On the Entropy of a One-Dimensional Gas with and without Mixing Using Sinai Billiard

A one-dimensional gas comprising N point particles undergoing elastic collisions within a finite space described by a Sinai billiard generating identical dynamical trajectories are calculated and analyzed with regard to strict extensivity of the entropy definitions of Boltzmann–Gibbs. Due to the collisions, trajectories of gas particles are strongly correlated and exhibit both chaotic and periodic properties. Probability distributions for the position of each particle in the one-dimensional gas can be obtained analytically, elucidating that the entropy in this special case is extensive at any given number N. Furthermore, the entropy obtained can be interpreted as a measure of the extent of interactions between molecules. The results obtained for the non-mixable one-dimensional system are generalized to mixable one- and two-dimensional systems, the latter by a simple example only providing similar findings.

Brief lectures on duality, integrability and deformations

We provide a pedagogical introduction to some aspects of integrability, dualities and deformations of physical systems in [Formula: see text] and in [Formula: see text] dimensions. In particular, we concentrate on the T-duality of point particles and strings as well as on the Ruijsenaars duality of finite many-body integrable models, we review the concept of the integrability and, in particular, of the Lax integrability and we analyze the basic examples of the Yang–Baxter deformations of nonlinear [Formula: see text]-models. The central mathematical structure which we describe in detail is the [Formula: see text]-model which is the dynamical system exhibiting all those three phenomena simultaneously. The last part of the paper contains original results, in particular, a formulation of sufficient conditions for strong integrability of non-degenerate [Formula: see text]-models.

Classical Covariance

Classical mechanics, from point particles through rigid objects and continuum mechanics is reviewed based on the notions of tensors, transformations, and the metric, as developed in the first two chapters. The geodesic equation on flat and curved spaces is introduced and solved in a classical setting. Motion in a potential, particularly a gravitational potential, is discussed. Galilean covariance and transformations are introduced. Time as a fourth dimension is shown to arise even in a classical setting, even if not as rigorous as it would be in relativity theory.

Tidal effects for spinning particles

Expanding on the recent derivation of tidal actions for scalar particles, we present here the action for a tidally deformed spin-1/2 particle. Focusing on operators containing two powers of the Weyl tensor, we combine the Hilbert series with an on-shell amplitude basis to construct the tidal action. With the tidal action in hand, we compute the leading-post-Minkowskian tidal contributions to the spin-1/2–spin-1/2 amplitude, arising at $$ \mathcal{O} $$ O (G2). Our amplitudes provide evidence that the observed long range spin-universality for the scattering of two point particles extends to the scattering of tidally deformed objects. From the scattering amplitude we find the conservative two-body Hamiltonian, linear and angular impulses, eikonal phase, spin kick, and aligned-spin scattering angle. We present analogous results in the electromagnetic case along the way.

Stochastic branching at the edge: individual-based modeling of tumor cell proliferation

An individual-based model of stochastic branching is proposed and studied, in which point particles drift in $$\bar{\mathbb {R}}_{+}:=[0,+\infty )$$ R ¯ + : = [ 0 , + ∞ ) toward the origin (edge) with unit speed, where each of them splits into two particles that instantly appear in $$\bar{\mathbb {R}}_{+}$$ R ¯ + at random positions. During their drift, the particles are subject to a random disappearance (death). The model is intended to capture the main features of the proliferation of tumor cells, in which trait $$x\in \bar{\mathbb {R}}_{+}$$ x ∈ R ¯ + of a given cell is time to its division and the death is caused by therapeutic factors. The main result of the paper is proving the existence of an honest evolution of this kind and finding a condition that involves the death rate and cell cycle distribution parameters, under which the mean size of the population remains bounded in time.

One example of epistemological idealization

In different branches of science we find idealizations. In physics we find frictionless surfaces, point-particles of molecules that don?t exert force on one another, in game theory we find fully rational agents who are aware of all of their preferences, and in biology we find infinite populations which do not migrate, and which do not suffer any sort of evolutionary pressure. It seems to us that similar idealizations can be found in epistemology. To be more specific, we mean the models of probabilistic coherentism. Exactly like scientific idealizations which disregard certain very real factors, because their goal is to shed light on the connections between important scientific concepts, this model of probabilistic coherence idealizes concepts of reliability of information sources and independence to show the connection between coherence and reliability.