Related papers: Legendre Transformation under Micro Canonical Ense…
The Legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in typical undergraduate or graduate courses, the power of motivation and…
We show how the Legendre transforms of the fundamental thermodynamic relation can be used to introduce different statistical ensembles.
Legendre transform between thermodynamic quantities such as the Helmholtz free energy and entropy plays a key role in the formulation of the canonical ensemble. In the standard treatment, the transform exchanges the independent variable…
This paper offers a pedestrian guide from the fundamental properties of entropy to the axioms of thermodynamics, which are a consequence of the axiom of statistical physics. It also dismantles flawed concepts, such as assigning physical…
A pedagogical approach for deriving the statistical mechanical partition function, in a manner that emphasizes the key role of entropy in connecting the microscopic states to thermodynamics, is introduced. The connections between the…
We discuss a subtlety involved in the calculation of multifractal spectra when these are expressed as Legendre-Fenchel transforms of functions analogous to free energy functions. We show that the Legendre-Fenchel transform of a free energy…
An extensive table of pairs of functions linked by the Legendre transformation is presented. Many special functions and formulas that are used in the sciences are included in the pairs. Formulations are provided for finding the Legendre…
The relationship between the Hamiltonian and Lagrangean functions in analytical mechanics is a type of duality. The two functions, while distinct, are both descriptive functions encoding the behavior of the same dynamical system. One…
In this paper we construct a q-analogue of the Legendre transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables).…
We study new Legendre transforms in classical mechanics and investigate some of their general properties. The behaviour of the new functions is analyzed under coordinate transformations.When invariance under different kinds of…
Nonstationary thermodynamic quantities depend on the full details of nonstationary probability distributions, making them difficult to measure directly in experiments and numerics. We propose a method to infer thermodynamic quantities in…
Legendre invariant metrics have been introduced in Geometrothermodynamics to take into account the important fact that the thermodynamic properties of physical systems do not depend on the choice of thermodynamic potential from a geometric…
The microcanonical ensemble has long been a starting point for the development of thermodynamics from statistical mechanics. However, this approach presents two problems. First, it predicts that the entropy is only defined on a discrete set…
The recent link discovered between generalized Legendre transforms and non-dually flat statistical manifolds suggests a fundamental reason behind the ubiquity of R\'{e}nyi's divergence and entropy in a wide range of physical phenomena.…
The inequivalence of thermodynamical ensembles related by a Legendre transformation is manifest in self-gravitating systems and in black hole thermodynamics. Using the Poincare's method of the linear series, we describe the mathematical…
Associated Legendre functions of fractional degree appear in the solution of boundary value problems in wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer,…
Equilibrium statistics of finite Hamiltonian systems is fundamentally described by the microcanonical ensemble (ME). Canonical, or grand-canonical partition functions are deduced from this by Laplace transform. Only in the thermodynamic…
A comparative analysis of two different versions of the Legendre transformation is presented. We provide an almost complete although somewhat superficial review of the geometric background for analytical mechanics. Complete coordinate…
The phenomenon of partial equivalence of statistical ensembles is illustrated by discussing two examples, the mean-field XY and the mean-field spherical model. The configurational parts of these systems exhibit partial equivalence of the…
Envariance -- entanglement assisted invariance -- is a recently discovered symmetry of composite quantum systems. We show that thermodynamic equilibrium states are fully characterized by their envariance. In particular, the microcanonical…