1,650 research outputs found
Topological approach to phase transitions and inequivalence of statistical ensembles
The relation between thermodynamic phase transitions in classical systems and
topology changes in their state space is discussed for systems in which
equivalence of statistical ensembles does not hold. As an example, the
spherical model with mean field-type interactions is considered. Exact results
for microcanonical and canonical quantities are compared with topological
properties of a certain family of submanifolds of the state space. Due to the
observed ensemble inequivalence, a close relation is expected to exist only
between the topological approach and one of the statistical ensembles. It is
found that the observed topology changes can be interpreted meaningfully when
compared to microcanonical quantities.Comment: 9 pages, 1 figur
When topology triggers a phase transition
Two mathematical mechanisms, responsible for the generation of a
thermodynamic singularity, are individuated. For a class of short-range,
confining potentials, a topology change in some family of configuration space
submanifolds is the only possible such mechanism. Two examples of systems in
which the phase transition is not accompanied by a such topology change are
discussed. The first one is a model with long-range interactions, namely the
mean-field phi^4-model, the second example is a one-dimensional system with a
non-confining potential energy function. For both these systems, the
thermodynamic singularity is generated by a maximization over one variable (or
one discrete index) of a smooth function, although the context in which the
maximization occurs is very different.Comment: Talk given at the Next-SigmaPhi conference in Kolymbari, Crete,
Greece, August 13-18, 200
Lyapunov exponents from geodesic spread in configuration space
The exact form of the Jacobi -- Levi-Civita (JLC) equation for geodesic
spread is here explicitly worked out at arbitrary dimension for the
configuration space manifold M_E = {q in R^N | V(q) < E} of a standard
Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric g_J. As
the Hamiltonian flow corresponds to a geodesic flow on (M_E,g_J), the JLC
equation can be used to study the degree of instability of the Hamiltonian
flow. It is found that the solutions of the JLC equation are closely resembling
the solutions of the standard tangent dynamics equation which is used to
compute Lyapunov exponents. Therefore the instability exponents obtained
through the JLC equation are in perfect quantitative agreement with usual
Lyapunov exponents. This work completes a previous investigation that was
limited only to two-degrees of freedom systems.Comment: REVTEX file, 10 pages, 2 figure
Topological aspects of geometrical signatures of phase transitions
Certain geometric properties of submanifolds of configuration space are
numerically investigated for classical lattice phi^4 models in one and two
dimensions. Peculiar behaviors of the computed geometric quantities are found
only in the two-dimensional case, when a phase transition is present. The
observed phenomenology strongly supports, though in an indirect way, a recently
proposed topological conjecture about a topology change of the configuration
space submanifolds as counterpart of a phase transition.Comment: REVTEX file, 4 pages, 5 figure
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