347 research outputs found
The evolution of radiation towards thermal equilibrium: A soluble model which illustrates the foundations of Statistical Mechanics
In 1916 Einstein introduced the first rules for a quantum theory of
electromagnetic radiation, and he applied them to a model of matter in thermal
equilibrium with radiation to derive Planck's black-body formula. Einstein's
treatment is extended here to time-dependent stochastic variables, which leads
to a master equation for the probability distribution that describes the
irreversible approach of Einstein's model towards thermal equilibrium, and
elucidates aspects of the foundation of statistical mechanics. An analytic
solution of this equation is obtained in the Fokker-Planck approximation which
is in excellent agreement with numerical results. At equilibrium, it is shown
that the probability distribution is proportional to the total number of
microstates for a given configuration, in accordance with Boltzmann's
fundamental postulate of equal a priori probabilities for these states. While
the counting of these configurations depends on particle statistics- Boltzmann,
Bose-Einstein, or Fermi-Dirac - the corresponding probability is determined
here by the dynamics which are embodied in the form of Einstein's quantum
transition probabilities for the emission and absorption of radiation. In a
special limit, it is shown that the photons in Einstein's model can act as a
thermal bath for the evolution of the atoms towards the canonical equilibrium
distribution of Gibbs. In this limit, the present model is mathematically
equivalent to an extended version of the Ehrenfests' ``dog-flea'' model, which
has been discussed recently by Ambegaokar and Clerk
Entropy and Time
The emergence of a direction of time in statistical mechanics from an
underlying time-reversal-invariant dynamics is explained by examining a simple
model. The manner in which time-reversal symmetry is preserved and the role of
initial conditions are emphasized. An extension of the model to finite
temperatures is also discussed.Comment: 9 pages, 8eps figures. To appear in the theme issue of the American
Journal of Physics on Statistical Physic
Coherent states and the classical-quantum limit considered from the point of view of entanglement
Three paradigms commonly used in classical, pre-quantum physics to describe
particles (that is: the material point, the test-particle and the diluted
particle (droplet model)) can be identified as limit-cases of a quantum regime
in which pairs of particles interact without getting entangled with each other.
This entanglement-free regime also provides a simplified model of what is
called in the decoherence approach "islands of classicality", that is,
preferred bases that would be selected through evolution by a Darwinist
mechanism that aims at optimising information. We show how, under very general
conditions, coherent states are natural candidates for classical pointer
states. This occurs essentially because, when a (supposedly bosonic) system
coherently exchanges only one quantum at a time with the (supposedly bosonic)
environment, coherent states of the system do not get entangled with the
environment, due to the bosonic symmetry.Comment: This is the definitive version of a paper entitled The
classical-quantum limit considered from the point of view of entanglement: a
survey (author T. Durt). The older version has been replaced by the
definitive on
The Area Quantum and Snyder Space
We show that in the Snyder space the area of the disc and of the sphere can
be quantized. It is also shown that the area spectrum of the sphere can be
related to the Bekenstein conjecture for the area spectrum of a black hole
horizon.Comment: 7 pages, in Press, Physics Letters
Einstein's quantum theory of the monatomic ideal gas: non-statistical arguments for a new statistics
In this article, we analyze the third of three papers, in which Einstein
presented his quantum theory of the ideal gas of 1924-1925. Although it failed
to attract the attention of Einstein's contemporaries and although also today
very few commentators refer to it, we argue for its significance in the context
of Einstein's quantum researches. It contains an attempt to extend and exhaust
the characterization of the monatomic ideal gas without appealing to
combinatorics. Its ambiguities illustrate Einstein's confusion with his initial
success in extending Bose's results and in realizing the consequences of what
later became to be called Bose-Einstein statistics. We discuss Einstein's
motivation for writing a non-combinatorial paper, partly in response to
criticism by his friend Ehrenfest, and we paraphrase its content. Its arguments
are based on Einstein's belief in the complete analogy between the
thermodynamics of light quanta and of material particles and invoke
considerations of adiabatic transformations as well as of dimensional analysis.
These techniques were well-known to Einstein from earlier work on Wien's
displacement law, Planck's radiation theory, and the specific heat of solids.
We also investigate the possible role of Ehrenfest in the gestation of the
theory.Comment: 57 pp
Exact Solution for the Time Evolution of Network Rewiring Models
We consider the rewiring of a bipartite graph using a mixture of random and
preferential attachment. The full mean field equations for the degree
distribution and its generating function are given. The exact solution of these
equations for all finite parameter values at any time is found in terms of
standard functions. It is demonstrated that these solutions are an excellent
fit to numerical simulations of the model. We discuss the relationship between
our model and several others in the literature including examples of Urn,
Backgammon, and Balls-in-Boxes models, the Watts and Strogatz rewiring problem
and some models of zero range processes. Our model is also equivalent to those
used in various applications including cultural transmission, family name and
gene frequencies, glasses, and wealth distributions. Finally some Voter models
and an example of a Minority game also show features described by our model.Comment: This version contains a few footnotes not in published Phys.Rev.E
versio
Quantum mechanical virial theorem in systems with translational and rotational symmetry
Generalized virial theorem for quantum mechanical nonrelativistic and
relativistic systems with translational and rotational symmetry is derived in
the form of the commutator between the generator of dilations G and the
Hamiltonian H. If the conditions of translational and rotational symmetry
together with the additional conditions of the theorem are satisfied, the
matrix elements of the commutator [G, H] are equal to zero on the subspace of
the Hilbert space. Normalized simultaneous eigenvectors of the particular set
of commuting operators which contains H, J^{2}, J_{z} and additional operators
form an orthonormal basis in this subspace. It is expected that the theorem is
relevant for a large number of quantum mechanical N-particle systems with
translational and rotational symmetry.Comment: 24 pages, accepted for publication in International Journal of
Theoretical Physic
Quantitative conditions do not guarantee the validity of the adiabatic approximation
In this letter, we point out that the widely used quantitative conditions in
the adiabatic theorem are insufficient in that they do not guarantee the
validity of the adiabatic approximation. We also reexamine the inconsistency
issue raised by Marzlin and Sanders (Phys. Rev. Lett. 93, 160408, 2004) and
elucidate the underlying cause.Comment: corrected typos. Eq. (32) is corrected. No other change
Stationary Distribution and Eigenvalues for a de Bruijn Process
We define a de Bruijn process with parameters n and L as a certain
continuous-time Markov chain on the de Bruijn graph with words of length L over
an n-letter alphabet as vertices. We determine explicitly its steady state
distribution and its characteristic polynomial, which turns out to decompose
into linear factors. In addition, we examine the stationary state of two
specializations in detail. In the first one, the de Bruijn-Bernoulli process,
this is a product measure. In the second one, the Skin-deep de Bruin process,
the distribution has constant density but nontrivial correlation functions. The
two point correlation function is determined using generating function
techniques.Comment: Dedicated to Herb Wilf on the occasion of his 80th birthda
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