25,502 research outputs found
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
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 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
Geometric Foundation of Thermo-Statistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit
A geometric foundation thermo-statistics is presented with the only axiomatic
assumption of Boltzmann's principle S(E,N,V)=k\ln W. This relates the entropy
to the geometric area e^{S(E,N,V)/k} of the manifold of constant energy in the
finite-N-body phase space. From the principle, all thermodynamics and
especially all phenomena of phase transitions and critical phenomena can
unambiguously be identified for even small systems. The topology of the
curvature matrix C(E,N) of S(E,N) determines regions of pure phases, regions of
phase separation, and (multi-)critical points and lines. Within
Boltzmann's principle, Statistical Mechanics becomes a geometric theory
addressing the whole ensemble or the manifold of all points in phase space
which are consistent with the few macroscopic conserved control parameters.
This interpretation leads to a straight derivation of irreversibility and the
Second Law of Thermodynamics out of the time-reversible, microscopic,
mechanical dynamics. This is all possible without invoking the thermodynamic
limit, extensivity, or concavity of S(E,N,V). The main obstacle against the
Second Law, the conservation of the phase-space volume due to Liouville is
overcome by realizing that a macroscopic theory like Thermodynamics cannot
distinguish a fractal distribution in phase space from its closure.Comment: 26 pages, 6 figure
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 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 \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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