29,671 research outputs found
Enumerative aspects of the Gross-Siebert program
We present enumerative aspects of the Gross-Siebert program in this
introductory survey. After sketching the program's main themes and goals, we
review the basic definitions and results of logarithmic and tropical geometry.
We give examples and a proof for counting algebraic curves via tropical curves.
To illustrate an application of tropical geometry and the Gross-Siebert program
to mirror symmetry, we discuss the mirror symmetry of the projective plane.Comment: A version of these notes will appear as a chapter in an upcoming
Fields Institute volume. 81 page
Continuous phase transitions with a convex dip in the microcanonical entropy
The appearance of a convex dip in the microcanonical entropy of finite
systems usually signals a first order transition. However, a convex dip also
shows up in some systems with a continuous transition as for example in the
Baxter-Wu model and in the four-state Potts model in two dimensions. We
demonstrate that the appearance of a convex dip in those cases can be traced
back to a finite-size effect. The properties of the dip are markedly different
from those associated with a first order transition and can be understood
within a microcanonical finite-size scaling theory for continuous phase
transitions. Results obtained from numerical simulations corroborate the
predictions of the scaling theory.Comment: 8 pages, 7 figures, to appear in Phys. Rev.
Microcanonical entropy for small magnetisations
Physical quantities obtained from the microcanonical entropy surfaces of
classical spin systems show typical features of phase transitions already in
finite systems. It is demonstrated that the singular behaviour of the
microcanonically defined order parameter and susceptibility can be understood
from a Taylor expansion of the entropy surface. The general form of the
expansion is determined from the symmetry properties of the microcanonical
entropy function with respect to the order parameter. The general findings are
investigated for the four-state vector Potts model as an example of a classical
spin system.Comment: 15 pages, 7 figure
Fragmentation phase transition in atomic clusters I --- Microcanonical thermodynamics
Here we first develop the thermodynamics of microcanonical phase transitions
of first and second order in systems which are thermodynamically stable in the
sense of van Hove. We show how both kinds of phase transitions can
unambiguously be identified in relatively small isolated systems of
atoms by the shape of the microcanonical caloric equation of state
I.e. within microcanonical thermodynamics one does not need to go to the
thermodynamic limit in order to identify phase transitions. In contrast to
ordinary (canonical) thermodynamics of the bulk microcanonical thermodynamics
(MT) gives an insight into the coexistence region. The essential three
parameters which identify the transition to be of first order, the transition
temperature , the latent heat , and the interphase surface
entropy can very well be determined in relatively small
systems like clusters by MT. The phase transition towards fragmentation is
introduced. The general features of MT as applied to the fragmentation of
atomic clusters are discussed. The similarities and differences to the boiling
of macrosystems are pointed out.Comment: Same as before, abstract shortened my e-mail address: [email protected]
Statistical mechanics of non-hamiltonian systems: Traffic flow
Statistical mechanics of a small system of cars on a single-lane road is
developed. The system is not characterized by a Hamiltonian but by a
conditional probability of a velocity of a car for the given velocity and
distance of the car ahead. Distribution of car velocities for various densities
of a group of cars are derived as well as probabilities of density fluctuations
of the group for different velocities. For high braking abilities of cars
free-flow and congested phases are found. Platoons of cars are formed for
system of cars with inefficient brakes. A first order phase transition between
free-flow and congested phase is suggested.Comment: 12 pages, 6 figures, presented at TGF, Paris, 200
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