundings. One motivation for this work is to take a view of 'statistical mechanics' which does away with probabilistic notions altogether.
To eliminate exceptions to the second law due to fluctuation phenomena, etc., the following changes are necessary. First we must state the laws of thermodynamics in terms of Gibbsian ensembles rather than of individual systems. Second we must sharpen our conception of the experimental procedure required for producing states of thermodynamic equilibrium. It will not suffice to isolate the system and wait. We must have the positive stirring action of thermal contact with a thermostat. Using the new definition of thermal equilibrium we can place statistical mechanics on a more satisfactory footing by showing that the thermostat experimental procedure leads to the canonical ensemble and maximizes v. Neumann's microscopic entropy. (Details of this proof appear in succeeding paper). Difficulties with his macroscopic definition of entropy are avoided by showing that his original microscopic definition is satisfactory when used in conjunction with a proper operational definition of thermodynamic equilibrium. In conclusion the second law of thermodynamics is derived from quantum statistics on the basis of the microscopic definition of entropy.
"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 Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular, in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
We examine the fundamental aspects of statistical mechanics, dividing the problem into a discussion purely about probability, which we analyse from a Bayesian standpoint. We argue that the existence of a unique maximising probability distribution {p(j∣K)} for states labelled by j given data K implies that the corresponding maximal value of the information entropy σ({(pj∣K)})=−∑j(pj∣K)ln(pj∣K) depends explicitly on the data at equilibrium and on the Hamiltonian of the system. As such, it is a direct measure of our uncertainty about the exact state of the body and can be identified with the traditional thermodynamic entropy. We consider the well known microcanonical, canonical and grand canonical methods and ensure that the fluctuations about mean values are generally minuscule for macroscopic systems before identifying these mean values with experimental observables and thereby connecting to many standard results from thermodynamics. Unexpectedly, we find that it is not generally possible for a quantum process to be both isentropic and reversibly adiabatic. This is in sharp contrast to traditional thermodynamics where it is assumed that isentropic, reversible adiabatic processes can be summoned up on demand and easily realised. By contrast, we find that linear relations between pressures Pj and energies Ej are necessary and sufficient conditions for a quasi-static and adiabatic change to be isentropic, but, of course, this relationship only holds for a few especially simple systems, such as the perfect gas, and is not generally true for more complicated systems. By considering the associated entropy increases up to second order in small volume changes we argue that the consequences are in practice negligible.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
This chapter discusses some comments, and the rules of quantum mechanics. The quantum mechanics is universally valid for all the ensembles to which the more general formalism just alluded to is applicable. Any ensemble of systems the specification of which involves no reference to the results of future measurements that are to be performed on these systems obeys the quantum rules. One of the views most often expressed by the supporters of the "Copenhagen interpretation" is that quantum mechanics is a complete theory. The main motivation for the assumption is that the splitting of ensemble and subensembles cannot be made in any straightforward manner. However, this assumption implies the existence of particular, hitherto unknown forces, and it is therefore quite generally considered as unattractive. Thus the probabilities, or statistical distributions, that conventional quantum mechanics introduces into physics are inherently different from those found in classical mechanics, for example, in the kinetic theory of gases.