39,108 research outputs found

### Asymptotic Freedom: From Paradox to Paradigm

Asymptotic freedom was developed as a response to two paradoxes: the
weirdness of quarks, and in particular their failure to radiate copiously when
struck; and the coexistence of special relativity and quantum theory, despite
the apparent singularity of quantum field theory. It resolved these paradoxes,
and catalyzed the development of several modern paradigms: the hard reality of
quarks and gluons, the origin of mass from energy, the simplicity of the early
universe, and the power of symmetry as a guide to physical law.Comment: 26 pages, 10 figures. Lecture on receipt of the 2004 Nobel Prize. v2:
typo (in Ohm's law) correcte

### Phase Transitions in "Small" Systems - A Challenge for Thermodynamics

Traditionally, phase transitions are defined in the thermodynamic limit only.
We propose a new formulation of equilibrium thermo-dynamics that is based
entirely on mechanics and reflects just the {\em geometry and topology} of the
N-body phase-space as function of the conserved quantities, energy, particle
number and others. This allows to define thermo-statistics {\em without the use
of the thermodynamic limit}, to apply it to ``Small'' systems as well and to
define phase transitions unambiguously also there. ``Small'' systems are
systems where the linear dimension is of the characteristic range of the
interaction between the particles. Also astrophysical systems are ``Small'' in
this sense. Boltzmann defines the entropy as the logarithm of the area
$W(E,N)=e^{S(E,N)}$ of the surface in the mechanical N-body phase space at
total energy E. The topology of S(E,N) or more precisely, of the curvature
determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial
N^2-(\partial^2S/\partial E\partial N)^2$ allows the classification of phase
transitions {\em without taking the thermodynamic limit}. The topology gives
further a simple and transparent definition of the {\em order parameter.}
Attention: Boltzmann's entropy S(E) as defined here is different from the
information entropy and can even be non-extensive and convex.Comment: 8 pages, 4 figures, Invited paper for CRIS200

### Microcanonical Thermostatistics, the basis for a New Thermodynamics, "heat can flow from cold to hot", and nuclear multifragmentation. The correct treatment of Phase Separation after 150 years of statistical mechanics

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 limit are they equivalent to ME for homogeneous systems.
Therefore ME is the only ensemble for non-extensive/inhomogeneous systems like
nuclei or stars where the $\lim_{N\to \infty,\rho=N/V=const}$ does not exist.
Conventional canonical thermo-statistic is inapplicable for non-extensive
systems. This has far reaching fundamental and quite counter-intuitive
consequences for thermo-statistics in general: Phase transitions of first order
are signaled by convexities of $S(E,N,Z,...)$ \cite{gross174}. Here the heat
capacity is {\em negative}. In these cases heat can flow from cold to hot! The
original task of thermodynamics, the description of boiling water in heat
engines can now be treated. Consequences of this basic peculiarity for nuclear
statistics as well for the fundamental understanding of Statistical Mechanics
in general are discussed. Experiments on hot nuclei show all these novel
phenomena in a rich variety. The close similarity to inhomogeneous astro
physical systems will be pointed out. \keyword{Microcanonical statistics, first
order transitions, phase separation, steam engines, nuclear multifragmentation,
negative heat capacity}Comment: 6 pages, 3 figures, Invited plenary talk at VI Latin American
Symposium on Nuclear Physics and Applications, Iguaz\'u, Argentina. October 3
to 7, 200

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