7,816 research outputs found
A Fresh Look at Entropy and the Second Law of Thermodynamics
This paper is a non-technical, informal presentation of our theory of the
second law of thermodynamics as a law that is independent of statistical
mechanics and that is derivable solely from certain simple assumptions about
adiabatic processes for macroscopic systems. It is not necessary to assume
a-priori concepts such as "heat", "hot and cold", "temperature". These are
derivable from entropy, whose existence we derive from the basic assumptions.
See cond-mat/9708200 and math-ph/9805005.Comment: LaTex file. To appear in the April 2000 issue of PHYSICS TODA
Quasi-Homogeneous Thermodynamics and Black Holes
We propose a generalized thermodynamics in which quasi-homogeneity of the
thermodynamic potentials plays a fundamental role. This thermodynamic formalism
arises from a generalization of the approach presented in paper [1], and it is
based on the requirement that quasi-homogeneity is a non-trivial symmetry for
the Pfaffian form . It is shown that quasi-homogeneous
thermodynamics fits the thermodynamic features of at least some
self-gravitating systems. We analyze how quasi-homogeneous thermodynamics is
suggested by black hole thermodynamics. Then, some existing results involving
self-gravitating systems are also shortly discussed in the light of this
thermodynamic framework. The consequences of the lack of extensivity are also
recalled. We show that generalized Gibbs-Duhem equations arise as a consequence
of quasi-homogeneity of the thermodynamic potentials. An heuristic link between
this generalized thermodynamic formalism and the thermodynamic limit is also
discussed.Comment: 39 pages, uses RevteX. Published version (minor changes w.r.t. the
original one
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
Off-diagonal long-range order, cycle probabilities, and condensate fraction in the ideal Bose gas
We discuss the relationship between the cycle probabilities in the
path-integral representation of the ideal Bose gas, off-diagonal long-range
order, and Bose--Einstein condensation. Starting from the Landsberg recursion
relation for the canonic partition function, we use elementary considerations
to show that in a box of size L^3 the sum of the cycle probabilities of length
k >> L^2 equals the off-diagonal long-range order parameter in the
thermodynamic limit. For arbitrary systems of ideal bosons, the integer
derivative of the cycle probabilities is related to the probability of
condensing k bosons. We use this relation to derive the precise form of the
\pi_k in the thermodynamic limit. We also determine the function \pi_k for
arbitrary systems. Furthermore we use the cycle probabilities to compute the
probability distribution of the maximum-length cycles both at T=0, where the
ideal Bose gas reduces to the study of random permutations, and at finite
temperature. We close with comments on the cycle probabilities in interacting
Bose gases.Comment: 6 pages, extensive rewriting, new section on maximum-length cycle
Spectral Equivalence of Bosons and Fermions in One-Dimensional Harmonic Potentials
Recently, Schmidt and Schnack (cond-mat/9803151, cond-mat/9810036), following
earlier references, reiterate that the specific heat of N non-interacting
bosons in a one-dimensional harmonic well equals that of N fermions in the same
potential. We show that this peculiar relationship between specific heats
results from a more dramatic equivalence between bose and fermi systems.
Namely, we prove that the excitation spectrums of such bose and fermi systems
are spectrally equivalent. Two complementary proofs are provided, one based on
an analysis of the dynamical symmetry group of the N-body system, the other
using combinatoric analysis.Comment: Six Pages, No Figures, Submitted to Phys. Rev.
Accessibility of physical states and non-uniqueness of entanglement measure
Ordering physical states is the key to quantifying some physical property of
the states uniquely. Bipartite pure entangled states are totally ordered under
local operations and classical communication (LOCC) in the asymptotic limit and
uniquely quantified by the well-known entropy of entanglement. However, we show
that mixed entangled states are partially ordered under LOCC even in the
asymptotic limit. Therefore, non-uniqueness of entanglement measure is
understood on the basis of an operational notion of asymptotic convertibility.Comment: 8 pages, 1 figure. v2: main result unchanged but presentation
extensively changed. v3: figure added, minor correction
The curve of lines on a prime Fano threefold of genus 8
We show that a general prime Fano threefold X of genus 8 can be reconstructed
from the pair , where is its Fano curve of lines and
is the theta-characteristic which gives a natural embedding
\Gamma \subset \matbb{P}^5.Comment: 24 pages, misprints corrected, to appear in International Journal of
Mathematic
Notes on the Third Law of Thermodynamics.I
We analyze some aspects of the third law of thermodynamics. We first review
both the entropic version (N) and the unattainability version (U) and the
relation occurring between them. Then, we heuristically interpret (N) as a
continuity boundary condition for thermodynamics at the boundary T=0 of the
thermodynamic domain. On a rigorous mathematical footing, we discuss the third
law both in Carath\'eodory's approach and in Gibbs' one. Carath\'eodory's
approach is fundamental in order to understand the nature of the surface T=0.
In fact, in this approach, under suitable mathematical conditions, T=0 appears
as a leaf of the foliation of the thermodynamic manifold associated with the
non-singular integrable Pfaffian form . Being a leaf, it cannot
intersect any other leaf const. of the foliation. We show that (N) is
equivalent to the requirement that T=0 is a leaf. In Gibbs' approach, the
peculiar nature of T=0 appears to be less evident because the existence of the
entropy is a postulate; nevertheless, it is still possible to conclude that the
lowest value of the entropy has to belong to the boundary of the convex set
where the function is defined.Comment: 29 pages, 2 figures; RevTex fil
Thermodynamic fluctuation relation for temperature and energy
The present work extends the well-known thermodynamic relation for the canonical ensemble. We start from the general
situation of the thermodynamic equilibrium between a large but finite system of
interest and a generalized thermostat, which we define in the course of the
paper. The resulting identity can account for thermodynamic states
with a negative heat capacity ; at the same time, it represents a
thermodynamic fluctuation relation that imposes some restrictions on the
determination of the microcanonical caloric curve . Finally, we comment briefly on the implications of the present
result for the development of new Monte Carlo methods and an apparent analogy
with quantum mechanics.Comment: Version accepted for publication in J. Phys. A: Math and The
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