A scientific theory consists of a symbolic framework containing laws and concepts that are derived by deductive reasoning from fundamental laws and primitive concepts, and complemented by the correspondence or the relation between the symbolic concepts and the real objects. The symbolic framework is understood as a representation of the real, constructed by our minds like a realistic painting. In accordance with this point of view, scientific laws are not discovered but conceived or created by our minds. We show how this approach is organized by rational thinking, which is understood as an innate ability of the mind that emerges between childhood and adolescence. We also examine with some detail the structure of the past scientific theories, pointing out their postulates, or fundamental laws, and the primitive concepts. We will restrict ourselves to some of the physical theories, starting with those that emerged in the Hellenistic period, and then proceed to examine other theories until the end of the seventeenth century.

We present an analysis of six deterministic models for epidemic spreading. The evolution of the number of individuals of each class is given by ordinary differential equations of the first order in time, which are set up by using the laws of mass action providing the rates of the several processes that define each model. The epidemic spreading is characterized by the frequency of new cases, which is the number of individuals that are becoming infected per unit time. It is also characterized by the basic reproduction number, which we show to be related to the largest eigenvalue of the stability matrix associated with the disease-free solution of the evolution equations. We also emphasize the analogy between the outbreak of an epidemic with a critical phase transition. When the density of the population reaches a critical value the spreading sets in, a result that was advanced by Kermack and McKendrick in their study of a model in which the recovered individuals acquire permanent immunization, which is one of the models analyzed here.

Abstract The stochastic thermodynamics provides a framework for the description of systems that are out of thermodynamic equilibrium. It is based on the assumption that the elementary constituents are acted by random forces that generate a stochastic dynamics, which is here represented by a Fokker-Planck-Kramers equation. We emphasize the role of the irreversible probability current, the vanishing of which characterizes the thermodynamic equilibrium and yields a special relation between fluctuation and dissipation. The connection to thermodynamics is obtained by the definition of the energy function and the entropy as well as the rate at which entropy is generated. The extension to quantum systems is provided by a quantum evolution equation which is a canonical quantization of the Fokker-Planck-Kramers equation. An example of an irreversible systems is presented which shows a nonequilibrium stationary state with an unceasing production of entropy. A relationship between the fluxes and the path integral is also presented.

Abstract We give an account and a critical analysis of the use of exact and inexact differentials in the early development of mechanics and thermodynamics, and the emergence of differential calculus and how it was applied to solve some mechanical problems, such as those related to the cycloidal pendulum. The Lagrange equations of motions are presented in the form they were originally obtained in terms of differentials from the principle of virtual work. The derivation of the conservation of energy in differential form as obtained originally by Clausius from the equivalence of heat and work is also examined.

Abstract The equipartition of energy in its simplest form, which is related to the translational motion of the molecules of a gas, was announced independently by Waterston in 1845 and by Clausius in 1857. In its more general form, it was formulated by Maxwell in 1860. Together with the relation between pressure and translational motion, given by the kinetic theory of gases, one can derive the equation of state of an ideal gas. One can also derive the Avogadro law, a fundamental law of physical chemistry as stated by Meyer and Mendeleev. From the equipartition of energy Boltzmann could explain the experimental ratio γ of the specific heats of diatomic gases, an explanation that was countered by Maxwell. We discuss these two conflicting explanation, and present an account and a critical analysis of the emergence of the law of equipartition of energy and other laws that preceded it but are understood as consequences or related to it. Our account includes also the Laplace relation between the speed of sound and γ, and the Clément and Desormes experiment to determine γ.

Abstract According to Sommerfeld, the well known Clausius and Kelvin statements of the second law of thermodynamics comprises two parts. The first part includes the Carnot principle that all Carnot engines operating between the same temperatures have the same efficiency. The second part contains the law of increase in entropy. Usually, the two parts are understood as a logical consequence of these statements, including the Carnot principle. Here, we argue that this principle need not be a derived law and may be considered as a fundamental law, without the need of demonstration. To this end we analyze the roots of the second law, which are contained in the memoir of Carnot on the production of work by heat, and its emergence in the papers of Clausius on heat.

We determined, by numerical simulations, the critical behavior one-dimensional systems with infinitely many absorbing states. These models are modified versions of the conserved lattice gas model in which two adjacent active particles instead of just one can jump to their neighboring sites. The models studied here are examples of systems with many absorbing states and order parameter coupled to a nondiffusive conserved field that show nontrivial critical behavior.

We have simulated a continuous time version of the Nagel-Schreckenberger model of vehicular traffic on highways and calculated the flux as a function of vehicle density. In the low density regime the flux increases linearly with density but becomes a power law when velocities are allowed to increase without bounds. We have simulated also a modified version in which the state of a vehicle depends on the velocity of the vehicle moving ahead. This model displays a phase transition from a state with nonzero flux to a jammed state at a critical density which is strictly less than the closed-packed density. We also study the relationship of this model with self-organized criticality.

A nonequilibrium system of classical particles interacting through nonconservative forces is studied by a stochastic Markovian process defined over the space of the particle positions. The velocities of the particles are treated as independent stochocastic variables with a given probability distribution. We deduce expressions for the rate in which energy is exchanged with the surrounding as well as the rate in which energy is dissipated. These results are applied to the special case in which the irrotational and solenoidal parts of the forces acting on the particles are orthogonal. We also solve exactly a model in which these two forces are linear functions of positions.