Gibbs

History shows that up to 1870’s, the thermodynamic cycles, particularly Carnot’s cycle, were the most important heuristic instruments as much to formulate the general laws of physics as well to deduce the experimental laws. From this moment on, this instrument falls into disuse with surprising rapidity. At the end of this decade emerges a new thermodynamic formulation, proposed by Gibbs, the thermodynamics of the potentials. This sudden transition from thermodynamic of cycles to potentials was triggered by the difficult to approach the emergence of the phase transition phenomena with the diagrammatic method. The main objective of the article is, then, to analyze the consequences of the substitution, by Gibbs, of the diagrammatic by the geometric method, particularly, its heuristic potential related to the proposal of the formulation of the thermodynamic of potentials.

We study the statistical mechanics of small clusters (N ~ 10 - 100) for two-level systems and harmonic oscillators. Both Boltzmann’s and Gibbs’s definitions of entropy are used. The properties of the studied systems are evaluated numerically but exactly; this means that Stirling’s approximation was not used in the calculation and that the discrete nature of energy was taken into account. Results show that, for the two-level system, using Gibbs entropy prevents temperatures from assuming negative values; however, they reach very high values that are not plausible in physical terms. In the case of harmonic oscillators, there are no significant differences when using either definition of entropy. Both systems show that for N = 100 the exact results evaluated with statistical mechanics coincide with those found in the thermodynamic limit. This suggests that thermodynamics can be applied to systems as small as these.

Researchers have used distributed constraint optimization problems (DCOPs) to model various multi-agent coordination and resource allocation problems. Very recently, Ottens et al. proposed a promising new approach to solve DCOPs that is based on confidence bounds via their Distributed UCT (DUCT) sampling-based algorithm. Unfortunately, its memory requirement per agent is exponential in the number of agents in the problem, which prohibits it from scaling up to large problems. Thus, in this paper, we introduce a new sampling-based DCOP algorithm called Distributed Gibbs, whose memory requirements per agent is linear in the number of agents in the problem. Additionally, we show empirically that our algorithm is able to find solutions that are better than DUCT; and computationally, our algorithm runs faster than DUCT as well as solve some large problems that DUCT failed to solve due to memory limitations.

Many rational aspects of some fields of physics, including those related to the process of discovery, may be better understood using methodological tools typical of the philosophical analysis of science. Throughout this paper we shall use the heuristic tradition of Descartes' analytical (or discovery) method for a rational reconstruction of Gibbs' thermodynamics, or thermodynamics of potentials, seeking to emphasize the potentialities of the representational character of science. Our contention is that Descartes's method of discovery illuminates the intellectual itinerary of the construction of the thermodynamics of potentials.