5,177 research outputs found
Normal and generalized Bose condensation in traps: One dimensional examples
We prove the following results. (i) One-dimensional Bose gases which interact
via unscaled integrable pair interactions and are confined in an external
potential increasing faster than quadratically undergo a complete generalized
Bose-Einstein condensation (BEC) at any temperature, in the sense that a
macroscopic number of particles are distributed on a o(N)number of one-particle
states. (ii) In a one dimensional harmonic trap the replacement of the
oscillator frequency \omega by \omega\ln N/N gives rise to a phase transition
at a=\hbar\omega\beta=1 in the noninteracting gas. For a<1 the limit
distribution of n_0/N^a is exponential and /N^a tends to 1. For a>1 there
is BEC with a condensate density /N going to 1-1/a. For a>=1, (\ln
N/N)(n_0-) is asymptotically distributed following Gumbel's law. For any
a>0 the free energy is -(\pi^2/6a\beta)N/\ln N+o(N/\ln N), with no singularity
at a=1. (iii) In Model (ii) both above and below the critical temperature the
the gas undergoes a complete generalized BEC, thus providing a coexistence of
ordinary and generalized condensates below the critical point. (iv) Adding an
interaction =o(N\ln N) to Model (ii) we prove that a complete generalized
BEC occurs at all temperatures.Comment: Published version with further improvement
Unknowns and unknown unknowns: from dark sky to dark matter and dark energy
Answering well-known fundamental questions is usually regarded as the major
goal of science. Discovery of other unknown and fundamental questions is,
however, even more important. Recognition that "we didn't know anything" is the
basic scientific driver for the next generation. Cosmology indeed enjoys such
an exciting epoch. What is the composition of our universe? This is one of the
well-known fundamental questions that philosophers, astronomers and physicists
have tried to answer for centuries. Around the end of the last century,
cosmologists finally recognized that "We didn't know anything". Except for
atoms that comprise slightly less than 5% of the universe, our universe is
apparently dominated by unknown components; 23% is the known unknown (dark
matter), and 72% is the unknown unknown (dark energy). In the course of
answering a known fundamental question, we have discovered an unknown, even
more fundamental, question: "What is dark matter? What is dark energy?" There
are a variety of realistic particle physics models for dark matter, and its
experimental detection may be within reach. On the other hand, it is fair to
say that there is no widely accepted theoretical framework to describe the
nature of dark energy. This is exactly why astronomical observations will play
a key role in unveiling its nature. I will review our current understanding of
the "dark sky", and then present on-going Japanese project, SuMIRe, to discover
even more unexpected questions.Comment: 11 pages, 7 figures, to appear in the proceedings of SPIE
Astronomical Instrumentation "Observational frontiesr of astronomy for the
new decade", based on a plenary talk on June 28, 201
- …